+++ /dev/null
-\input ../lecnotes.tex
-
-\prednaska{0}{Ukázka pou¾ití maker}{(zapsal Martin Mare¹)}
-
-% Takhle vypadá nadpis
-\h{Pøedposlední kapitola}
-
-% \> zaøídí, aby se odstavec neodsadil. Pomocí \I se sází kurzíva, \foot vyrobí poznámku pod èarou.
-\>Je svaèveèer. {\I Lysperní jezeleni} se vírnì vrtáèejí v~mokøavì.\foot{Viz Lewis Caroll: Jabberwocky.}
-
-\itemize\ibull
-\:Takhle.
-\:Vypadají.
-\:Odrá¾ky.
-\endlist
-
-\h{Druhá kapitola}
-
-\s{Vìta:} Paøez není strom.
-
-\proof
-Strom je speciálním pøípadem lesa. Les je (podle definice) graf bez kru¾nic.
-Paøez obsahuje alespoò jednu kru¾nici. Proto paøez nemù¾e být les, a~tedy ani
-strom.
-\qed
-
-\s{Algoritmus:} (tøídìní posloupnosti $a_1,\ldots,a_n$ pomocí {\sc Stupidsort}u)
-
-\algo
-\:$\pi \leftarrow \hbox{identická permutace na~mno¾inì $\{1,\ldots,n\}$}$.
-\:Opakuj:
-\::Ovìø, zda je posloupnost $a_{\pi(1)},a_{\pi(2)},\ldots,a_{\pi(n)}$ uspoøádána vzestupnì.
- Pokud je, vra» ji jako výsledek.
-\::Nahraï $\pi$ jejím lexikografickým následníkem.
-\endalgo
-
-\bye
+++ /dev/null
-P=0-demo
-
-include ../Makerules
+++ /dev/null
-\input ../lecnotes.tex
-\input epsf.tex
-
-\def\N{N}
-
-\prednaska{1}{Eukleidùv algoritmus}{(zapsaly T.~Klimo¹ová a K.~B\"ohmová)}
-
-\h{}
-\>{\bf Problém hledání nejvìt¹ího spoleèného dìlitele }
-
-Cílem je najít nejvìt¹ího spoleèného dìlitele zadaných èísel
-$a_0, b_0 \in \N $, dále jen $\gcd(a_0, b_0)$, tedy Greatest Common
-Divisor.
-
-\s{Algoritmus:} (Eukleidùv algoritmus verze $1.0$; snad 200 let pøed Eukleidem)
-
-\algo
-\algin èísla $a_0$, $b_0$.
-\:$a\leftarrow a_0, b\leftarrow b_0$.
-\:while $a\not=b$ :
-\::if $a > b \Rightarrow a \leftarrow a-b $
-\::\rlap{else}\phantom{if }$\phantom{a > b} \Rightarrow b \leftarrow b-a $
-\algout $a=\gcd(a_0,b_0)$.
-\endalgo
-
-\noindent{\sl Dùkaz správnosti: }
-
-\>{\bo 1) EA se zastaví}
-
-V ka¾dém prùchodu cyklem se buï $a$ zmen¹í o hodnotu $b$ anebo se $b$ zmen¹í o~hodnotu $a$. Proto souèet $a+b$ klesne poka¾dé alespoò o $1$. Tedy celkem
-se uskuteèní nanejvý¹ $a+b$ prùchodù cyklem a pak algoritmus skonèí.
-
-\>{\bo 2) EA vydá $\gcd(a_0, b_0)$}
-
-Pøi dùkazech správnosti algoritmu se èasto pou¾ívá {\I invariant.}
-To je nìco, co se v prùbìhu algoritmu nemìní (obvykle hodnota nìjakého výrazu nebo jeho vlastnost, napø. dìlitelnost dvìma).
-
-Doká¾eme, ¾e platí invariant: $\gcd(a, b) = \gcd(a_0, b_0)$.
-
-Algoritmus pøi svém bìhu postupnì mìní hodnoty $a$ a $b$.
-Tedy
-$(a_0, b_0) \rightarrow (a_1, b_1) \rightarrow
-\dots \rightarrow (a_k, b_k)$.
-A jakmile je splnìna rovnost $a_k = b_k$, tak skonèí.
-My chceme ukázat, ¾e platí
-$\gcd(a_0, b_0) = \gcd(a_1, b_1) = \dots = \gcd(a_k, b_k)$.
-
-K~tomu bude tøeba nìkolik pozorování:
-
-\>{\I Pozorování 1:} $\gcd(a, a) = a$ (pro $a>0$).
-
-Nahlédneme snadno. Nejvìt¹í spoleèný dìlitel jediného èísla je ono samo.
-
-\>{\I Pozorování 2:} $\gcd(a, b) = \gcd(b, a)$.
-
-Takté¾ jednoduché. Pøi urèování nejvìt¹ího spoleèného dìlitele
-nám na poøadí èísel nezále¾í.
-
-\>{\I Pozorování 3:} $\gcd(a, b) = \gcd(a \pm b, b)$.
-
-%%Rozmyslíme si, ¾e a¾ doká¾eme {\I Pozorování 3 }, máme vyhráno.
-
-Doká¾eme silnìj¹í vìc -- platnost ekvivalence
-$k\vert a, k\vert b \Leftrightarrow k\vert a, k\vert(a\pm b)$.
-
-\>$\Rightarrow$: Jeliko¾ $k\vert a, k\vert b$, dá se $a$ vyjádøit jako $k$-násobek nìjakého $a'$. Tedy $a = k\cdot a'$.
-Obdobnì $b = k\cdot b'$, pro nìjaká èísla $a',b'\in \N$.
-Tak¾e souèet $a\pm b$ si mù¾eme rozepsat jako $(a'\pm b')\cdot k$ z èeho¾ je
-ji¾ vidìt, ¾e $k\vert(a\pm b)$.
-
-\>$\Leftarrow$: Podobnì $k\vert (a\pm b), k\vert b \Leftrightarrow k\vert a$.
-
-Uvìdomíme si, ¾e tato tøi pozorování nám staèí k tomu, aby invariant
-platil. Algoritmus tedy opravdu vydá $\gcd(a_0,b_0)$.
-\qed
-
-\goodbreak
-
-\>{3) \bo Rychlost algoritmu}
-
-Algoritmus provede v~nehor¹ím pøípadì øádovì $a+b$ iterací. To je velmi pomalé,
-a proto vyvstává otázka, dá-li se nìjak zrychlit.
-
-V¹imnìme si, ¾e opakované odèítání jednoho èísla od~druhého lze nahradit výpoètem
-zbytku po~dìlení. Algoritmus tedy mu¾eme upravit následovnì:
-
-\s{Algoritmus:} (Eukleidùv alg. v$2.0$; neznámý Eukleidùv pokraèovatel, neznámo kdy)
-
-\algo
-\algin èísla $a_0$, $b_0$.
-\:$a\leftarrow a_0, b\leftarrow b_0$.
-\:while $b \neq 0$:
-\::$a \leftarrow a \mod b$
-\::$a \leftrightarrow b$
-\algout $a=\gcd(a_0,b_0)$.
-\endalgo
-
-\>{\sl Dùkaz správnosti: }
-
-Algoritmus urèitì vydá správný výsledek, proto¾e poèítá toté¾, co pøedchozí verze.
-Uká¾eme, ¾e poèet krokù, který provede verze 2.0, je øádovì $\log(\max(a,b))$.
-K~tomu se nám bude hodit následující:
-
-{\narrower
-\s{Lemma:} Za dva prùchody cyklem se buï hodnota $a$ nebo hodnota $b$
-zmen¹í alespoò dvakrát.
-
-\proof
-Rozebereme, co se stane po dvojím projití cyklem.
-Na zaèátku máme èísla $a_0$ a $b_0$. Bez újmy na obecnosti pøedpokládejme,
-¾e $a_0>b_0$. Po~prvním prùchodu získáme $a_1=b_0; b_1=a_0 \mod b_0$, co¾ je ménì
-ne¾ $b_0$. Po druhém prùchodu: $a_2=b_1; b_2=a_1 \mod b_1 = b_0 \mod b_1$. Chceme
-ukázat, ¾e kdy¾ je $b_1 < b_0$, tak $b_0 \mod b_1 \leq b_0/2$.
-
-\itemize\ibull
-\:Buï platí $b_1 \leq b_0/2$. Pak jistì $b_0 \mod b_1 < b_1 \leq b_0/2.$
-\:Nebo nastane druhá mo¾nost $b_1 > b_0/2$.
-Pak tedy $b_0 \mod b_1 = b_0-b_1$, co¾ je takté¾ maximálnì $b_0/2$.
-\qed
-\endlist
-
-}
-
-Nyní staèí lemma ¹ikovnì vyu¾ít:
-Dvojic prùchodù, které zmen¹ují hodnotu $a$, je nejvý¹e $\lceil \log_2
-a \rceil$. Obdobnì tìch sni¾ujících hodnotu $b$ je nejvý¹e $\lceil \log_2 b \rceil$.
-Celkový poèet prùchodù tedy je nejvý¹e $2(\lceil \log_2 a \rceil+\lceil
-\log_2 b\rceil)$.
-
-Je¹tì bychom chtìli dokázat, ¾e ná¹ horní odhad na~poèet prùchodù není pøíli¹
-velkorysý. Zkusme se podívat, co se stane, kdy¾ algoritmus necháme spoèítat
-$\gcd(F_n, F_{n-1})$, kde~$F_i$ je $i$-té Fibonacciho èíslo. Tehdy v¹echna
-modula budou ve~skuteènosti odèítání a algoritmus bude poèítat
-$\gcd(F_n, F_{n-1}) \rightarrow \gcd(F_{n-1}, F_{n-2}) \rightarrow
-\dots \rightarrow \gcd(1, 1)$.
-
-Díky exponenciální velikosti Fibonacciho èísel tento výpoèet bude
-opravdu trvat logaritmicky dlouho vzhledem k~velikosti vìt¹ího ze
-zadaných èísel.
-(Zároveò je to také pìkný dùkaz toho, ¾e Fibonacciho èísla jsou
-navzájem nesoudìlná.)
-
-\h{Výpoèetní model}
-Kdy¾ chceme studovat algoritmy, mìli bychom se zaèít zaobírat otázkou, co takový algoritmus vlastnì je.
-®ádná obecnì uznávaná definice algoritmu neexistuje, nadefinujeme si proto alespoò výpoèetní model
-a za~algoritmy budeme pova¾ovat programy v tomto modelu.
-
-\s{Základní vlastnosti výpoèetního modelu:}
-
-V první øadì potøebujeme modelu sdìlit, s èím má pracovat, a pak se
-dovìdìt, jak to dopadlo. Bez újmy na obecnosti mù¾eme pova¾ovat $n$
-èísel za {\sl vstup} a za {\sl výstup} $m$ èísel $($mù¾eme se omezit
-na èísla, nebo» cokoli jiného umíme do èísel zakódovat$)$.
-
-Dále by mìl výpoèetní model umìt provádìt {\sl elementární operace},
-co¾
-jsou nejen ty aritmetické $(+,-,*,/,{\rm mod}, \dots)$ a logické (negace,
-and, or, \dots), ale také øídící operace (skoky, podmínìné skoky, halt,
-\dots) a práci s pamìtí.
-
-\s{Definice:}
-\itemize\ibull
-\:{\I Èas bìhu algoritmu} $t(x)$ pro vstup~$x$ mìøíme jako poèet
-elementárních operací, které program provedl pøi zpracování vstupu
-$x$.
-\:{\I Prostor bìhu algoritmu} $s(x)$ je analogicky poèet pamì»ových
-bunìk spotøebovaných pøi výpoètu se vstupem~$x$.
-\:{\I Èasová slo¾itost} (v~nejhor¹ím pøípadì) je:
-$$T(n) := \max \{t(x) ; \hbox{$x$ je vstup délky $n$}\}.$$
-\:{\I Prostorová slo¾itost} (v~nejhor¹ím pøípadì) je:
-$$S(n) := \max \{s(x) ; \hbox{$x$ je vstup délky $n$}\}.$$
-\endlist
-
-\s{Díra v definici:} Co jsou to èísla?
-Kdy¾ mohou být libovolnì velká, pak se dá podvádìt pøi poèítání èasové i prostorové
-slo¾itosti. Napravit to mù¾eme buïto tak, ¾e vstup budeme mìøit v~bitech a operace
-s~$k$-bitovými èísly bude trvat øádovì $\log k$ krokù, nebo ¾e omezíme èísla
-na~polynomiálnì velká vzhledem k~velikosti vstupu. Pro normální algoritmy tyto dvì
-definice dávají èasové slo¾itosti li¹ící se v¾dy logaritmem velikosti vstupu, tak¾e
-pro srovnávání algoritmù nemusíme tento rozdíl uva¾ovat. Pro tuto pøedná¹ku si vybereme
-omezení èísel na~polynomiálnì velká.
-
-\h{Panorama slo¾itostí}
-
-Abychom mìli pøedstavu o~tom, jak pou¾itelné jsou algoritmy se kterou
-èasovou slo¾itostí, zkusme si pár slo¾itostí srovnat za pøedpokladu,
-¾e ná¹ poèítaè za sekundu provede $10^9$ operací:
-\def\[#1]{\,\hbox{#1}}
-$$\vbox{
-\halign{$#$ \hfil&\quad\hfil $#$&\quad\hfil $#$&\quad\hfil
-$#$&\quad\hfil $#$&\quad\hfil $#$&\quad\hfil $#$&\quad\hfil $#$\cr
-\hbox{funkce / $n=$\hskip-2em}& 10 & 20 & 30
-& 50 & 100 & 1\,000 & 1\,000\,000 \cr
-\noalign{\medskip\hrule\bigskip}
-\log_2 n & 3.3\[ns] & 4.3\[ns] & 4.9\[ns]
-& 5.6\[ns] & 6.6\[ns] & 10.0\[ns] & 19.9\[ns] \cr
-n & 10\[ns] & 20\[ns] & 30\[ns]
-& 50\[ns] & 100\[ns] & 1\[$\mu$s] & 1\[ms] \cr
-n\cdot\log_2 n\hskip-1em & 33\[ns] & 86\[ns]
-& 147\[ns] & 282\[ns] & 664\[ns] & 10\[$\mu$s] & 20\[ms]
-\cr
-n^2 & 100\[ns] & 400\[ns] & 900\[ns]
-& 2.5\[$\mu$s] & 100\[$\mu$s] & 1\[ms] & 1\,000\[s] \cr
-n^3 & 1\[$\mu$s] & 8\[$\mu$s] & 27\[$\mu$s]
-& 125\[$\mu$s] & 1\[ms] & 1\[s] & 10^9\[s] \cr
-2^n & 1\[$\mu$s] & 1\[ms] & 1\[s]
-& 10^6\[s] & 10^{21}\[s] & 10^{292}\[s] & \approx\infty \cr
-n! & 10^6\[s] & 10^{18}\[s] & 10^{32}\[s]
-& 10^{64}\[s] & 10^{158}\[s] & 10^{2567}\[s] & \approx\infty \cr
-}}$$
-\halign{#\hfil&\quad #\hfil\cr
-Pro pøedstavu: &$1\,000\[s]$ je asi tak ètvrt hodiny\cr
- &$1\,000\,000\[s]$ je necelých 12 dní\cr
- &$10^9\[s]$ je 31 let\cr
- &$10^{18}\[s]$ je asi tak stáøí Vesmíru. \cr}
-
-Poèítaèe se nám ale zrychlují\foot{\uv{Ka¾dé dva roky se výkon poèítaèù
-zdvojnásobí} -- Gordon Moore (prý)} -- není tedy lep¹í chvíli poèkat
-ne¾ vymý¹let lep¹í algoritmus? :-) Podívejme se, jak velký vstup
-doká¾eme zpracovat za $10^6$ a $2\cdot 10^6$ operací:
-$$\vbox{\halign{\hfil$#$\quad&\hfil$#$\quad&\hfil$#$\qquad&#\hfil\cr
-\hbox{funkce} & 10^6\;\hbox{op.} & 2\cdot10^6\;\hbox{op.} & zmìna \cr
-\noalign{\medskip\hrule\bigskip}
-n &1\,000\,000 &2\,000\,000 &2-krát více \cr
-n^2 &1\,000 &1\,414 &$\sqrt{2}\approx 1.41$-krát
-více \cr
-n^3 &100 &126 &$\root3\of2\approx 1.26$-krát
-více \cr
-2^n &20 &21 &o 1 více \cr
-}}$$
-Na nìkterých èasových slo¾itostech se tedy exponenciální zrychlování
-poèítaèù projevuje jen minimálnì.
-
-Jak je vidìt na~následujícím grafu, konstanty v~èasové slo¾itosti
-vùbec nehrají pøíli¹ velkou roli, proto se vyplácí najít nejdøíve
-algoritmus, jeho¾ slo¾itost je asymptoticky dobrá, a teprve pak
-optimalizovat konstanty.
-
-\>\epsfxsize=1\hsize\epsfbox{../slides/funcs.eps}
-
-V~praxi také èasto pomù¾e zpracovávat velké vstupy asymptoticky rychlým
-algoritmem a pro malé vstupy pøepnout na~\uv{hloupý} algoritmus, který
-má malé multiplikativní konstanty.
-
-\bye
+++ /dev/null
-P=1-euklid
-
-include ../Makerules
+++ /dev/null
-\input ../lecnotes.tex
-
-\prednaska{10}{Nejkrat¹í cesty}{(zapsali L. Banáková, O. Hoferek, J. Bøeèka)}
-
-\s{Definice:} {\I Vzdálenost vrcholù} $d(u,v)$ je délka nejkrat¹í cesty $(u,v)$,
-pokud existuje, nebo $+\infty$, pokud taková cesta neexistuje.
-
-\s{Problém:} Je dán graf G a funkce $l: E(G) \rightarrow {\bb R}$ pøiøazující
-hranám jejich délky. Dále je dán startovací a cílový vrchol
-$s,c~\in~V(G)$. Chceme najít cestu $s=v_1, v_2, \dots, v_k=c$ takovou, aby délka cesty
-$$l(v_1, v_2) + l(v_2, v_3) + \dots + l(v_{k-1}, v_k)$$
-byla minimální. Takovéto cestì budeme øíkat {\I nejkrat¹í cesta} z~$s$~do~$c$.
-
-\noindent
-Bez újmy na obecnosti budeme pøedpokládat, ¾e graf $G$ je orientovaný, bez smyèek a
-násobných hran.
-
-\noindent
-Nejprve uvá¾íme pøípady, kdy jsou v¹echny hrany stejnì dlouhé. Tehdy mù¾eme pou¾ít
-algoritmus prohledávání do ¹íøky (BFS). K jeho implementaci je zapotøebí fronta $Q$ s
-metodami $Put(Q, v)$ (pøidání prvku $v$ na konec fronty) a $Get(Q)$ (odebrání
-prvku ze zaèátku fronty). Algoritmus vrátí pole \uv{znaèek} $z$: $z(v) = 1$ právì tehdy
-pokus je vrchol $v$ dostupný ze startovacího vrcholu $s$, jinak $z(v) = 0$.
-
-\s{Algoritmus:} (Prohledávání do ¹íøky (BFS))
-
-\algo
-\:$z(*) \leftarrow 0, z(s) \leftarrow 1$
-\:$Q \leftarrow \{s\}$
-\:while $Q \not= \emptyset :$
-\::$v \leftarrow Get(Q)$
-\::for $\forall w: (v,w)\in E$
-\:::if $z(w) = 0$ then
-\::::$z(w) \leftarrow 1$
-\::::$Put(Q, w)$
-\endalgo
-
-\s{Lemma È (èasové):} BFS dobìhne v~èase $\O(m+n)$.
-
-\proof
-Ka¾dý vrchol se do fronty dostane maximálnì jednou a ka¾dou hranu zpracujeme
-maximálnì jednou.
-\qed
-
-\s{Lemma D (dosa¾itelnost):} BFS oznaèí právì vrcholy dostupné ze $s$.
-
-\proof
-
-\noindent
-\uv{$\Rightarrow$}: Indukcí podle bìhu algoritmu.
-
-\noindent
-\uv{$\Leftarrow$}: Doká¾eme sporem. Uva¾me, ¾e existuje vrchol dostupný, ale algoritmem
-neoznaèený. Vezmìme takovýto \uv{¹patný} vrchol $v$, který je $s$ nejblí¾e. Uva¾me
-nejkrat¹í cestu $(s,v)$: $s,\dots,u,v$. Pøedchozí vrchol na této cestì $u$
-musí být \uv{dobrý}. Vrchol $u$ se dostane do fronty, pak je z~ní vybrán a tím
-se zpracuje i vrchol $v$ $\Rightarrow$ spor.
-
-Pøedchozím algoritmem jsme pouze zjistili, které vrcholy jsou z~$s$ dosa¾itelné.
-Nyní si tento algoritmus modifikujeme tak, abychom na¹li $d(s,v)$ pro v¹echna
-$v~\in~V(G)$.
-
-\s{Roz¹íøený algoritmus:} (Prohledávání do ¹íøky (BFS))
-
-\algo
-\:$z(*) \leftarrow 0, z(s) \leftarrow 1$
-\:$D(*) \leftarrow +\infty, D(s) \leftarrow 0$
-\:$Q \leftarrow \{s\}$
-\:while $Q \not= \emptyset :$
-\::$v \leftarrow Get(Q)$
-\::for $\forall w: (v,w)\in E$
-\:::if $z(w) = 0$ then
-\::::$z(w) \leftarrow 1$
-\::::$D(w) \leftarrow D(v) + 1$
-\::::$Put(Q, w)$
-\endalgo
-
-\noindent
-Prùbìh tohoto algoritmu si mù¾eme ukázat na následujícím obrázku (\uv{praseèí graf}):
-
-\figure{praseci-graf.eps}{Praseèí graf}{75mm}
-
-\s{Definice:} {\I Vrstva} $ L_i\subseteq V $ : $ L_0 = \{s\} $ , $ L_{i+1} = $
-$ \{ w$ : $ w $ oznaèen pøi procházení vrcholu $ v \in L_i $$ \} $
-
-\s{Lemma V (vzdálenosti):} Na konci BFS je $\forall v: D(v) = d(s,v)$.
-
-\proof
-
-\noindent
-a) pro nedosa¾itelné vrcholy: OK (algoritmus je neoznaèí $\Rightarrow D(v) = \infty$)
-
-\noindent
-b) pro dosa¾itelné vrcholy: Víme, ¾e pro vrchol $v \in L_i$ platí $D(v)=i$,
-staèí tedy zjistit, zda $\forall i: \forall v \in L_i$ je $d(s,v)=i$. To ovìøíme matematickou indukcí
-podle~$i$:
-
-pro $i=0$: evidentní,
-
-pro $i>0$: Pøi procházení vrstvy $L_{i-1}$ musí být vrchol $v$ takový, ¾e $d(s,v)=i$, oznaèen
-a zároveò nemohl být oznaèen døíve a tudí¾ patøí do $L_i$.
-\qed
-
-\noindent
-V¹imnìme si, ¾e v~grafu existují pouze tøi typy hran vzhledem k na¹emu algoritmu:
-\itemize\ibull
-\:dopøedu o jednu vrstvu
-\:dozadu o jednu nebo víc vrstev
-\:ve vrstvì
-\endlist
-
-Abychom na¹li i samotné cesty, nejen jejich délky, modifikujeme svùj algoritmus tak,
-¾e si pro ka¾dý vrchol $v$, který oznaèíme, budeme pamatovat jeho pøedchùdce $p(v)$:
-
-\algo
-\:$z(*) \leftarrow 0, z(s) \leftarrow 1$
-\:$D(*) \leftarrow +\infty, D(s) \leftarrow 0$
-\:$p(*) \leftarrow$ ?
-\:$Q \leftarrow \{s\}$
-\:while $Q \not= \emptyset :$
-\::$v \leftarrow Get(Q)$
-\::for $\forall w: (v,w)\in E$
-\:::if $z(w) = 0$ then
-\::::$z(w) \leftarrow 1$
-\::::$D(w) \leftarrow D(v) + 1$
-\::::$p(w) \leftarrow v$
-\::::$Put(Q, w)$
-\endalgo
-
-\s{Pozorování:} Pokud $p(v) \not=$ ? $\Rightarrow D(v) = D(p(v)) + 1$.
-
-\s{Lemma S (stromové):} Graf $(W,F)$, kde $W$ = dosa¾itelná èást $V$, $F = \{(v,p(v))\}$
-je strom orientovaný do~koøene $s$. Navíc ukazuje nejkrat¹í cestu pro ka¾dý dosa¾itelný
-vrchol (pro ka¾dý vrchol v tomto grafu, existuje jediná orientovaná cesta do koøene
-$s$, která je zároveò nejkrat¹í cesta z $s$ do tohoto vrcholu).
-
-\noindent
-Nyní uva¾me, ¾e délky hran $l$ nejsou v¹echny stejnì dlouhé:
-
-Uká¾eme si trik pro ohodnocené hrany: hranu délky $l$
-podrozdìlíme na $l$ hran délky 1 a pak spustíme BFS. Algoritmus pobì¾í s èasovou
-slo¾itostí $O((m+n)l_{max})$. To není zrovna moc výhodné, a proto si uká¾eme tzv. {\I Budíkovou taktiku}:
-
-Místo toho, abychom ka¾dou hranu podrozdìlovali, si budeme pro ka¾dý vrchol pamatovat \uv{èas budíku} $b$.
-Èasem $b(v)$ budíku vrcholu~$v$ rozumíme èas, kdy se do tohoto vrcholu vlna dostane, kdy¾ se bude ¹íøit po
-dosud známých hranách. Na zaèátku nastavíme $\forall v \in V: b(v) = l(s,v)$ a pokud pøi bìhu
-algoritmu objevíme hranu, která nám umo¾ní dostat se do $v$ rychleji, tak hodnotu $b(v)$ zmìníme.
-Algoritmus pak bude pracovat tak, ¾e bude spát, dokud nezazvoní nìjaký budík (co¾ znamená, ¾e
-jsme dorazili do nìjakého vrcholu).
-
-Tato taktika nás pøivede k Dijkstrovu algoritmu, kde se ze dvoustavového $z(v)$ stane tøístavové
-$z(v)) \in \{{\bf{N}}, {\bf{V}}, {\bf{H}}\}$, kde {\bf{N}} znamená nevidìn, {\bf{V}} vidìn a {\bf{H}} hotov. Podle hodnoty $z(v)$ se
-pak $D(v)$ rovná $+\infty$ pro $z(v) = {\bf{N}}$, èasu budíku pro $z(v) = {\bf{V}}$ nebo vzdálenosti $d(s,v)$ pro $z(v) = {\bf{H}}$.
-
-\s{Algoritmus:} (Dijkstrùv algoritmus)
-
-\algo
-\:$z(*) \leftarrow {\bf{N}}, z(s) \leftarrow {\bf{V}}$
-\:$D(*) \leftarrow +\infty , D(s) \leftarrow 0$
-\:$p(*) \leftarrow $?
-\:while $\exists v : z(v) = {\bf{V}} :$
-\::$v \leftarrow $ vrchol, pro nìj¾ $z(v)={\bf{V}}$ a zároveò $D(v)$ je minimální
-\::$z(v) \leftarrow {\bf{H}}$
-\::for $\forall w : (v,w) \in E$:
-\:::if $D(w)>D(v) + l(v,w)$ then
-\::::$D(w) \leftarrow D(v) + l(v,w)$
-\::::$z(w) \leftarrow {\bf{V}}$
-\::::$p(w) \leftarrow v$
-\endalgo
-
-\s{Vìta:} Pokud $\forall v,w: l(v,w) \in \bb{Z}^+ \cup \{\infty\}$, pak Dijkstrùv algoritmus
-najde nejkrat¹í cesty v èase $\O(n^2)$.
-
-\proof
-V prùbìhu algoritmu se maximálnì $n$-krát hledá vrchol $v$ s minimálním $D(v)$,
-maximálnì $m$-krát se pøenastavuje $D(v)$. Pokud bude struktura $D$ ulo¾ena jako
-pole (hledání minima v $\O(n)$, zmìna hodnoty $\O(1)$), bude celková
-èasová slo¾itost Dijkstrova algoritmu $\O(n^2 + m) = \O(n^2)$. Fakt, ¾e se algoritmus
-zastaví je evidentní, proto¾e se stav ¾ádného vrcholu nevrací z {\bf{H}} do {\bf{V}}.
-
-Úplný dùkaz správnosti a èasové slo¾itosti i pro hrany ohodnocené reálnými èísly
-bude na pøí¹tí pøedná¹ce.
-\qed
-
-Nahlédnìme nyní, zda by vedlo k~vylep¹ení èasové slo¾itosti, kdybychom pøi~implementaci
-Dijkstrova algoritmu pou¾ili jako datovou strukturu pro ukládání vidìných vrcholù rùzné typy hald.
-Provádìli bychom s~nimi operace \<DeleteMin> (maximálnì $n$-krát) a \<Decrease> (maximálnì $m$-krát). Zmínìné
-operace mají pro rùzné typy hald následující èasové slo¾itosti:
-$$\vbox{\halign{\strut # \quad & # \quad & # \quad & #\cr
-& \it binární halda & \it $k$-regulární halda & \it Fibonacciho halda\cr
-\noalign{\smallskip}
-\<DeleteMin> & $\O(\log{n})$ & $\O(k\log_k{n})$ & $\O(\log{n})$ \cr
-\<Decrease> & $\O(\log{n})$ & $\O(\log_k{n})$ & $\O(1)$ \cr
-{\I celková slo¾itost} & $\O((m+n)\log{n})$ & $\O(m\log_{m/n}{n})$ & $\O(n\log{n}+m)$ \cr
-}}$$
-
-Pou¾ijeme-li klasickou haldu, dostaneme celkovou èasovou slo¾itost $\O((m+n)\log{n})$, co¾ znamená,
-¾e jsme si pomohli v pøípadech, kdy je graf \uv{øídký}. U $k$-regulární haldy se celková èasová slo¾itost
-rovná $\O(nk\log_k{n}+m\log_k{n})$, ideálnì pro $k=m/n$: $\O(m\log_{m/n}{n})$. Prozkoumejme,
-jak se nám v tomto pøípadì bude mìnit èasová slo¾itost pro rùznì \uv{husté} grafy:
-\itemize\ibull
-\:$m \approx n \rightarrow \O((m+n)\log{n})$
-\:$m \approx n^2 \rightarrow \O(m)$
-\:$m \approx n^{1+\epsilon} \rightarrow \O(m) \dots (k=m/n= n^\epsilon)$
-\endlist
-\noindent
-Pøi pou¾ití Fibonacciho haldy je pak celková èasová slo¾itost rovna $\O(n\log{n}+m)$.
-
-\medskip
-
-Nyní si pøedvedeme trochu jiný algoritmus na prohledávání grafu. Mìjme matici sousednosti $A$
-ke grafu $G = (V, E)$: $ A_{ij} = 1 \Leftrightarrow (v_i, v_j) \in E $, jinak $A_{ij} = 0$. Potom
-$A^2_{ij} = \sum_{k=1}^n A_{ik}A_{kj} $. Tento souèet je nenulový právì tehdy, kdy¾
-vrcholy $i$,$k$,$j$ tvoøí sled délky 2. Jinými slovy $A^2_{ij}$ se rovná poètu sledù
-délky 2 v $G$ z $i$ do $j$. Obdobnì $A^k_{ij}$ = poèet sledù délky $k$. Pokud
-k matici $A$ pøièteme jednotkovou matici ${\bb E}_n$ (èím¾ si pøimyslíme smyèky),
-dostaneme po umocnìní:
-
-$ (A+{\bb E}_n)^k_{ij} \neq 0 \Leftrightarrow \exists$ $i,j$-sled délky maximálnì $k$.
-
-Výstupem algoritmu tedy bude matice $(A + {\bb E}_n)^n$. Mù¾eme si také v¹imnout, ¾e
-pokud matici $A + {\bb E}_n$ umocníme na èíslo vìt¹í ne¾ $n$, tak výsledek bude poøád
-správný, jen bude výpoèet trvat déle. Tak¾e algoritmus mù¾e pracovat
-v iteracích umocòováním matice na druhou: $M_0 = A + {\bb E}_n$, $M_1 = M^2_0$, $M_{i+1} = M^2_i$.
-Výstupní matice bude tudí¾:
-
-$ M_{\lceil \log{n} \rceil (ij)} \neq 0 \Leftrightarrow \exists$ cesta z $i$ do $j$
-
-Na výpoèet dosa¾itelnosti vrcholù v grafu tedy staèí $\O(\log{n})$ násobení matic,
-a pokud na toto pou¾ijeme Strassenùv algoritmus, dostaneme
-ve výsledku algoritmus se slo¾itostí $\O(n^{\log_2{7}}\log{n})$.
-
-\bye
+++ /dev/null
-P=10-cesty
-
-include ../Makerules
+++ /dev/null
-\input ../lecnotes.tex
-
-\prednaska{11}{Nekrat¹í cesty podruhé}{(zapsali Du¹an Renát a Radek Tupec)}
-
-Na~této pøedná¹ce budeme studovat problém hledání nejkrat¹ích cest
-v~grafech ohodnocených reálnými èísly.
-
-\s{Situace:} Máme orientovany graf~$G$ a funkce $l : E(G) \rightarrow {\bb R}$
-pøiøazující hranám jejich ohodnocení (délky). Pro vrcholy $u,v\in V(G)$ budeme
-chtít spoèítat jejich vzdálenost $d(u,v)$, co¾ bude délka nejkrat¹í cesty z~$u$
-do~$v$ nebo $\infty$, pokud ¾ádná cesta neexistuje.
-
-Aby se vzdálenosti chovaly \uv{rozumnì} (tj. jako metrika), budeme chtít, aby platily
-následující vlastnosti:
-
-\itemize\ibull
-\:$d(u, u) = 0$,
-\:$d(u, v) \leq d(u, w) + d(w, v)$ (trojúhelníková nerovnost).
-\endlist
-
-To nemusí obecnì platit (kazí nám to záporné cykly), proto budeme studovat
-pouze grafy, v~nich¾ záporné cykly neexistují. Pak u¾ budou obì vlastnosti
-splnìny, jak plyne napøíklad z~následujícího lemmatu:
-
-\s{Lemma:}
-V~grafu bez záporných cyklù existuje ke~ka¾dému nejkrat¹ímu sledu z~$u$ do~$v$ stejnì dlouhá $uv$-cesta.
-
-\proof
-Máme-li nejkrat¹í $uv$-sled, který není cestou, opakuje se v~nìm nìjaký vrchol $w \in V(G)$.
-Délka cyklu $l(c) \geq 0 \Rightarrow l(u\dots v\dots w) \leq l(pùvodní sled)$. Tento postup mù¾eme opakovat a po koneèném poètu krokù dostaneme cestu, tedy platí trojúhelníková nerovnost.
-\qed
-
-Opakování: Dijkstrùv algoritmus
-\itemize\ibull
-\:$D(v)$ \dots doèasná vzdálenost z $s$ do $v$
-\:Znaèky $Z(v)$ \itemize\ibull
-\::Nevidìn
-\::Vidìn
-\::Hotov
-\endlist
-\endlist
-
-%druha strana zapisku
-\algo
-\:$D(*) \leftarrow +\infty , D(s) \leftarrow 0$
-\:$Z(*) \leftarrow Neviden, Z(s) \leftarrow V$
-\:while $\exists v : Z(v) = V$
-\::vybereme $v : Z(v) = V, D(v) = min$
-\::$Z(v) = H$
-\::for $\forall w : (v, w) \in E(G)$
-\:::$D(w) \leftarrow min( D(w), D(w) + l(v, w))$
-\::if $Z(w) = N \Rightarrow Z(w) \leftarrow V$
-\endalgo
-
-\s{Vìta:} Pokud $G$ je nezápornì ohodnocený graf, pak se Dijkstrùv algoritmus zastaví a vydá $\forall v : D(v) = d(s, v)$ (tedy správné hodnoty).
-
-\proof Následující posloupností lemmat.\qed
-
-\s{Lemma 1:} Pokud $v_0,\dots,v_k$ je nejkrat¹í $v_0v_k$-cesta, pak $v_0,\dots,v_{k-1}$ je nejkrat¹í $v_0v_{k-1}$-cesta.
-
-\proof
-Pokud by tomu tak nebylo, mù¾eme $v_0,\dots,v_{k-1}$ vymìnit za~krat¹í cestu, a~tím
-získat krat¹í $v_0v_k$-cestu.
-\qed
-
-\s{Lemma 2:} Algoritmus se zastaví po $\leq n$ prùchodech cyklem.
-
-\proof
-Zøejmé z~toho, ¾e ka¾dý vrchol uzavøeme nejvý¹e jednou.
-\qed
-
-\s{Lemma 3:} Po zastavení jsou hotovy právì vrcholy dosa¾itelné z $s$.
-
-\proof
-Viz minulá pøedná¹ka.
-\qed
-
-\s{Lemma 4:} $D(v)$ uzavíraných vrcholù tvoøí neklesající posloupnost.
-
-\proof
-V okam¾iku, kdy uzavíráme $v$ platí $\forall w \notin H : D(w) \geq D(v)$, pøípadnì pøepoèítáme $D(w)$ na $D(v) - l(v, w) \geq D(v)$.
-\qed
-
-\s{Lemma 5:} Pokud $v \in H$, pak $D(v)$ se u¾ nezmìní.
-
-\proof
-Indukcí podle bìhu algoritmu.
-\qed
-
-\s{Lemma 6:} Pro $\forall v D(v)$ je délka nejkrat¹í $sv$-cesty, její¾ vnitøní vrcholy le¾í v¹echny v $H$.
-
-\proof
-\itemize\ibull
-\:po 1. prùchodu OK
-\:uzavíráme-li dal¹í vrchol $v$:
-\itemize\relax
-\:a) $D(w)$ pro $w \in H$
-Podle Lemma 5 se $D(w)$ nemìní. Musíme nahlédnout, ¾e se opravdu zmìnit nemá.
-\:b) $D(w)$ pro $w \notin H$
-$D(w) = min{D(v) + l(v, w)}$
-\endlist
-\endlist
-\qed
-
-Existuje pomalej¹í algoritmus pro grafy se zápornými hranami bez záporných cyklù.
-\s{Bellman-Fordùv algoritmus}
-\algo
-\:$D(*) \leftarrow \infty, D(s) \leftarrow 0$
-\:Opakuji
-\::Pro $\forall v \in V$
-\:::$prozkoumej(v)$
-\:::(koukne se, jestli nejde vylep¹it cestu¨
-\:::do sousedù $v$)
-\:dokud se nìjaké $D(\dots)$ mìní
-\endalgo
-
-\s{Lemma 1:} Pokud $D(v) < \infty$, pak existuje sled z $s$ do $v$ délky $D(v)$.
-(speciálnì z toho plyne $\forall v D(v) \geq d(s,v)$)
-
-\s{Lemma 2:} $\forall v D(v)$ nikdy neroste.
-\s{Lemma 3:} Po $k$ fázích je $\forall v D(v) \leq$ délka nejkrat¹ího $sv$-sledu o $\leq k$ hranách.
-%ctvrta stranka
-\proof
-Indukcí\dots pro $k \geq 0$ OK
-Indukèní krok: Dobìhla $k-1$ fáze, pou¹tíme $k$-tou
-
-\qed
-\s{Vìta:} Pro graf bez záporných cyklù se Bellman-Fordùv algoritmus zastaví po nejvý¹e $m$ fázích a vydá $D(v) = d(s, v)$ pro v¹echna $v$. (zøejmé z lemmat)
-
-\s{Lemma 4:} Pokud v grafu existuje záporný cyklus dosa¾itelný z $s$, algoritmus se nezastaví. (Zajímavý test na to, zda graf obsahuje záporný cyklus.)
-
-Èasová slo¾itost Bellman-Fordova algoritmu : $O(n*m)$
-
-\s{Floyd-Warshallùv algoritmus}
-$G$ je graf se záporným ohodnocením hran, bez záporných cyklù
-$D_{i,j}^k$ = délka nejkrat¹í cesty z $v_i$ do $v_j$ pøes $v_1 \dots v_k$
-%pozorovani
-$D_{ij}^0 = l(v_i, v_j)$, $D_{ii}^0 = 0$
-$D_{ij}^n = d(v_i, v_j)$ - skuteèná vzdálenost v grafu
-\algo
-\:$D_{i,j} \rightarrow l(v_i, v_j), D_{i,i} \rightarrow 0 \forall i, j$
-\:for $k = 1$ to n
-\::for $i = 1$ to n
-\:::for $j = 1$ to n
-\::::$D_{i,j} = min(D_{i,j}, D_{i,k} + D_{k,j})$
-\endalgo
-
-\s{Vìta:} Floyd-Warshallùv algoritmus spoèítá $D_{i,j} = d(v_i, v_j)$ v èase $O(n^3)$.
-
-\>Shrnutí:
-
-\s{$s \rightarrow *$}
-Dijkstrùv algoritmus $O(n^2)$
-\dots s haldou $O((n+m)\log m)$
-\dots s regulární haldou $O(n + m*log n)$
-\dots s Fibbonaciho haldou $O(n*log n + m)$
-Bellman-Ford $O(n*m)$
-
-\s{$* \rightarrow *$}
-Floyd-Warshall $O(n^3)$
-
-\bye
+++ /dev/null
-P=11-dijkstra
-
-include ../Makerules
+++ /dev/null
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-%%Title: 05.eps
-%%CreationDate: Sun May 20 18:08:37 2007
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-% Use own dictionary to avoid conflicts
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-{currentfile /ASCII85Decode filter /RunLengthDecode filter rstr readstring pop}
-{currentfile /ASCII85Decode filter /RunLengthDecode filter gstr readstring pop}
-{currentfile /ASCII85Decode filter /RunLengthDecode filter bstr readstring pop}
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-\input lecnotes.tex
-
-\prednaska{12}{Problém minimální kostry}{(zapsali K. Ka¹èák a M. Vachna)}
-
-\s{Zadání úlohy:} Pro neorientovaný graf $G$ s~ohodnocením hran {\I váhami} $w: E(G) \rightarrow \bb R$,
-chceme najít kostru $T$ s minimálním ohodnocením $w(T)=\sum_{e\in E(T)} w(e)$.
-
-\s{Navíc pøedpokládáme:} (bez újmy na~obecnosti)
-\itemize\ibull
-\:Graf $G$ je souvislý (jinak ho nejprve rozlo¾íme na komponenty).
-\:$\forall e,f \in E(G$) : $e\neq f \Rightarrow w(e)\neq w(f)$.
-\endlist
-
-Nyní si uká¾eme tøi algoritmy pro hledání minimální kostry, konkrétnì se jedná
-o~Jarníkùv, Borùvkùv a Kruskalùv algoritmus.
-
-\h{Jarníkùv algoritmus}
-
-\s{Algoritmus:}
-
-\algo
-\algin Graf~$G$ s~ohodnocením~$w$.
-\:Zvolíme libovolný vrchol $v_0\in V(G)$.
-\:$T\leftarrow(\left\{v_0\right\},\emptyset)$ (zatím jednovrcholový strom)
-\:Dokud $\vert V(T) \vert \neq n$:
-\::Vybereme hranu $uv\in E(G) : u\in V(T), v\notin V(T)$ tak, aby $w(uv)$ byla minimálni.
-\::$T\leftarrow T+uv$.
-\algout Minimální kostra~$T$.
-\endalgo
-
-\s{Vìta:} Jarníkùv algoritmus se zastaví po maximálnì $n$ iteracích a vydá minimální kostru grafu $G$.
-
-\proof
-Pøi ka¾dé iteraci algoritmus pøidá jeden vrchol do~$T$, a~proto se po~maximálnì $n$ iteracích zastaví.
-Vydaný graf je strom, proto¾e se stále pøidává list k ji¾ existujícímu stromu, a~jeliko¾ má $n$~vrcholù,
-je to kostra. Zbývá nám u¾ jen dokázat, ¾e nalezená kostra je minimální. K~tomu pomu¾e následující lemma:
-
-{\narrower
-
-\s{Definice:} {\I Øez} v~grafu $G=(V,E)$ je mno¾ina hran $F\subseteq E$ taková, ¾e $\exists U\subset V$ :
-$F=\left\{uv\in E:u\in U, v\notin U \right\}$.
-
-\s{Lemma:} Pokud $G$ je graf, $w$ jeho prosté ohodnocení, $F$ je øez v grafu $G$ a $f$ je nejlehèí hrana v øezu
-$F$, pak pro ka¾dou minimální kostru $T$ grafu $G$ je $f\in E(T)$.
-
-\proof
-Buï $T$ kostra a $f=uv\notin E(T)$. Pak existuje cesta $P\subseteq T$ spojující $u$ a $v$.
-Cesta musí øez alespoò jednou pøekroèit. Proto existuje $e\in P \cap F$ a navíc víme, ¾e $w(e) > w(f)$. Uva¾me $T'=T-e+f$.
-Tento graf je rovnì¾ kostra grafu $G$, proto¾e odebraním hrany $e$ se graf rozpadne na dvì komponenty a pøidáním
-hrany $f$ se tyto komponenty opìt spojí. Navíc $w(T')=w(T)-w(e)+w(f)<w(T)$.
-\qed
-
-}
-
-\>V~dùkazu korektnosti Jarníkova algoritmu toto lemma vyu¾ijeme tak, ¾e si v¹imneme, ¾e hrany mezi
-vrcholy stromu~$T$ a zbytkem grafu tvoøí øez a algoritmus nejlehèí hranu tohoto øezu pøidá
-do~$T$. Podle lemmatu tedy v¹echny hrany~$T$ musí být souèástí ka¾dé minimální kostry a jeliko¾~$T$ je strom,
-musí být minimální kostrou.
-
-\qed
-
-
-\s{Dùsledky:} Graf $G$ s prostým ohodnocením má pravì jednu minimální kostru. Minimální kostra je
-jednoznaènì urèená lineárním uspoøádáním hran.
-
-\s{Implementace:}
-\itemize\ibull
-\:Pøímoèará: pamatujeme si, které vrcholy a hrany jsou v kostøe $T$ a které ne. Èasová slo¾itost je $\O(nm)$.
-\:Chytøej¹í: Pro $v\notin V(T)$ si pamatujeme $D(v)=\min\left\{w(uv):u\in T\right\}$. Pøi ka¾dém
-prùchodu hlavním cyklem pak procházíme v¹echna~$D(v)$ (to v¾dy trvá $\O(n)$) a pøi pøidání vrcholu do~$T$ kontrolujeme
-okolní~$D(w)$ pro $vw\in E$ a pøípadnì je sni¾ujeme (za~ka¾dou hranu~$\O(1)$). Èasovou slo¾itost tím celkovì
-zlep¹íme na~$\O(n^2+m)=\O(n^2)$.
-\:Také se dá pou¾ít halda pro uchovávání hran nebo hodnot~$D(v)$.
-\endlist
-
-\h{Borùvkùv algoritmus}
-
-\s{Algoritmus:}
-
-\algo
-\algin Graf~$G$ s~ohodnocením~$w$.
-\:$F\leftarrow(V(G),\emptyset)$
-\:Dokud $F$ má alespoò dvì komponenty:
-\::Pro ka¾dou komponentu $T_i$ grafu $F$ vybereme nejlehèí incidentní hranu $t_i$.
-\::V¹echny hrany $t_i$ pøidáme do $F$.
-\algout Minimální kostra~$F$.
-\endalgo
-
-\s{Vìta:} Borùvkùv algoritmus se zastaví po $\left\lceil \log_2 n\right\rceil$ iteracích a vydá minimální kostru grafu $G$.
-
-\proof
-V¹imnìme si nejprve, ¾e po~$k$ iteracích mají v¹echny komponenty grafu~$F$ minimálnì $2^k$ vrcholù
-(na~poèátku jsou v¹echny komponenty jednovrcholové, v~ka¾dé dal¹í iteraci se komponenty sluèují do~vìt¹ích,
-ka¾dá s~alespoò jednou sousední, tak¾e se velikosti komponent minimálnì zdvojnásobí).
-Proto nejpozdìji po~$\left\lceil \log_2 n\right\rceil$ iteracích u¾~velikost komponenty dosáhne poètu v¹ech vrcholù a algoritmus
-se zastaví.
-
-Hrany mezi ka¾dou komponentou a~zbytkem grafu tvoøí øez, tak¾e podle pøedchozího lemmatu v¹echny
-hrany pøidané do~$F$ musí být souèástí (jednoznaènì urèené) minimální kostry. Graf $F\subseteq G$
-je tedy v¾dy les a a¾ se algoritmus zastaví, bude roven minimální kostøe.
-\qed
-
-\s{Implementace:}
-\itemize\ibull
-\:Inicializace pøímoèará.
-\:Pomocí DFS rozlo¾íme les na komponenty. Pro ka¾dý vrchol si pamatujeme èíslo komponenty.
-\:Pro ka¾dou hranu zjistíme, do které komponenty patøí, a pro ka¾dou komponentu
-si uchováme nejlehèí hranu.
-\endlist
-
-\>Takto doká¾eme ka¾dou iteraci provést v~èase $\O(m)$ a celý algoritmus dobìhne v~$\O(m\log n)$.
-
-\h{Kruskalùv neboli hladový algoritmus}
-
-\s{Algoritmus:}
-
-\algo
-\algin Graf~$G$ s~ohodnocením~$w$.
-\:Setøídíme v¹echny hrany z $E(G)$ tak, aby: $w(e_1)<...<w(e_m)$.
-\:$F\leftarrow (V(G),\emptyset)$.
-\:Pro $i=1$ do $m$:
-\::Pokud $F+e_i$ je acyklický, provedeme $F\leftarrow F+e_i$.
-\algout Minimální kostra~$F$.
-\endalgo
-
-\s{Vìta:} Kruskalùv algoritmus se zastaví po~$m$ iteracích a vydá minimální kostru.
-
-\proof
-Ka¾dá iterace algoritmu zpracovává jednu hranu, tak¾e iterací je~$m$. Indukcí doká¾eme,
-¾e~$F$ je v¾dy podgrafem minimální kostry: prázdné poèáteèní~$F$ je podgrafem èehokoliv,
-ka¾dá hrana, kterou pak pøidáme, je minimální v~øezu oddìlujícím nìjakou
-komponentu~$F$ od~zbytku grafu (ostatní hrany tohoto øezu je¹tì nebyly
-zpracovány, a~tudí¾ jsou tì¾¹í). Naopak ¾ádná hrana, kterou jsme se rozhodli
-do~$F$ nepøidat, nemù¾e být souèástí minimální kostry, jeliko¾ s~hranami,
-o~kterých ji¾ víme, ¾e v~minimální kostøe le¾í, tvoøí kru¾nici. \qed
-
-\s{Implementace:}
-\itemize\ibull
-\:Setøídìní v èase $\O(m\log m)=\O(m\log n)$.
-\:Pak potøebujeme udr¾ovat komponenty souvislosti grafu~$F$, abychom umìli rychle
- urèit, jestli právì zpracovávaná hrana vytvoøí kru¾nici. Potøebujeme tedy strukturu
- pro udr¾ování komponent souvislosti, které se $m$-krát zeptáme, zda dva vrcholy
- le¾í v~té¾e komponentì (tomu budeme øíkat operace \<Find>), a~$(n-1)$-krát spojíme
- dvì komponenty do jedné (\<Union>).
-\endlist
-
-\>Kruskalùv algoritmus tedy pobì¾í v~èase $\O(m\log n + mT_f + nT_u)$, kde~$T_u$ je
-èas na~provedení jedné operace \<Union> a $T_f$ na~operaci \<Find>.
-
-\s{Jednoduchá struktura pro komponenty:}
-Budeme pamatovat v~poli èísla komponent, ve~kterých le¾í jednotlivé vrcholy. \<Find> zvládneme v~èase $\O(1)$,
-ale \<Union> bude stát $\O(n)$. Celý algoritmus pak pobì¾í v~èase $\O(m\log n+ m + n^2) = \O(m\log n+n^2)$.
-
-\s{Chytøej¹í struktura:} Ka¾dou komponentou si ulo¾íme jako strom orientovaný smìrem ke koøeni
--- ka¾dý vrchol si pamatuje svého otce, navíc ka¾dý koøen si pamatuje velikost komponenty.
-Operace \<Find> vystoupá z~obou vrcholù ke~koøeni a koøeny porovná. \<Union> rovnì¾ najde
-koøeny a pøipojí koøen men¹í komponenty pod koøen té vìt¹í. Obojí zvládneme v~èase lineárním
-v~hloubce stromu a jak si uká¾eme, tato hloubka je v¾dy nejvý¹e logaritmická, a proto celý Kruskalùv
-algoritmus pobì¾í v~èase $\O(m\log n + m\log n + n\log n) = \O(m\log n)$.\foot{%
-Drobnou úpravou bychom mohli dosáhnout daleko efektivnìj¹í struktury, ale tu bychom neupotøebili,
-jeliko¾ by nás stejnì brzdilo tøídìní, a analýza slo¾itosti by byla \dots\ inu, slo¾itìj¹í.}
-
-\s{Lemma:} Strom hloubky $h$ má alespoò $2^h$ prvkù.
-
-\proof
-Indukcí: Pokud \<Union> spojí strom s~hloubkou $h$ s jiným s~hloubkou men¹í ne¾ $h$, pak hloubka výsledného
-stromu zùstává~$h$. Pokud spojuje dva stromy stejné hloubky~$h$, pak má výsledný strom hloubku $h+1$.
-Z~indukèního pøedpokladu víme, ¾e strom hloubky $h$ má minimálnì $2^h$ vrcholù,
-a~tedy výsledný strom hloubky $h+1$ má alespoò $2^{h+1}$ vrcholù. \qed
-
-\h{Èerveno-èerné stromy}
-
-Ve~zbytku pøedná¹ky se vrátíme k~vyhledávacím stromùm a uká¾eme si je¹tì jeden zpùsob,
-jak udr¾et logaritmickou hloubku stromu.
-
-\s{Definice:} {\I Èerveno-èerný strom} je binární vyhledávací strom, jeho¾ vrcholy jsou
-obarvené (nìkteré èervenì, zbylé èernì), a ke~kterému byly pøidány externí vrcholy v¹ude
-tam, kde pùvodnímu vrcholu chybí nìkterý ze~synù. Listy è-è stromu jsou tedy v¾dy externí.
-Navíc platí následující axiomy:
-
-\s{Axiomy:}
-\itemize\ibull
-\:Koøen a externí vrcholy jsou èerné.
-\:Otec èerveného vrcholu je èerný.
-\:Na v¹ech cestách z~koøene do~externího vrcholu je stejný poèet èerných vrcholù.
-\endlist
-
-\s{Vìta:} Èerveno-èerný strom o~$n$ vrcholech má hloubku $\O(\log n)$.
-
-\proof
-Zkomprimujeme v¹echny hrany spojující èerný a èervený vrchol, èím¾ v¹echny èervené
-vrcholy \uv{zasuneme} do~jejich èerných otcù. Vznikne $2,4$-strom a ten, jak u¾ víme,
-má hloubku $\O(\log n)$. Pùvodní è-è strom má tedy hloubku nejvý¹e $2\log n$, co¾ je
-asymptoticky taky $\O(\log n)$.
-\qed
-
-\noindent{\sl Alternativní dùkaz:} \foot{\uv{Vy máte protipøíklad? Nevadí, já mám dva dùkazy :)}}
-Pøedpokládejme, ¾e $T$ je èerveno-èerný strom, který má na cestì z koøene do listu
-$k$ èerných vrcholù. Pak pro poèet vrcholù stromu $T$ platí $2^k-1\leq n \leq 2^{2k}-1$. Nejmen¹í takový strom má v¹echny vrcholy obarvené èernì a je to úplný
-pravidelný binární strom o hloubce $k-1$, co¾ dává dolní odhad. Nejvìt¹í takový strom má v¹echny vrcholy v sudých hladinách obarveny èervenì a v lichých hladinách èernì, je to úplný pravidelný binární strom o hloubce $2k-1$ a tím je dán horní odhad. Tedy $k\leq 1+\log n$. Z vlastností èerveno-èerných stromù plyne, ¾e $k\leq \<hloubka>(T) \leq 2k$.
-\qed
-
-Nyní si rozebereme vkládaní do èerveno-èerných stromù.
-
-\s{Insert:}
-Vlo¾íme hodnotu do stromu jako do standardního binárního vyhledávacího stromu, tedy jako list,
-a obarvíme èervenì. První ani tøetí axiom jsme neporu¹ili a pokud je pøedek èerný, tak ani
-axiom druhý, èili jsme hotovi:
-\fig{01.eps}{2in}
-Jestli¾e je otec~$o$ na¹eho vrcholu~$t$ také èervený, budeme øíkat, ¾e do¹lo k~{\I poru¹e}
-a budeme se ji sna¾it opravit. Situace musí vypadat následovnì:
-
-\fig{02.eps}{1.5in}
-
-Nyní rozli¹íme následující tøi pøípady:
-\itemize\ibull
-\:Pokud strýc~$s$ vrcholu~$t$ je èervený, pak pouze pøebarvíme $o$, $d$ a $s$ podle následujícího obrázku.
-Nyní $d$ mù¾e být porucha, ov¹em posunutá o 2 hladiny vý¹e, tak¾e pøi nejhor¹ím pokraèujeme
-do~koøene a poslední poruchu opravíme pøebarvením koøene na~èerno.
-\fig{03.eps}{3.5in}
-\:Strýc $s$ je èerný. Pak zále¾í na tom, zda hodnota $t$ le¾í mezi hodnotami $o$ a $d$ nebo ne. Jinými slovy, zda cesta $t-o-d$ obsahuje zatáèku.
- \numlist\nalpha
- \:Bez zatáèky: Provedeme rotaci a pøebarvíme. Splnìny budou axiomy 1, 2 i 3, tedy máme èervenoèerný strom:
- \fig{04.eps}{4in}
- \:Se zatáèkou: Provedeme dvojitou (LR) rotaci a pøebarvíme. Opìt získáme korektní èervenoèerný strom:
- \fig{05.eps}{4in}
- \endlist
-\endlist
-
-\bye
+++ /dev/null
-P=12-kostry
-
-include ../Makerules
+++ /dev/null
-\input lecnotes.tex
-
-\prednaska{13}{Hashování}{(zapsali Martin Francù a Bohdan Maslowski)}
-
-V~pøedchozích kapitolách jsme ukázali, ¾e aèkoliv v~obecném pøípadì
-nelze tøídit rychleji ne¾ v~èase $\Theta(n\log n)$, pro nìkteré speciální
-druhy hodnot (malá celá èísla, øetìzce apod.) existuje efektivnìj¹í lineární
-algoritmus. Nyní si podobným zpùsobem posvítíme na~reprezentaci mno¾in --
-navrhneme datovou strukturu, která bude umìt s~vhodnými typy prvkù provádìt
-základní mno¾inové operace (\<Insert>, \<Delete>, \<Find>) v~{\I prùmìrnì}
-konstantním èase (pomocí vyhledávacích stromù jsme to zatím umìli v~èase $\O(\log n)$
-na operaci).
-
-Jaké máme mo¾nosti? Zaènìme od~tìch nejjednodu¹¹ích a oznaème si~$n$ velikost
-mno¾iny a $U$ velikost univerza, z~nìj¾ prvky mno¾iny vybíráme.
-
-\s{1. Pole indexované od $0$ do $U-1$}
-
-\indent \<Insert>, \<Delete> i \<Find> pracují v~èase $\O(1)$, ale platíme za~to
-pamì»ovou slo¾itostí $\Theta(U)$.
-
-\s{2. Èíslicový strom (radix tree)}
-
-Zvolíme základ soustavy $b$, prvky universa zapisujeme jako $k$-tice èíslic. Èili:
-$$x\in U \mapsto [b]^{k}, k=\lceil\log_b U\rceil.$$
-Tím nám vznikne strom s~hloubkou $k$, na $k$-té úrovni se rozhodujeme podle $k$-té èíslice.
-
-Operace \<Find> bude trvat $O(\log_b U)$, operace \<Insert> a \<Delete> pak
-$O(b \log_b U)$. Strom zabere prostor velikosti $O(nb\log_b U)$.
-
-Èím bude základ $b$ men¹í, tím lep¹í pamì»ová a hor¹í èasová slo¾itost,
-a obrácenì. Èíslicový strom tedy vy¾aduje peèlivé nastavení parametrù.
-
-\h{3. Písmenkový strom }
-
-Pou¾ívá se pro øezìzce, za základ soustavy $b$ zvolíme abecedu.
-
-\medskip
-
-$\vert x\vert $...délka øetìzce
-
-Find, Insert, Delete: $O(\vert x\vert )$
-
-Pamì»: $O(\sum_{i} \vert x\vert )$
-
-\bigskip
-
-\h{4. Hashování }
-
-
-$h$ ... hashovací funkce $[U] \rightarrow [m]$
-
-$m$ ... poèet pøihrádek, v ka¾dé pøihrádce spojový seznam.
-
-\medskip
-
-Perfektní hashovací funkce pøiøadí do ka¾dé z $m$ pøihrádek jediný prvek z $[U]$.
-
-\s{Pøíklad:} Jaká je pravdìpodobnost, ¾e 2 z k lidí mají narozeniny ve stejný den? Staèí 23 lidí, aby nastala kolize s pravdìpodobností vìt¹í ne¾ jedna polovina. Aby nenastala, potøebovali bychom vìt¹í poèet pøihrádek: $p = cm^2$
-
-
-
-
-\h{Hashování se separovými øetezci}
-
-\medskip
-Insert, Delete i Find pracují v èase $O(\vert x\vert )$, pamì»ová slo¾itost je $O(m+n)$
-\medskip
-
-vlo¾ili jsme $x_1 \ldots x_n \in [U]$
-
-nyní pracujeme s $x \in [U]$
-
-pøedpokládejme, ¾e $h(x_i)$ a $h(x)$ jsou náhodné
-
-$p:=h(x)$
-
-$N_p:=\sharp$ : $h(x_1)=h(x)$
-
-$N_{i,p} := 0$ pokud $h(x_i)\neq p$
-
-$N_{i,p} := 1$ pokud $h(x_i)= p$
-
-$N_p = \sum_{i=1}^n N_{i,p}$
-
-$E N_{i,p} = Pr[h(x_i)=p] = {1 \over m} $
-
-$E N_p = n {1\over m} = {n \over m} $
-
-\medskip
-
-Za pøedpokladu, ¾e hash pøidìluje náhodnou pøihrádku, je v ka¾dé pøihrádce ${n \over m}$ prvkù.
-
-\bigskip
-
-
-\h{Druhá pùlka}
-
-
-\noindent {\sc Vìta:} {\it Pokud jsme do hashovací tabulky vlo¾ili
-hodnoty $x_1,\dots,x_m$ tak, ¾e $h_{x_i}$ jsou v¹echny stejnì
-pravdìpodobné, pak následující Insert, Delete, Find pracují v èase
-${\cal O}\,({n \over m}+1)$. (Za pøedpokladu, ¾e výpoèet $h$ trvá
-${\cal O}\,(1)$.)}
-
-\medskip
-
-{\it Insert}: Spoèítá si $h(x)$ a vlo¾í jej do spojového seznamu v patøièné pøihrádce.
-
-{\it Delete}: Spoèítá si $h(x)$ a najde jej v spojovém seznamu v patøièné pøihrádce a
-odstraní jej.
-
-{\it Find}: Spoèítá si $h(x)$ a najde jej v spojovém seznamu v patøièné pøihrádce.
-
-\medskip
-
-\noindent Tato metoda má dva háèky:
-
-1. vysoká koliznost
-
-2. musíme dopøedu vìdìt, kolik prvkù budeme vkládat
-
-\bigskip
-
-{\large \bf Nafukovací hashovací tabulka}
-
-na poèátku: $n=0;m=4$
-
-v ka¾dém okam¾iku budeme udr¾ovat pomìr ${n\over m}\in \langle
-{1\over 2},1)$
-
-tedy: (pokud nastane)
-$$n=4 \Rightarrow m:=8 $$
-$$n=8 \Rightarrow m:=16$$
-Pøi ka¾dém zvìt¹ování tabulky musíme zvìt¹it obor hodnot hashovací funkce a v¹echny prvky ji¾
-v tabulce ulo¾ené znovu "zhashovat".
-
-Nafukování trvá ${\cal O}\, (n)$, ale mezi dvìma nafouknutími je
-minimálnì $ {n\over 2} $ Insertù.
-
-Pokud tedy ka¾dý Insert pøispìje konstantou, tak to máme zadarmo!
-
-\bigskip
-
-PROBLÉM: Jak udìlat funkci Delete? Jednoduché zmen¹ování tabulky pøi pomìru
-${n\over m}={1\over 2}$ není dostaèující, nebo» by se mohlo stát, ¾e bychom nafukovali a
-zfukovali hned po sobì.
-
-\bigskip
-
-{\bf Verze s Insert a Delete}
-
-Udr¾ujeme ${n\over m}\in \langle {1\over 4},1)$
-
-pøi zfukování tabulku pùlíme, pøi nafukování zdvojnásobujeme
-
-\medskip
-
-Po pøehashování je pomìr ${n\over m}={1\over 2} \Rightarrow$mezi dvìma pøehashováními
-nastane minimálnì ${m\over 2}$ Insertù èi ${m\over 4}$ Deletù, tedy alespoò ${m\over 4}$
-operací.
-
-Zároveò zabereme v¾dy lineárnì prostoru vzhledem k $n$.
-
-\bigskip
-
-\noindent {\sc Vìta:} {\it Nafukovací hashování (se separovanými øetìzci) za pøedpokladu
-rovnomìrné pravdìpodobnosti pracuje v prùmìrném èase ${\cal O}\,(1)$ na operaci a v prostoru
-${\cal O}\,(n)$ (kde $n$ je poèet pøítomných prvkù).}
-
-\medskip
-
-Kde pøijít k rovnomìrné hashovací funkci?
-
-\medskip
-
-\noindent Dìdeèkovo vyprávìní:
-
-a) $h(x)= x$ $mod (m)$ -rychlá, snadná kolize, sudé x v¾dy v sudých pøihrádkách. Na náhodných
-èíslech funguje v pohodì.
-
-\bigskip
-
-\noindent POZOROVÁNÍ: $\alpha \in(0,1)$ iracionální;
-
-$x \in{\sc N}$ pak $\alpha x $ $ mod (1)$ se chová "náhodnì".
-
-\medskip
-
- b) $h(x)=\lfloor m(\alpha x $ $mod(1))\rfloor$
- $$ {A\over W} \sim \alpha \Rightarrow h(x)= \lfloor m(Ax \ mod(W)) \rfloor $$
-
- \medskip
-
- Jako velmi výhodné se jeví pou¾ívat $\alpha = {{\sqrt{5} -1}\over{2}} $ - pøevrácená hodnota zlatého øezu.
-
-\medskip
-
-c) funkce k hashování øetìzcù
-$$ h(\emptyset)=0 $$
-$$ h(x_1, \dots , x_n)=c(h(x_1, \dots ,x_{n-1})+x_n)$$
-$$ \Rightarrow h(x_1, \dots , x_n)=x_n + cx_{n-1} + c^2 x_{n-2} +
-\dots $$ (polynom)
-
-\medskip
-
-d)Pokud $x$ je $k$-tice (celých) èísel. Pak mù¾eme jako hashovací funkci pou¾ít skalární
-souèín s náhodnì vybraným prvkem z vektorového prostoru $Z_p^k$ (nad koneèným tìlesem):
-$$ h(x)= \overrightarrow{a} * \overrightarrow{x} mod (p) $$
-
-\medskip
-
-Pravdìpodobnost pøes v¹echny volby $\overrightarrow{a}$, ¾e $x,y$ se zahashují stejnì je
-pak:
-$$ p[h(x)=h(y)] = p[\overrightarrow{a}*\overrightarrow{x} =
-\overrightarrow{a}*\overrightarrow{y}]= p[\overrightarrow{a}*
-(\overrightarrow{x}-\overrightarrow{y})=0)] $$
-
-Pokud platí $p[\overrightarrow{a}* (\overrightarrow{x}-\overrightarrow{y}=0)]$, pak
-$\overrightarrow{a}$ je prvkem ortogonálního doplòku k $(x-y)$, ten má $ dim (W^\bot) =
-(k-1)$
-
-$$ p= {{W^\bot}\over{Z_p^k}}= {{p^{k-1}}\over{p^k}} = {1\over p}={1\over m}$$
-
-kde $m$ je poèet pøihrádek. Co¾ je dostaèující.
-
-\bye
-
+++ /dev/null
-P=13-hash
-
-include ../Makerules
+++ /dev/null
-\input ../lecnotes.tex
-
-\prednaska{2}{Rozdìl a panuj}{(zapsali J. Záloha a P. Ba¹ista)}
-
-Na~minulé pøedná¹ce jsme si zavedli výpoèetní model a èasovou slo¾itost.
-Nyní se pustíme do~analýzy slo¾itosti rùzných algoritmù, zejména rekurzivních
-zalo¾ených na metodì Rozdìl a panuj {\sl (Divide et Impera)}.
-
-\h{Asymptotická notace}
-
-Jak u¾ víme, pøi porovnávání algoritmù je obvykle rozhodující asymptotické
-chování a multiplikativní konstanty mù¾eme zanedbávat (beztak závisí na~konkrétním
-poèítaèi). Proto si zavedeme takzvanou asymptotickou notaci, která slou¾í
-pro právì takové porovnávání rùstu funkcí:
-
-\s{Definice:} Pro funkce $f,g: {\bb N} \rightarrow {\bb R}^+$ øekneme,
-¾e $f$ je $\O(g)$ právì tehdy, kdy¾ $\exists c>0: \forall ^{*} n \in {\bb N}: f(n) \leq c \cdot g(n)$.
-Zde $\forall^* n \in {\bb N}$ je zkratka za \uv{$\exists n_0 \in {\bb N}: \forall n \geq n_0$}, tedy
-\uv{pro v¹echna~$n$ a¾ na~koneènì mnoho výjimek.}
-
-\s{Poznámka:} $\O$-notace tedy vyjadøuje, ¾e funkce~$f$ je skoro v¹ude men¹í nebo nejvý¹e rovná
-nìjakému reálnému násobku funkce~$g$. Aèkoliv zápis vypadá jako rovnost, rozhodnì
-není symetrický: napøíklad platí $\log n=\O(n)$, ale neplatí $n=\O(\log n)$. Formálnì by bylo lep¹í pova¾ovat $\O(g)$
-za tøídu funkcí, pro které platí, ¾e se dají shora odhadnout kladným násobkem funkce~$g$, a~psát tedy~$f\in\O(g)$,
-ale zvyk je bohu¾el ¾elezná ko¹ile.
-
-\s{Pøíklady:} $2.5n^{2} = \O(n^{2})$, $2.5n^{2}+30n = \O(n^{2})$.
-
-\>Také platí:
-$$
-\O(f)+\O(g) \in \O(f+g),
-$$
-èím¾ myslíme, ¾e pokud vezmeme libovolnou $f'=O(f)$ a $g'=O(g)$, bude $f'+g'=O(f+g)$.
-To platí, jeliko¾ skoro v¹ude je $f' \leq cf$, $g'\leq dg$, a~tedy $f'+g' \le cf+dg \le (c+d)(f+g)$.
-
-\s{Cvièení:} Uka¾te, ¾e:
-\itemize\ibull
-\:$\O(f) \cdot \O(g)=\O(f \cdot g)$,
-\:$\O(f+g)=\O(\max(f,g))$,
-\:$\O(n^{2})+\O(n)=\O(n^{2}+n)=\O(n^{2})$.
-\endlist
-
-$\O$-notace popisuje horní odhad asymptotického chování funkce. Mnohdy v¹ak
-potøebujeme také odhandout funkci zespodu (chceme-li øíci, ¾e algoritmus potøebuje
-{\I alespoò} nìjaké mno¾ství èasu nebo pamìti), pøípadnì z~obou stran:
-
-\s{Definice:}
-
-\itemize\ibull
-\:$f=\Omega(g) \equiv \exists c>0:\forall^* n \in {\bb N}: f(n) \geq c\cdot g(n)$.
-
-$\Omega$-notace tedy øíká, ¾e hodnota funkce $f$ je v¾dy stejná nebo vy¹¹í ne¾ nìjaký $c$-násobek funkce $g$, a tedy $g=\O(f)$.
-\:$f=\Theta(g) \equiv f=O(g) \wedge f=\Omega(g)$
-
-nebo výøeènìji:
-
-$f=\Theta(g) \equiv \exists$ $c_{1},c_{2} > 0: c_{1}\cdot g(n) \leq f(n) \leq c_{2}\cdot g(n)$ To znamená, ¾e existují nezáporné reálne konstanty $c_{1},c_{2}$ takové, ¾e se funkce $f(n)$ dá ohranièit $c_{1}$- a $c_{2}$-násobkem funkce $g(n)$.
-\endlist
-
-\s{Porovnání rùstu funkcí:} (aneb jak moc máme algoritmy rádi podle jejich chování od~nejlep¹ích k~nejhor¹ím)
-
-\itemize\ibull
-\:$\Theta(1) \ldots$ funkce zespoda i shora ohranièené konstantami
-\:$\Theta(\log{( \log{n} )})$
-\:$\Theta(\log{n})$ \dots\ logaritmická
-\:$\Theta(n^{\varepsilon}), \varepsilon \in (0,1)$ \dots\ sublineární
-\:$\Theta(n)$ \dots\ lineární
-\:$\Theta(n^{2})$ \dots\ kvadratická
-\:$\Theta(n^{k})$ \dots\ polynomiální
-\:$\Theta(2^{n})$ \dots\ exponenciální pøi základu $2$
-\:$\Theta(3^{n})$ \dots\ exponenciální pøi základu $3$
-\:$\Theta(k^{n})$ \dots\ exponenciální pøi základu $k>1$
-\:$\Theta(n!)$ \dots\ faktoriálová
-\:$\Theta(n^{n})$
-\:\dots\ nekoneènì mnoho dal¹ích tøíd (i mezi tìmi vý¹e uvedenými)
-\endlist
-
-\>{\sl Poznámka:} Pokud se v~odhadu slo¾itosti vyskytne logaritmus (jinde ne¾
-v~exponentu), nezále¾í na tom, jaký má základ, proto¾e platí:
-$$
-\log_k{n}={{\log_c{n}}\over{\log_c{k}}}={{1}\over{\log_c{k}}}\cdot \log_c{n},
-$$
-kde $1/\log_c{k}$ je jen konstanta, tak¾e ji mù¾eme \uv{schovat do~$\O$.}
-
-\s{Pøíklad:} Eukleidùv algoritmus:
-Pokud jej pustíme na $2$ èísla o $n$ bitech, poèet iterací bude $\O(n)$, ka¾dá
-iterace trvá $\O(n^{2})$ krokù. Tak¾e celkovì má tento algoritmus èasovou
-slo¾itost $\O(n^{3})$.
-
-\h{Metoda Rozdìl a panuj}
-
-Nyní pøestaòme chodit kolem horké ka¹e a øekòeme si, co to ono vý¹e zmiòované
-\uv {rozdìl a panuj} znamená. Mìjme nìjaký problém, který má tu vlastnost, ¾e
-kdy¾ jej rozdìlíme na nìjaké podproblémy, které mají stejný charakter, a ty
-vyøe¹íme, slo¾ením jejich øe¹ení mù¾eme získat øe¹ení pùvodního problému.
-Algoritmus tedy bude rekurzivnì volat sám sebe, ne¾ se dostane k~podproblému
-nìjaké konstantní velikosti, který u¾ umí vyøe¹it triviálnì, a pak se zaène
-z~rekurze vracet a skládat jednotlivá dílèí øe¹ení.
-
-\s{Pøíklad 1 -- MergeSort:} Algoritmus {\I MergeSort} pro tøídìní posloupnosti vypadá takto:
-vstup rozdìlíme na~dvì (skoro) stejné èásti, ty rekurzivním voláním setøídíme,
-a~nakonec výsledné dvì posloupnosti slijeme do~jedné. Rozdìlování a slévání nám
-trvá lineárnì dlouho, tak¾e pro èasovou slo¾itost MergeSortu platí tato
-rekurentní rovnice: $$T(n)=2 \cdot T({{n}/{2}})+\O(n).$$ Jak ji vyøe¹íme? Tøeba
-tak, ¾e si pøedstavíme strom rekurzivních volání. Ka¾dý vrchol má dva syny
-(dìlíme vstup na~dvì èásti), v~nich¾ jsou vstupy polovièní velikosti. V~ka¾dém
-vrcholu trávíme èas lineární s~velikostí jeho vstupu, souèet velikostí vstupù
-pøes ka¾dou hladinu je~$n$ a hloubka stromu musí být $\O(\log n)$. Vyjde nám
-tedy, ¾e $T(n)=\O(n\log n)$.
-
-\s{Pøiklad 2 -- násobení èísel:} Pokud násobíme dvì èísla $X$ a $Y$ zpùsobem,
-který nás uèili na základní ¹kole, dostaneme se na èasovou slo¾itost $\Theta(n^2)$.
-Proto¾e se jedná o~dost èastou operaci, zamysleme se, jestli by ne¹la zrychlit.
-Nasmìrujme na¹e úvahy na postup \uv{rozdìl a panuj}. Rozdìlíme ka¾dého èinitele
-na dvì stejnì dlouhé èásti a pro jednoduchost pøedpokládejme, ¾e toto
-roz¹tìpení èinitele probìhne v¾dy bez zbytku:
-$$
-X=A \cdot 10^{{n}/2}+B, \qquad Y=C \cdot10^{{n}/{2}}+D.
-$$
-Zde $A, B, C, D$ jsou u¾ jen $n/2$-ciferná èísla. Pùvodní souèin získáme jako:
-$$
-XY=(A\cdot 10^{{n}/{2}}+B) (C\cdot 10^{{n}/{2}}+D)=AC \cdot 10^{n}+(AD+BC)\cdot 10^{{n}/{2}}+BD.
-$$
-Nyní, jak vidíme, staèí spoèítat souèin ètyø $n/2$-ciferných èísel. Uva¾me,
-jakou bude mít tento algoritmus èasovou slo¾itost:
-$$T(n) = 4T(n/2)+\Theta(n).$$
-Jak takovou rekurenci vyøe¹íme?
-
-\>{\sl 1. zpùsob: Øe¹ení pomocí rozepisování:}
-$$\eqalign{
-T(n)&= 4T(n/2)+\Theta(n) = \cr
- &= 4T(n/2)+cn = \cr
- &= 4\cdot (4T(n/4)+cn/2)+cn = 4^2T(n/4)+2cn+cn = 4^2T(n/4)+3cn = \cr
- &= 4^2\cdot (2T(n/8)+cn/4)+3cn = 4^3T(n/8)+4cn+3cn = 4^3T(n/8)+7cn = \cr
- &\ldots\cr
-}$$
-Odtud snadno vypozorujeme, ¾e jednotlivé vztahy se vyvíjí podle vzorce
-$T(n)=4^kT(n/2^k) + (2^k-1)cn.$ Pro $k=\lceil\log_2 n\rceil$ je ov¹em
-$2^k\le 1$, tak¾e $T(n/2^k)=\Theta(1)$ a dostaneme (horní celou èást zanedbáme,
-ta ovlivní jen konstanty):
-$$
-T(n) = 4^{\log_2 n}\Theta(1) + (2^{\log_2 n}-1)cn = n^2\Theta(1) + (n-1)cn = \Theta(n^2).
-$$
-
-\>{\sl 2. zpùsob: Úvaha o~stromu:} Nakreslíme si strom rekurzivních volání
-na¹eho algoritmu:
-\fig{figure.eps}{4in}
-Na~$i$-té hladinì stromu máme $4^i$ vrcholù, v~nich jsou vstupy velikosti
-$n/2^i$, tak¾e na~celé hladinì trávíme èas celkem $\Theta(4^i\cdot n/2^i)
-= \Theta(2^in)$. Velikosti vstupù klesají exponenciálnì, tak¾e celý strom
-je hluboký $k=\log_2 n$ (opìt si dovolíme zapomenout na~horní celou èást).
-Celkem tedy trávíme èas $\sum_{i=0}^k \Theta(2^in) = \Theta(n\cdot\sum_{i=0}^k 2^i) = \Theta(n^2)$.
-
-Oba zpùsoby analýzy se tedy shodují, ¾e ná¹ algoritmus má kvadratickou èasovou
-slo¾itost a ¾e jsme si oproti klasickému algoritmu nikterak nepomohli.
-Podívejme se je¹tì jednou na~to, jak se ná¹ algoritmus vìtví:
-$$\vbox{\halign{\hfil#\hfil \quad & \hfil#\hfil \quad &\hfil#\hfil\cr
-hloubka & poèet úloh & velikost podúlohy\cr
-\noalign{\smallskip\hrule\medskip}
-0 & $4^{0}$ & ${n}/{2^{0}}$\cr
-1 & $4^{1}$ & ${n}/{2^{1}}$\cr
-2 & $4^{2}$ & ${n}/{2^{2}}$\cr
-3 & $4^{3}$ & ${n}/{2^{3}}$\cr
-\vdots & \vdots & \vdots\cr
-$k$ & $4^{k}$ & ${n}/{2^{k}}$\cr}}$$
-Naskýtá se otázka, jestli bychom nemohli èasovou slo¾itost zlep¹it. Toho bychom
-mohli dosáhnout buïto zlep¹ením èlenu $cn$ v~na¹í rekurenci, èili zefektivnìním
-spojování podúloh. To ov¹em není pøíli¹ nadìjné (pokud ètenáø nevìøí, mù¾e si to dokázat),
-tak¾e místo toho vyu¾ijeme druhou ¹anci, a~to omezení vìtvení ze~ètyø vìtví na~tøi.
-Pøipomeòme si, ¾e potøebujeme spoèítat:
-$$
-XY=AC\cdot 10^{n}+(AD+BC)\cdot 10^{n/2}+BD.
-$$
-Pøitom ale nepotøebujeme znát souèiny $AD$ ani $BC$ samostatnì, nebo» nám staèí
-zjistit celý èlen $AD+BC$. Kdybychom poèítali $AC$, $BD$
-a potom $(A+B)(C+D)=AC+AD+BC+BD$, tak odèítáním $(AC+BD)$ od $AC+AD+BC+BD$ dostaneme
-hledaný prostøední èlen $AD+BC$. Nyní nám ji¾ staèí jen tøi
-násobení, ale potøebujeme tøi sèítání a jedno odèítání navíc.
-Uká¾eme, ¾e tato komplikace je zanedbatelná oproti práci u¹etøené
-men¹ím vìtvením. Podívejme se opìt na~tabulku:
-$$\vbox{\halign{\hfil#\hfil \quad & \hfil#\hfil \quad &\hfil#\hfil\cr
-hloubka & poèet úloh & velikost podúlohy\cr
-\noalign{\smallskip\hrule\medskip}
-0 & $3^{0}$ & ${n}/{2^{0}}$\cr
-1 & $3^{1}$ & ${n}/{2^{1}}$\cr
-2& $3^{2}$ & ${n}/{2^{2}}$\cr
-3 & $3^{3}$ & ${n}/{2^{3}}$\cr
-\vdots & \vdots & \vdots\cr
-$k$ & $3^{k}$ & ${n}/{2^{k}}$\cr}}$$
-Opìt uva¾me, kolik práce spotøebujeme v~souètu pøes v¹echny hladiny (hloubka stromu
-$k$ je opìt $\lceil\log_2 n\rceil$ a horní celou èást zanedbáme):
-$$\sum_{i=0}^{k}3^{i}\cdot {{n}\over{2^{i}}}=\sum_{i=0}^{k} \left( {{3}\over{2}} \right) ^{i}\cdot n=n\cdot \sum_{i=0}^{k} \left( {{3}\over{2}} \right) ^{i}=n\cdot {{ \left( {{3}\over{2}} \right) ^{k+1}-1}\over{{{3}\over{2}}-1}}=
-$$
-$$
-=n\cdot {{ \left( {3}\over{2} \right) ^{k+1}-1}\over{{{1}\over{2}}}}=2\cdot n\cdot \left[ \left( {{3}\over{2}} \right) ^{k+1}-1 \right] = \O \left( n\cdot \left( {{3}\over{2}} \right) ^{\log_2{n}} \right) =
-$$
-$$
-=\O \left( n\cdot {{3^{\log_2{n}}}\over{2^{\log_2{n}}}} \right)=\O \left( n\cdot {{3^{\log_2{n}}}\over{n}} \right)=\O \left( 3^{\log_2{n}} \right)=\O \left( (2^{\log_2{3}})^{\log_2{n}} \right)=
-$$
-$$
-=\O \left( 2^{(\log_2{n}) \cdot \log_2{3}} \right)=\O \left( (2^{\log_2{n}})^{\log_2{3}} \right)=\O \left( n^{\log_2{3}} \right) =\O \left( n^{1.585} \right).
-$$
-Upravený algoritmus u¾ tedy má lep¹í èasovou slo¾itost, konkrétnì $\O(n^{1.585})$.
-V~praxi bychom samozøejmì pro èinitele ne¹tìpili a¾ na jednociferná èísla,
-ale zastavili se u~nìjaké dostateènì malé délky (øeknìme 50~cifer) a tam
-pøepnuli na~kvadratický algoritmus, který má men¹í re¾ii.
-
-(Mimochodem, pro násobení èísel existuje je¹tì efektívnìj¹í algoritmus, který
-dosahuje èasové slo¾itosti $\O(n \log{n})$, av¹ak na~to u¾ jsou potøeba trochu
-pokroèilej¹í techniky, jako je diskrétní Fourierova transformace, tak¾e
-si jej necháme na~pøí¹tí semestr.)
-
-\h{Master Theorem}
-
-Metody øe¹ení rekurentních rovnic pro slo¾itosti rozdìlovacích a panovacích algoritmù
-by asi fungovaly i na~jiné algoritmy ne¾ jen na¹e dva pøíklady, ale proè se poka¾dé
-moøit upravováním výrazù? Radìji si doká¾eme obecnou vìtu, která pùjde na~vìt¹inu
-takových rekurencí nasadit. Øíká se jí Master Theorem nebo také (vzhledem k~tomu,
-jak se pou¾ívá) Kuchaøková vìta.
-
-\s{Vìta:} \>{\sl (Master Theorem)} Pøedpokládejme, ¾e $T(1)=\O(1)$ a $T(n)=a\cdot T(\lceil {{n}\over{b}} \rceil)+\O(n^d)$, kde $a \geq 1$, $b>1$ a $d \geq 0$. Potom $T(n)$ je:
-
-\smallskip
-
-\halign{#&#&#\cr
-\indent & $\O(n^d)$ & kdy¾ $a<b^d$,\cr
-& $\O(n^d\cdot \log{n})$ & kdy¾ $a=b^d$,\cr
-& $\O(n^{\log_b{a}})$ & kdy¾ $a>b^d$.\cr}
-
-\proof Pøedpokládejme nejdøíve, ¾e $n=b^m, m \in \bb{N}$, aby platilo $\lceil
-{{n}\over{b}} \rceil = {{n}\over{b}}$. Pou¾ijeme opìt \uv{dùkaz stromem}.
-Strom rekurzivních volání se v¾dy vìtví na stejný poèet vìtví, konkrétnì~$a$,
-a~velikosti vstupù klesají $b$-krát. Podívejme se na~tabulku:
-$$\vbox{\halign{\hfil#\hfil \quad & \hfil#\hfil \quad & \hfil#\hfil & \hfil#\hfil \cr
-hloubka & poèet podúloh & velikost podúlohy & èas na hladinì \cr
-\noalign{\medskip\hrule\medskip}
-0 & $1$ & $n$ & $\O(n^d)$\cr
-1 & $a$ & $n/{b^1}$ & ${\O(({n/{b^1}})^d) \cdot a^1}$\cr
-2 & $a^2$ & $n/{b^2}$ & ${\O(({n/{b^2}})^d) \cdot a^2}$\cr
-3 & $a^3$ & $n/{b^3}$ & ${\O(({n/{b^3}})^d) \cdot a^3}$\cr
-\vdots & \vdots & \vdots\cr
-$k$ & $a^k$ & $n/{b^k}$ & ${\O(({n/{b^k}})^d) \cdot a^k}$\cr}}$$
-
-\noindent
-Zkoumejme èas potøebný na vyøe¹ení v¹ech podúloh na jedné hladinì:
-$$
-\O \left( \left( {{n}\over{b^k}} \right) ^d \right) \cdot a^k=\O \left( a^k\cdot n^d \left( {{1}\over{b^k}} \right) ^d \right)=\O \left( a^k \cdot n^d \left( {{1}\over{b^d}} \right) ^k \right)=\O \left( n^d \left( {{a}\over{b^d}} \right) ^k \right).
-$$
-Celkem je tedy èas potøebný na vyøe¹ení v¹ech podúloh na v¹ech hladinách (to
-znamená celé úlohy):
-$$
-T(n)=\sum_{k=0}^{\log_b{n}}\O \left( n^d \left( {{a}\over{b^d}} \right) ^k \right)=\O \left( n^d \sum_{k=0}^{\log_b{n}} \left( {{a}\over{b^d}} \right) ^k \right)
-$$
-V¹imnìme si èlenu ${a}\over{b^d}$. Na jeho hodnotì závisí hodnota výsledné sumy, proto¾e se vlastnì jedná o kvocient geometrické øady. Proto rozli¹me následující pøípady:
-
-\>{\I 1.} ${{a}\over{b^d}}<1$ -- jinými slovy, práce na jednotlivých hladinách exponenciálnì ubývá a souèet sumy, i kdyby
-byla nekoneèná, se dá omezit nìjakou konstantou, která se schová do $\O$, a tak je $T(n)=\O(n^d)$.
-
-\>{\I 2.} ${{a}\over{b^d}}=1$ -- práce na jednotlivých hladinách je stejnì, to znamená, ¾e souèet sumy je právì $\log_b n$, a tedy $T(n) = \O(n^d \cdot \log_b(n))$.
-
-\>{\I 3.} ${{a}\over{b^d}}>1$ -- to znamená, ¾e práce na jednotlivých hladinách
-pøibývá, tak¾e musíme geometrickou øadu seèíst poctivì: $T(n) = \O(n^d \cdot
-({{a}\over{b^d}})^{\log_b{n}})$. Tento výraz vypadá ponìkud o¹klivì,
-ale je¹tì ho trochu (alespoò kosmeticky) upravíme:
-$$
-\O\left(n^d \cdot \left({{a}\over{b^d}}\right)^{\log_b{n}}\right)=\O\left({ a^{\log_b{n}} \cdot n^d \over (b^d)^{\log_b{n}}}\right)=\O\left({\left(b^{\log_b{a}}\right)^{\log_b{n}} \cdot n^d \over{\left(b^d\right)^{\log_b{n}}}}\right)=
-$$
-$$
-=\O\left({\left(b^{\log_b{n}}\right)^{\log_b{a}} \cdot n^d \over{\left(b^{\log_b{n}}\right)^d}}\right)
-=\O\left({n^{\log_b{a}} \cdot n^d \over{n^d}}\right)
-=\O\left(n^{\log_b{a}}\right).
-$$
-Tyto tøi pøípady pøesnì odpovídají rozdìlení pøípadu v~tvrzení vìty.
-
-Vra»me se nyní k~mo¾nosti, kdy $n$ není mocnina~$b$.
-Tehdy platí $b^l<n<b^{l+1}$ pro nìjaké $l \in \bb{N}$. V tomto pøípadì
-zaokrouhleme $n$ a~poèítejme s $n^{\prime}=b^{l+1} < bn$. Jeliko¾ funkce
-$T(n)$ je neklesající, dostaneme $T(n)\le T(n')$ a ve~v¹ech tøech pøípadech
-je výsledek pro~$n'$ maximálnì konstanta-krát vìt¹í ne¾ pro~$n$, co¾ se
-\uv{schová do~$\O$.} Vìta tedy platí i v~tomto pøípadì.
-\qed
-
-\s{Pøíklad:}
-Porovnejme si nìkteré známé algoritmy a jejich èasovou slo¾itost pomocí \>{\sl Master Theoremu}:
-$$\vbox{\halign{# \quad \quad & # \quad \quad & # \quad \quad & # \quad \quad & #\cr
-algoritmus & $a$ & $b$ & $d$ & èasová slo¾itost\cr
-\noalign{\smallskip\hrule\medskip}
-Mergesort & 2 & 2 & 1 & $\O({n \cdot \log{n}})$\cr
-Násobení I. & 4 & 2 & 1 & $\O(n^2)$\cr
-Násobení II. & 3 & 2 & 1 & $\O(n^{\log_2{3}})$\cr
-Binární vyhledávání & 1 & 2 & 0 & $\O(\log{n})$\cr}}$$
-
-\s{Domácí úkol nakonec:} Vymyslete algoritmus, který by z $n$ zadaných bodù v rovinì (prostoru) na¹el takové dva, které~jsou od sebe nejménì vzdálené (zde existuje takový algoritmus s èasovou slo¾itostí $\O(n \cdot \log{n})$).
-
-\bye
+++ /dev/null
-P=2-rozdel
-
-include ../Makerules
+++ /dev/null
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-%%EOF
--- /dev/null
+\input ../lecnotes.tex
+
+\prednaska{0}{Ukázka pou¾ití maker}{(zapsal Martin Mare¹)}
+
+% Takhle vypadá nadpis
+\h{Pøedposlední kapitola}
+
+% \> zaøídí, aby se odstavec neodsadil. Pomocí \I se sází kurzíva, \foot vyrobí poznámku pod èarou.
+\>Je svaèveèer. {\I Lysperní jezeleni} se vírnì vrtáèejí v~mokøavì.\foot{Viz Lewis Caroll: Jabberwocky.}
+
+\itemize\ibull
+\:Takhle.
+\:Vypadají.
+\:Odrá¾ky.
+\endlist
+
+\h{Druhá kapitola}
+
+\s{Vìta:} Paøez není strom.
+
+\proof
+Strom je speciálním pøípadem lesa. Les je (podle definice) graf bez kru¾nic.
+Paøez obsahuje alespoò jednu kru¾nici. Proto paøez nemù¾e být les, a~tedy ani
+strom.
+\qed
+
+\s{Algoritmus:} (tøídìní posloupnosti $a_1,\ldots,a_n$ pomocí {\sc Stupidsort}u)
+
+\algo
+\:$\pi \leftarrow \hbox{identická permutace na~mno¾inì $\{1,\ldots,n\}$}$.
+\:Opakuj:
+\::Ovìø, zda je posloupnost $a_{\pi(1)},a_{\pi(2)},\ldots,a_{\pi(n)}$ uspoøádána vzestupnì.
+ Pokud je, vra» ji jako výsledek.
+\::Nahraï $\pi$ jejím lexikografickým následníkem.
+\endalgo
+
+\bye
--- /dev/null
+P=0-demo
+
+include ../Makerules
--- /dev/null
+\input ../lecnotes.tex
+\input epsf.tex
+
+\def\N{N}
+
+\prednaska{1}{Eukleidùv algoritmus}{(zapsaly T.~Klimo¹ová a K.~B\"ohmová)}
+
+\h{}
+\>{\bf Problém hledání nejvìt¹ího spoleèného dìlitele }
+
+Cílem je najít nejvìt¹ího spoleèného dìlitele zadaných èísel
+$a_0, b_0 \in \N $, dále jen $\gcd(a_0, b_0)$, tedy Greatest Common
+Divisor.
+
+\s{Algoritmus:} (Eukleidùv algoritmus verze $1.0$; snad 200 let pøed Eukleidem)
+
+\algo
+\algin èísla $a_0$, $b_0$.
+\:$a\leftarrow a_0, b\leftarrow b_0$.
+\:while $a\not=b$ :
+\::if $a > b \Rightarrow a \leftarrow a-b $
+\::\rlap{else}\phantom{if }$\phantom{a > b} \Rightarrow b \leftarrow b-a $
+\algout $a=\gcd(a_0,b_0)$.
+\endalgo
+
+\noindent{\sl Dùkaz správnosti: }
+
+\>{\bo 1) EA se zastaví}
+
+V ka¾dém prùchodu cyklem se buï $a$ zmen¹í o hodnotu $b$ anebo se $b$ zmen¹í o~hodnotu $a$. Proto souèet $a+b$ klesne poka¾dé alespoò o $1$. Tedy celkem
+se uskuteèní nanejvý¹ $a+b$ prùchodù cyklem a pak algoritmus skonèí.
+
+\>{\bo 2) EA vydá $\gcd(a_0, b_0)$}
+
+Pøi dùkazech správnosti algoritmu se èasto pou¾ívá {\I invariant.}
+To je nìco, co se v prùbìhu algoritmu nemìní (obvykle hodnota nìjakého výrazu nebo jeho vlastnost, napø. dìlitelnost dvìma).
+
+Doká¾eme, ¾e platí invariant: $\gcd(a, b) = \gcd(a_0, b_0)$.
+
+Algoritmus pøi svém bìhu postupnì mìní hodnoty $a$ a $b$.
+Tedy
+$(a_0, b_0) \rightarrow (a_1, b_1) \rightarrow
+\dots \rightarrow (a_k, b_k)$.
+A jakmile je splnìna rovnost $a_k = b_k$, tak skonèí.
+My chceme ukázat, ¾e platí
+$\gcd(a_0, b_0) = \gcd(a_1, b_1) = \dots = \gcd(a_k, b_k)$.
+
+K~tomu bude tøeba nìkolik pozorování:
+
+\>{\I Pozorování 1:} $\gcd(a, a) = a$ (pro $a>0$).
+
+Nahlédneme snadno. Nejvìt¹í spoleèný dìlitel jediného èísla je ono samo.
+
+\>{\I Pozorování 2:} $\gcd(a, b) = \gcd(b, a)$.
+
+Takté¾ jednoduché. Pøi urèování nejvìt¹ího spoleèného dìlitele
+nám na poøadí èísel nezále¾í.
+
+\>{\I Pozorování 3:} $\gcd(a, b) = \gcd(a \pm b, b)$.
+
+%%Rozmyslíme si, ¾e a¾ doká¾eme {\I Pozorování 3 }, máme vyhráno.
+
+Doká¾eme silnìj¹í vìc -- platnost ekvivalence
+$k\vert a, k\vert b \Leftrightarrow k\vert a, k\vert(a\pm b)$.
+
+\>$\Rightarrow$: Jeliko¾ $k\vert a, k\vert b$, dá se $a$ vyjádøit jako $k$-násobek nìjakého $a'$. Tedy $a = k\cdot a'$.
+Obdobnì $b = k\cdot b'$, pro nìjaká èísla $a',b'\in \N$.
+Tak¾e souèet $a\pm b$ si mù¾eme rozepsat jako $(a'\pm b')\cdot k$ z èeho¾ je
+ji¾ vidìt, ¾e $k\vert(a\pm b)$.
+
+\>$\Leftarrow$: Podobnì $k\vert (a\pm b), k\vert b \Leftrightarrow k\vert a$.
+
+Uvìdomíme si, ¾e tato tøi pozorování nám staèí k tomu, aby invariant
+platil. Algoritmus tedy opravdu vydá $\gcd(a_0,b_0)$.
+\qed
+
+\goodbreak
+
+\>{3) \bo Rychlost algoritmu}
+
+Algoritmus provede v~nehor¹ím pøípadì øádovì $a+b$ iterací. To je velmi pomalé,
+a proto vyvstává otázka, dá-li se nìjak zrychlit.
+
+V¹imnìme si, ¾e opakované odèítání jednoho èísla od~druhého lze nahradit výpoètem
+zbytku po~dìlení. Algoritmus tedy mu¾eme upravit následovnì:
+
+\s{Algoritmus:} (Eukleidùv alg. v$2.0$; neznámý Eukleidùv pokraèovatel, neznámo kdy)
+
+\algo
+\algin èísla $a_0$, $b_0$.
+\:$a\leftarrow a_0, b\leftarrow b_0$.
+\:while $b \neq 0$:
+\::$a \leftarrow a \mod b$
+\::$a \leftrightarrow b$
+\algout $a=\gcd(a_0,b_0)$.
+\endalgo
+
+\>{\sl Dùkaz správnosti: }
+
+Algoritmus urèitì vydá správný výsledek, proto¾e poèítá toté¾, co pøedchozí verze.
+Uká¾eme, ¾e poèet krokù, který provede verze 2.0, je øádovì $\log(\max(a,b))$.
+K~tomu se nám bude hodit následující:
+
+{\narrower
+\s{Lemma:} Za dva prùchody cyklem se buï hodnota $a$ nebo hodnota $b$
+zmen¹í alespoò dvakrát.
+
+\proof
+Rozebereme, co se stane po dvojím projití cyklem.
+Na zaèátku máme èísla $a_0$ a $b_0$. Bez újmy na obecnosti pøedpokládejme,
+¾e $a_0>b_0$. Po~prvním prùchodu získáme $a_1=b_0; b_1=a_0 \mod b_0$, co¾ je ménì
+ne¾ $b_0$. Po druhém prùchodu: $a_2=b_1; b_2=a_1 \mod b_1 = b_0 \mod b_1$. Chceme
+ukázat, ¾e kdy¾ je $b_1 < b_0$, tak $b_0 \mod b_1 \leq b_0/2$.
+
+\itemize\ibull
+\:Buï platí $b_1 \leq b_0/2$. Pak jistì $b_0 \mod b_1 < b_1 \leq b_0/2.$
+\:Nebo nastane druhá mo¾nost $b_1 > b_0/2$.
+Pak tedy $b_0 \mod b_1 = b_0-b_1$, co¾ je takté¾ maximálnì $b_0/2$.
+\qed
+\endlist
+
+}
+
+Nyní staèí lemma ¹ikovnì vyu¾ít:
+Dvojic prùchodù, které zmen¹ují hodnotu $a$, je nejvý¹e $\lceil \log_2
+a \rceil$. Obdobnì tìch sni¾ujících hodnotu $b$ je nejvý¹e $\lceil \log_2 b \rceil$.
+Celkový poèet prùchodù tedy je nejvý¹e $2(\lceil \log_2 a \rceil+\lceil
+\log_2 b\rceil)$.
+
+Je¹tì bychom chtìli dokázat, ¾e ná¹ horní odhad na~poèet prùchodù není pøíli¹
+velkorysý. Zkusme se podívat, co se stane, kdy¾ algoritmus necháme spoèítat
+$\gcd(F_n, F_{n-1})$, kde~$F_i$ je $i$-té Fibonacciho èíslo. Tehdy v¹echna
+modula budou ve~skuteènosti odèítání a algoritmus bude poèítat
+$\gcd(F_n, F_{n-1}) \rightarrow \gcd(F_{n-1}, F_{n-2}) \rightarrow
+\dots \rightarrow \gcd(1, 1)$.
+
+Díky exponenciální velikosti Fibonacciho èísel tento výpoèet bude
+opravdu trvat logaritmicky dlouho vzhledem k~velikosti vìt¹ího ze
+zadaných èísel.
+(Zároveò je to také pìkný dùkaz toho, ¾e Fibonacciho èísla jsou
+navzájem nesoudìlná.)
+
+\h{Výpoèetní model}
+Kdy¾ chceme studovat algoritmy, mìli bychom se zaèít zaobírat otázkou, co takový algoritmus vlastnì je.
+®ádná obecnì uznávaná definice algoritmu neexistuje, nadefinujeme si proto alespoò výpoèetní model
+a za~algoritmy budeme pova¾ovat programy v tomto modelu.
+
+\s{Základní vlastnosti výpoèetního modelu:}
+
+V první øadì potøebujeme modelu sdìlit, s èím má pracovat, a pak se
+dovìdìt, jak to dopadlo. Bez újmy na obecnosti mù¾eme pova¾ovat $n$
+èísel za {\sl vstup} a za {\sl výstup} $m$ èísel $($mù¾eme se omezit
+na èísla, nebo» cokoli jiného umíme do èísel zakódovat$)$.
+
+Dále by mìl výpoèetní model umìt provádìt {\sl elementární operace},
+co¾
+jsou nejen ty aritmetické $(+,-,*,/,{\rm mod}, \dots)$ a logické (negace,
+and, or, \dots), ale také øídící operace (skoky, podmínìné skoky, halt,
+\dots) a práci s pamìtí.
+
+\s{Definice:}
+\itemize\ibull
+\:{\I Èas bìhu algoritmu} $t(x)$ pro vstup~$x$ mìøíme jako poèet
+elementárních operací, které program provedl pøi zpracování vstupu
+$x$.
+\:{\I Prostor bìhu algoritmu} $s(x)$ je analogicky poèet pamì»ových
+bunìk spotøebovaných pøi výpoètu se vstupem~$x$.
+\:{\I Èasová slo¾itost} (v~nejhor¹ím pøípadì) je:
+$$T(n) := \max \{t(x) ; \hbox{$x$ je vstup délky $n$}\}.$$
+\:{\I Prostorová slo¾itost} (v~nejhor¹ím pøípadì) je:
+$$S(n) := \max \{s(x) ; \hbox{$x$ je vstup délky $n$}\}.$$
+\endlist
+
+\s{Díra v definici:} Co jsou to èísla?
+Kdy¾ mohou být libovolnì velká, pak se dá podvádìt pøi poèítání èasové i prostorové
+slo¾itosti. Napravit to mù¾eme buïto tak, ¾e vstup budeme mìøit v~bitech a operace
+s~$k$-bitovými èísly bude trvat øádovì $\log k$ krokù, nebo ¾e omezíme èísla
+na~polynomiálnì velká vzhledem k~velikosti vstupu. Pro normální algoritmy tyto dvì
+definice dávají èasové slo¾itosti li¹ící se v¾dy logaritmem velikosti vstupu, tak¾e
+pro srovnávání algoritmù nemusíme tento rozdíl uva¾ovat. Pro tuto pøedná¹ku si vybereme
+omezení èísel na~polynomiálnì velká.
+
+\h{Panorama slo¾itostí}
+
+Abychom mìli pøedstavu o~tom, jak pou¾itelné jsou algoritmy se kterou
+èasovou slo¾itostí, zkusme si pár slo¾itostí srovnat za pøedpokladu,
+¾e ná¹ poèítaè za sekundu provede $10^9$ operací:
+\def\[#1]{\,\hbox{#1}}
+$$\vbox{
+\halign{$#$ \hfil&\quad\hfil $#$&\quad\hfil $#$&\quad\hfil
+$#$&\quad\hfil $#$&\quad\hfil $#$&\quad\hfil $#$&\quad\hfil $#$\cr
+\hbox{funkce / $n=$\hskip-2em}& 10 & 20 & 30
+& 50 & 100 & 1\,000 & 1\,000\,000 \cr
+\noalign{\medskip\hrule\bigskip}
+\log_2 n & 3.3\[ns] & 4.3\[ns] & 4.9\[ns]
+& 5.6\[ns] & 6.6\[ns] & 10.0\[ns] & 19.9\[ns] \cr
+n & 10\[ns] & 20\[ns] & 30\[ns]
+& 50\[ns] & 100\[ns] & 1\[$\mu$s] & 1\[ms] \cr
+n\cdot\log_2 n\hskip-1em & 33\[ns] & 86\[ns]
+& 147\[ns] & 282\[ns] & 664\[ns] & 10\[$\mu$s] & 20\[ms]
+\cr
+n^2 & 100\[ns] & 400\[ns] & 900\[ns]
+& 2.5\[$\mu$s] & 100\[$\mu$s] & 1\[ms] & 1\,000\[s] \cr
+n^3 & 1\[$\mu$s] & 8\[$\mu$s] & 27\[$\mu$s]
+& 125\[$\mu$s] & 1\[ms] & 1\[s] & 10^9\[s] \cr
+2^n & 1\[$\mu$s] & 1\[ms] & 1\[s]
+& 10^6\[s] & 10^{21}\[s] & 10^{292}\[s] & \approx\infty \cr
+n! & 10^6\[s] & 10^{18}\[s] & 10^{32}\[s]
+& 10^{64}\[s] & 10^{158}\[s] & 10^{2567}\[s] & \approx\infty \cr
+}}$$
+\halign{#\hfil&\quad #\hfil\cr
+Pro pøedstavu: &$1\,000\[s]$ je asi tak ètvrt hodiny\cr
+ &$1\,000\,000\[s]$ je necelých 12 dní\cr
+ &$10^9\[s]$ je 31 let\cr
+ &$10^{18}\[s]$ je asi tak stáøí Vesmíru. \cr}
+
+Poèítaèe se nám ale zrychlují\foot{\uv{Ka¾dé dva roky se výkon poèítaèù
+zdvojnásobí} -- Gordon Moore (prý)} -- není tedy lep¹í chvíli poèkat
+ne¾ vymý¹let lep¹í algoritmus? :-) Podívejme se, jak velký vstup
+doká¾eme zpracovat za $10^6$ a $2\cdot 10^6$ operací:
+$$\vbox{\halign{\hfil$#$\quad&\hfil$#$\quad&\hfil$#$\qquad&#\hfil\cr
+\hbox{funkce} & 10^6\;\hbox{op.} & 2\cdot10^6\;\hbox{op.} & zmìna \cr
+\noalign{\medskip\hrule\bigskip}
+n &1\,000\,000 &2\,000\,000 &2-krát více \cr
+n^2 &1\,000 &1\,414 &$\sqrt{2}\approx 1.41$-krát
+více \cr
+n^3 &100 &126 &$\root3\of2\approx 1.26$-krát
+více \cr
+2^n &20 &21 &o 1 více \cr
+}}$$
+Na nìkterých èasových slo¾itostech se tedy exponenciální zrychlování
+poèítaèù projevuje jen minimálnì.
+
+Jak je vidìt na~následujícím grafu, konstanty v~èasové slo¾itosti
+vùbec nehrají pøíli¹ velkou roli, proto se vyplácí najít nejdøíve
+algoritmus, jeho¾ slo¾itost je asymptoticky dobrá, a teprve pak
+optimalizovat konstanty.
+
+\>\epsfxsize=1\hsize\epsfbox{../slides/funcs.eps}
+
+V~praxi také èasto pomù¾e zpracovávat velké vstupy asymptoticky rychlým
+algoritmem a pro malé vstupy pøepnout na~\uv{hloupý} algoritmus, který
+má malé multiplikativní konstanty.
+
+\bye
--- /dev/null
+P=1-euklid
+
+include ../Makerules
--- /dev/null
+\input ../lecnotes.tex
+
+\prednaska{10}{Nejkrat¹í cesty}{(zapsali L. Banáková, O. Hoferek, J. Bøeèka)}
+
+\s{Definice:} {\I Vzdálenost vrcholù} $d(u,v)$ je délka nejkrat¹í cesty $(u,v)$,
+pokud existuje, nebo $+\infty$, pokud taková cesta neexistuje.
+
+\s{Problém:} Je dán graf G a funkce $l: E(G) \rightarrow {\bb R}$ pøiøazující
+hranám jejich délky. Dále je dán startovací a cílový vrchol
+$s,c~\in~V(G)$. Chceme najít cestu $s=v_1, v_2, \dots, v_k=c$ takovou, aby délka cesty
+$$l(v_1, v_2) + l(v_2, v_3) + \dots + l(v_{k-1}, v_k)$$
+byla minimální. Takovéto cestì budeme øíkat {\I nejkrat¹í cesta} z~$s$~do~$c$.
+
+\noindent
+Bez újmy na obecnosti budeme pøedpokládat, ¾e graf $G$ je orientovaný, bez smyèek a
+násobných hran.
+
+\noindent
+Nejprve uvá¾íme pøípady, kdy jsou v¹echny hrany stejnì dlouhé. Tehdy mù¾eme pou¾ít
+algoritmus prohledávání do ¹íøky (BFS). K jeho implementaci je zapotøebí fronta $Q$ s
+metodami $Put(Q, v)$ (pøidání prvku $v$ na konec fronty) a $Get(Q)$ (odebrání
+prvku ze zaèátku fronty). Algoritmus vrátí pole \uv{znaèek} $z$: $z(v) = 1$ právì tehdy
+pokus je vrchol $v$ dostupný ze startovacího vrcholu $s$, jinak $z(v) = 0$.
+
+\s{Algoritmus:} (Prohledávání do ¹íøky (BFS))
+
+\algo
+\:$z(*) \leftarrow 0, z(s) \leftarrow 1$
+\:$Q \leftarrow \{s\}$
+\:while $Q \not= \emptyset :$
+\::$v \leftarrow Get(Q)$
+\::for $\forall w: (v,w)\in E$
+\:::if $z(w) = 0$ then
+\::::$z(w) \leftarrow 1$
+\::::$Put(Q, w)$
+\endalgo
+
+\s{Lemma È (èasové):} BFS dobìhne v~èase $\O(m+n)$.
+
+\proof
+Ka¾dý vrchol se do fronty dostane maximálnì jednou a ka¾dou hranu zpracujeme
+maximálnì jednou.
+\qed
+
+\s{Lemma D (dosa¾itelnost):} BFS oznaèí právì vrcholy dostupné ze $s$.
+
+\proof
+
+\noindent
+\uv{$\Rightarrow$}: Indukcí podle bìhu algoritmu.
+
+\noindent
+\uv{$\Leftarrow$}: Doká¾eme sporem. Uva¾me, ¾e existuje vrchol dostupný, ale algoritmem
+neoznaèený. Vezmìme takovýto \uv{¹patný} vrchol $v$, který je $s$ nejblí¾e. Uva¾me
+nejkrat¹í cestu $(s,v)$: $s,\dots,u,v$. Pøedchozí vrchol na této cestì $u$
+musí být \uv{dobrý}. Vrchol $u$ se dostane do fronty, pak je z~ní vybrán a tím
+se zpracuje i vrchol $v$ $\Rightarrow$ spor.
+
+Pøedchozím algoritmem jsme pouze zjistili, které vrcholy jsou z~$s$ dosa¾itelné.
+Nyní si tento algoritmus modifikujeme tak, abychom na¹li $d(s,v)$ pro v¹echna
+$v~\in~V(G)$.
+
+\s{Roz¹íøený algoritmus:} (Prohledávání do ¹íøky (BFS))
+
+\algo
+\:$z(*) \leftarrow 0, z(s) \leftarrow 1$
+\:$D(*) \leftarrow +\infty, D(s) \leftarrow 0$
+\:$Q \leftarrow \{s\}$
+\:while $Q \not= \emptyset :$
+\::$v \leftarrow Get(Q)$
+\::for $\forall w: (v,w)\in E$
+\:::if $z(w) = 0$ then
+\::::$z(w) \leftarrow 1$
+\::::$D(w) \leftarrow D(v) + 1$
+\::::$Put(Q, w)$
+\endalgo
+
+\noindent
+Prùbìh tohoto algoritmu si mù¾eme ukázat na následujícím obrázku (\uv{praseèí graf}):
+
+\figure{praseci-graf.eps}{Praseèí graf}{75mm}
+
+\s{Definice:} {\I Vrstva} $ L_i\subseteq V $ : $ L_0 = \{s\} $ , $ L_{i+1} = $
+$ \{ w$ : $ w $ oznaèen pøi procházení vrcholu $ v \in L_i $$ \} $
+
+\s{Lemma V (vzdálenosti):} Na konci BFS je $\forall v: D(v) = d(s,v)$.
+
+\proof
+
+\noindent
+a) pro nedosa¾itelné vrcholy: OK (algoritmus je neoznaèí $\Rightarrow D(v) = \infty$)
+
+\noindent
+b) pro dosa¾itelné vrcholy: Víme, ¾e pro vrchol $v \in L_i$ platí $D(v)=i$,
+staèí tedy zjistit, zda $\forall i: \forall v \in L_i$ je $d(s,v)=i$. To ovìøíme matematickou indukcí
+podle~$i$:
+
+pro $i=0$: evidentní,
+
+pro $i>0$: Pøi procházení vrstvy $L_{i-1}$ musí být vrchol $v$ takový, ¾e $d(s,v)=i$, oznaèen
+a zároveò nemohl být oznaèen døíve a tudí¾ patøí do $L_i$.
+\qed
+
+\noindent
+V¹imnìme si, ¾e v~grafu existují pouze tøi typy hran vzhledem k na¹emu algoritmu:
+\itemize\ibull
+\:dopøedu o jednu vrstvu
+\:dozadu o jednu nebo víc vrstev
+\:ve vrstvì
+\endlist
+
+Abychom na¹li i samotné cesty, nejen jejich délky, modifikujeme svùj algoritmus tak,
+¾e si pro ka¾dý vrchol $v$, který oznaèíme, budeme pamatovat jeho pøedchùdce $p(v)$:
+
+\algo
+\:$z(*) \leftarrow 0, z(s) \leftarrow 1$
+\:$D(*) \leftarrow +\infty, D(s) \leftarrow 0$
+\:$p(*) \leftarrow$ ?
+\:$Q \leftarrow \{s\}$
+\:while $Q \not= \emptyset :$
+\::$v \leftarrow Get(Q)$
+\::for $\forall w: (v,w)\in E$
+\:::if $z(w) = 0$ then
+\::::$z(w) \leftarrow 1$
+\::::$D(w) \leftarrow D(v) + 1$
+\::::$p(w) \leftarrow v$
+\::::$Put(Q, w)$
+\endalgo
+
+\s{Pozorování:} Pokud $p(v) \not=$ ? $\Rightarrow D(v) = D(p(v)) + 1$.
+
+\s{Lemma S (stromové):} Graf $(W,F)$, kde $W$ = dosa¾itelná èást $V$, $F = \{(v,p(v))\}$
+je strom orientovaný do~koøene $s$. Navíc ukazuje nejkrat¹í cestu pro ka¾dý dosa¾itelný
+vrchol (pro ka¾dý vrchol v tomto grafu, existuje jediná orientovaná cesta do koøene
+$s$, která je zároveò nejkrat¹í cesta z $s$ do tohoto vrcholu).
+
+\noindent
+Nyní uva¾me, ¾e délky hran $l$ nejsou v¹echny stejnì dlouhé:
+
+Uká¾eme si trik pro ohodnocené hrany: hranu délky $l$
+podrozdìlíme na $l$ hran délky 1 a pak spustíme BFS. Algoritmus pobì¾í s èasovou
+slo¾itostí $O((m+n)l_{max})$. To není zrovna moc výhodné, a proto si uká¾eme tzv. {\I Budíkovou taktiku}:
+
+Místo toho, abychom ka¾dou hranu podrozdìlovali, si budeme pro ka¾dý vrchol pamatovat \uv{èas budíku} $b$.
+Èasem $b(v)$ budíku vrcholu~$v$ rozumíme èas, kdy se do tohoto vrcholu vlna dostane, kdy¾ se bude ¹íøit po
+dosud známých hranách. Na zaèátku nastavíme $\forall v \in V: b(v) = l(s,v)$ a pokud pøi bìhu
+algoritmu objevíme hranu, která nám umo¾ní dostat se do $v$ rychleji, tak hodnotu $b(v)$ zmìníme.
+Algoritmus pak bude pracovat tak, ¾e bude spát, dokud nezazvoní nìjaký budík (co¾ znamená, ¾e
+jsme dorazili do nìjakého vrcholu).
+
+Tato taktika nás pøivede k Dijkstrovu algoritmu, kde se ze dvoustavového $z(v)$ stane tøístavové
+$z(v)) \in \{{\bf{N}}, {\bf{V}}, {\bf{H}}\}$, kde {\bf{N}} znamená nevidìn, {\bf{V}} vidìn a {\bf{H}} hotov. Podle hodnoty $z(v)$ se
+pak $D(v)$ rovná $+\infty$ pro $z(v) = {\bf{N}}$, èasu budíku pro $z(v) = {\bf{V}}$ nebo vzdálenosti $d(s,v)$ pro $z(v) = {\bf{H}}$.
+
+\s{Algoritmus:} (Dijkstrùv algoritmus)
+
+\algo
+\:$z(*) \leftarrow {\bf{N}}, z(s) \leftarrow {\bf{V}}$
+\:$D(*) \leftarrow +\infty , D(s) \leftarrow 0$
+\:$p(*) \leftarrow $?
+\:while $\exists v : z(v) = {\bf{V}} :$
+\::$v \leftarrow $ vrchol, pro nìj¾ $z(v)={\bf{V}}$ a zároveò $D(v)$ je minimální
+\::$z(v) \leftarrow {\bf{H}}$
+\::for $\forall w : (v,w) \in E$:
+\:::if $D(w)>D(v) + l(v,w)$ then
+\::::$D(w) \leftarrow D(v) + l(v,w)$
+\::::$z(w) \leftarrow {\bf{V}}$
+\::::$p(w) \leftarrow v$
+\endalgo
+
+\s{Vìta:} Pokud $\forall v,w: l(v,w) \in \bb{Z}^+ \cup \{\infty\}$, pak Dijkstrùv algoritmus
+najde nejkrat¹í cesty v èase $\O(n^2)$.
+
+\proof
+V prùbìhu algoritmu se maximálnì $n$-krát hledá vrchol $v$ s minimálním $D(v)$,
+maximálnì $m$-krát se pøenastavuje $D(v)$. Pokud bude struktura $D$ ulo¾ena jako
+pole (hledání minima v $\O(n)$, zmìna hodnoty $\O(1)$), bude celková
+èasová slo¾itost Dijkstrova algoritmu $\O(n^2 + m) = \O(n^2)$. Fakt, ¾e se algoritmus
+zastaví je evidentní, proto¾e se stav ¾ádného vrcholu nevrací z {\bf{H}} do {\bf{V}}.
+
+Úplný dùkaz správnosti a èasové slo¾itosti i pro hrany ohodnocené reálnými èísly
+bude na pøí¹tí pøedná¹ce.
+\qed
+
+Nahlédnìme nyní, zda by vedlo k~vylep¹ení èasové slo¾itosti, kdybychom pøi~implementaci
+Dijkstrova algoritmu pou¾ili jako datovou strukturu pro ukládání vidìných vrcholù rùzné typy hald.
+Provádìli bychom s~nimi operace \<DeleteMin> (maximálnì $n$-krát) a \<Decrease> (maximálnì $m$-krát). Zmínìné
+operace mají pro rùzné typy hald následující èasové slo¾itosti:
+$$\vbox{\halign{\strut # \quad & # \quad & # \quad & #\cr
+& \it binární halda & \it $k$-regulární halda & \it Fibonacciho halda\cr
+\noalign{\smallskip}
+\<DeleteMin> & $\O(\log{n})$ & $\O(k\log_k{n})$ & $\O(\log{n})$ \cr
+\<Decrease> & $\O(\log{n})$ & $\O(\log_k{n})$ & $\O(1)$ \cr
+{\I celková slo¾itost} & $\O((m+n)\log{n})$ & $\O(m\log_{m/n}{n})$ & $\O(n\log{n}+m)$ \cr
+}}$$
+
+Pou¾ijeme-li klasickou haldu, dostaneme celkovou èasovou slo¾itost $\O((m+n)\log{n})$, co¾ znamená,
+¾e jsme si pomohli v pøípadech, kdy je graf \uv{øídký}. U $k$-regulární haldy se celková èasová slo¾itost
+rovná $\O(nk\log_k{n}+m\log_k{n})$, ideálnì pro $k=m/n$: $\O(m\log_{m/n}{n})$. Prozkoumejme,
+jak se nám v tomto pøípadì bude mìnit èasová slo¾itost pro rùznì \uv{husté} grafy:
+\itemize\ibull
+\:$m \approx n \rightarrow \O((m+n)\log{n})$
+\:$m \approx n^2 \rightarrow \O(m)$
+\:$m \approx n^{1+\epsilon} \rightarrow \O(m) \dots (k=m/n= n^\epsilon)$
+\endlist
+\noindent
+Pøi pou¾ití Fibonacciho haldy je pak celková èasová slo¾itost rovna $\O(n\log{n}+m)$.
+
+\medskip
+
+Nyní si pøedvedeme trochu jiný algoritmus na prohledávání grafu. Mìjme matici sousednosti $A$
+ke grafu $G = (V, E)$: $ A_{ij} = 1 \Leftrightarrow (v_i, v_j) \in E $, jinak $A_{ij} = 0$. Potom
+$A^2_{ij} = \sum_{k=1}^n A_{ik}A_{kj} $. Tento souèet je nenulový právì tehdy, kdy¾
+vrcholy $i$,$k$,$j$ tvoøí sled délky 2. Jinými slovy $A^2_{ij}$ se rovná poètu sledù
+délky 2 v $G$ z $i$ do $j$. Obdobnì $A^k_{ij}$ = poèet sledù délky $k$. Pokud
+k matici $A$ pøièteme jednotkovou matici ${\bb E}_n$ (èím¾ si pøimyslíme smyèky),
+dostaneme po umocnìní:
+
+$ (A+{\bb E}_n)^k_{ij} \neq 0 \Leftrightarrow \exists$ $i,j$-sled délky maximálnì $k$.
+
+Výstupem algoritmu tedy bude matice $(A + {\bb E}_n)^n$. Mù¾eme si také v¹imnout, ¾e
+pokud matici $A + {\bb E}_n$ umocníme na èíslo vìt¹í ne¾ $n$, tak výsledek bude poøád
+správný, jen bude výpoèet trvat déle. Tak¾e algoritmus mù¾e pracovat
+v iteracích umocòováním matice na druhou: $M_0 = A + {\bb E}_n$, $M_1 = M^2_0$, $M_{i+1} = M^2_i$.
+Výstupní matice bude tudí¾:
+
+$ M_{\lceil \log{n} \rceil (ij)} \neq 0 \Leftrightarrow \exists$ cesta z $i$ do $j$
+
+Na výpoèet dosa¾itelnosti vrcholù v grafu tedy staèí $\O(\log{n})$ násobení matic,
+a pokud na toto pou¾ijeme Strassenùv algoritmus, dostaneme
+ve výsledku algoritmus se slo¾itostí $\O(n^{\log_2{7}}\log{n})$.
+
+\bye
--- /dev/null
+P=10-cesty
+
+include ../Makerules
--- /dev/null
+\input ../lecnotes.tex
+
+\prednaska{11}{Nekrat¹í cesty podruhé}{(zapsali Du¹an Renát a Radek Tupec)}
+
+Na~této pøedná¹ce budeme studovat problém hledání nejkrat¹ích cest
+v~grafech ohodnocených reálnými èísly.
+
+\s{Situace:} Máme orientovany graf~$G$ a funkce $l : E(G) \rightarrow {\bb R}$
+pøiøazující hranám jejich ohodnocení (délky). Pro vrcholy $u,v\in V(G)$ budeme
+chtít spoèítat jejich vzdálenost $d(u,v)$, co¾ bude délka nejkrat¹í cesty z~$u$
+do~$v$ nebo $\infty$, pokud ¾ádná cesta neexistuje.
+
+Aby se vzdálenosti chovaly \uv{rozumnì} (tj. jako metrika), budeme chtít, aby platily
+následující vlastnosti:
+
+\itemize\ibull
+\:$d(u, u) = 0$,
+\:$d(u, v) \leq d(u, w) + d(w, v)$ (trojúhelníková nerovnost).
+\endlist
+
+To nemusí obecnì platit (kazí nám to záporné cykly), proto budeme studovat
+pouze grafy, v~nich¾ záporné cykly neexistují. Pak u¾ budou obì vlastnosti
+splnìny, jak plyne napøíklad z~následujícího lemmatu:
+
+\s{Lemma:}
+V~grafu bez záporných cyklù existuje ke~ka¾dému nejkrat¹ímu sledu z~$u$ do~$v$ stejnì dlouhá $uv$-cesta.
+
+\proof
+Máme-li nejkrat¹í $uv$-sled, který není cestou, opakuje se v~nìm nìjaký vrchol $w \in V(G)$.
+Délka cyklu $l(c) \geq 0 \Rightarrow l(u\dots v\dots w) \leq l(pùvodní sled)$. Tento postup mù¾eme opakovat a po koneèném poètu krokù dostaneme cestu, tedy platí trojúhelníková nerovnost.
+\qed
+
+Opakování: Dijkstrùv algoritmus
+\itemize\ibull
+\:$D(v)$ \dots doèasná vzdálenost z $s$ do $v$
+\:Znaèky $Z(v)$ \itemize\ibull
+\::Nevidìn
+\::Vidìn
+\::Hotov
+\endlist
+\endlist
+
+%druha strana zapisku
+\algo
+\:$D(*) \leftarrow +\infty , D(s) \leftarrow 0$
+\:$Z(*) \leftarrow Neviden, Z(s) \leftarrow V$
+\:while $\exists v : Z(v) = V$
+\::vybereme $v : Z(v) = V, D(v) = min$
+\::$Z(v) = H$
+\::for $\forall w : (v, w) \in E(G)$
+\:::$D(w) \leftarrow min( D(w), D(w) + l(v, w))$
+\::if $Z(w) = N \Rightarrow Z(w) \leftarrow V$
+\endalgo
+
+\s{Vìta:} Pokud $G$ je nezápornì ohodnocený graf, pak se Dijkstrùv algoritmus zastaví a vydá $\forall v : D(v) = d(s, v)$ (tedy správné hodnoty).
+
+\proof Následující posloupností lemmat.\qed
+
+\s{Lemma 1:} Pokud $v_0,\dots,v_k$ je nejkrat¹í $v_0v_k$-cesta, pak $v_0,\dots,v_{k-1}$ je nejkrat¹í $v_0v_{k-1}$-cesta.
+
+\proof
+Pokud by tomu tak nebylo, mù¾eme $v_0,\dots,v_{k-1}$ vymìnit za~krat¹í cestu, a~tím
+získat krat¹í $v_0v_k$-cestu.
+\qed
+
+\s{Lemma 2:} Algoritmus se zastaví po $\leq n$ prùchodech cyklem.
+
+\proof
+Zøejmé z~toho, ¾e ka¾dý vrchol uzavøeme nejvý¹e jednou.
+\qed
+
+\s{Lemma 3:} Po zastavení jsou hotovy právì vrcholy dosa¾itelné z $s$.
+
+\proof
+Viz minulá pøedná¹ka.
+\qed
+
+\s{Lemma 4:} $D(v)$ uzavíraných vrcholù tvoøí neklesající posloupnost.
+
+\proof
+V okam¾iku, kdy uzavíráme $v$ platí $\forall w \notin H : D(w) \geq D(v)$, pøípadnì pøepoèítáme $D(w)$ na $D(v) - l(v, w) \geq D(v)$.
+\qed
+
+\s{Lemma 5:} Pokud $v \in H$, pak $D(v)$ se u¾ nezmìní.
+
+\proof
+Indukcí podle bìhu algoritmu.
+\qed
+
+\s{Lemma 6:} Pro $\forall v D(v)$ je délka nejkrat¹í $sv$-cesty, její¾ vnitøní vrcholy le¾í v¹echny v $H$.
+
+\proof
+\itemize\ibull
+\:po 1. prùchodu OK
+\:uzavíráme-li dal¹í vrchol $v$:
+\itemize\relax
+\:a) $D(w)$ pro $w \in H$
+Podle Lemma 5 se $D(w)$ nemìní. Musíme nahlédnout, ¾e se opravdu zmìnit nemá.
+\:b) $D(w)$ pro $w \notin H$
+$D(w) = min{D(v) + l(v, w)}$
+\endlist
+\endlist
+\qed
+
+Existuje pomalej¹í algoritmus pro grafy se zápornými hranami bez záporných cyklù.
+\s{Bellman-Fordùv algoritmus}
+\algo
+\:$D(*) \leftarrow \infty, D(s) \leftarrow 0$
+\:Opakuji
+\::Pro $\forall v \in V$
+\:::$prozkoumej(v)$
+\:::(koukne se, jestli nejde vylep¹it cestu¨
+\:::do sousedù $v$)
+\:dokud se nìjaké $D(\dots)$ mìní
+\endalgo
+
+\s{Lemma 1:} Pokud $D(v) < \infty$, pak existuje sled z $s$ do $v$ délky $D(v)$.
+(speciálnì z toho plyne $\forall v D(v) \geq d(s,v)$)
+
+\s{Lemma 2:} $\forall v D(v)$ nikdy neroste.
+\s{Lemma 3:} Po $k$ fázích je $\forall v D(v) \leq$ délka nejkrat¹ího $sv$-sledu o $\leq k$ hranách.
+%ctvrta stranka
+\proof
+Indukcí\dots pro $k \geq 0$ OK
+Indukèní krok: Dobìhla $k-1$ fáze, pou¹tíme $k$-tou
+
+\qed
+\s{Vìta:} Pro graf bez záporných cyklù se Bellman-Fordùv algoritmus zastaví po nejvý¹e $m$ fázích a vydá $D(v) = d(s, v)$ pro v¹echna $v$. (zøejmé z lemmat)
+
+\s{Lemma 4:} Pokud v grafu existuje záporný cyklus dosa¾itelný z $s$, algoritmus se nezastaví. (Zajímavý test na to, zda graf obsahuje záporný cyklus.)
+
+Èasová slo¾itost Bellman-Fordova algoritmu : $O(n*m)$
+
+\s{Floyd-Warshallùv algoritmus}
+$G$ je graf se záporným ohodnocením hran, bez záporných cyklù
+$D_{i,j}^k$ = délka nejkrat¹í cesty z $v_i$ do $v_j$ pøes $v_1 \dots v_k$
+%pozorovani
+$D_{ij}^0 = l(v_i, v_j)$, $D_{ii}^0 = 0$
+$D_{ij}^n = d(v_i, v_j)$ - skuteèná vzdálenost v grafu
+\algo
+\:$D_{i,j} \rightarrow l(v_i, v_j), D_{i,i} \rightarrow 0 \forall i, j$
+\:for $k = 1$ to n
+\::for $i = 1$ to n
+\:::for $j = 1$ to n
+\::::$D_{i,j} = min(D_{i,j}, D_{i,k} + D_{k,j})$
+\endalgo
+
+\s{Vìta:} Floyd-Warshallùv algoritmus spoèítá $D_{i,j} = d(v_i, v_j)$ v èase $O(n^3)$.
+
+\>Shrnutí:
+
+\s{$s \rightarrow *$}
+Dijkstrùv algoritmus $O(n^2)$
+\dots s haldou $O((n+m)\log m)$
+\dots s regulární haldou $O(n + m*log n)$
+\dots s Fibbonaciho haldou $O(n*log n + m)$
+Bellman-Ford $O(n*m)$
+
+\s{$* \rightarrow *$}
+Floyd-Warshall $O(n^3)$
+
+\bye
--- /dev/null
+P=11-dijkstra
+
+include ../Makerules
--- /dev/null
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--- /dev/null
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+%%Creator: GIMP PostScript file plugin V 1,17 by Peter Kirchgessner
+%%Title: 05.eps
+%%CreationDate: Sun May 20 18:08:37 2007
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
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+%%EndComments
+%%BeginProlog
+% Use own dictionary to avoid conflicts
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+%%EndProlog
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+{currentfile /ASCII85Decode filter /RunLengthDecode filter rstr readstring pop}
+{currentfile /ASCII85Decode filter /RunLengthDecode filter gstr readstring pop}
+{currentfile /ASCII85Decode filter /RunLengthDecode filter bstr readstring pop}
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+showpage
+%%Trailer
+end
+%%EOF
--- /dev/null
+\input lecnotes.tex
+
+\prednaska{12}{Problém minimální kostry}{(zapsali K. Ka¹èák a M. Vachna)}
+
+\s{Zadání úlohy:} Pro neorientovaný graf $G$ s~ohodnocením hran {\I váhami} $w: E(G) \rightarrow \bb R$,
+chceme najít kostru $T$ s minimálním ohodnocením $w(T)=\sum_{e\in E(T)} w(e)$.
+
+\s{Navíc pøedpokládáme:} (bez újmy na~obecnosti)
+\itemize\ibull
+\:Graf $G$ je souvislý (jinak ho nejprve rozlo¾íme na komponenty).
+\:$\forall e,f \in E(G$) : $e\neq f \Rightarrow w(e)\neq w(f)$.
+\endlist
+
+Nyní si uká¾eme tøi algoritmy pro hledání minimální kostry, konkrétnì se jedná
+o~Jarníkùv, Borùvkùv a Kruskalùv algoritmus.
+
+\h{Jarníkùv algoritmus}
+
+\s{Algoritmus:}
+
+\algo
+\algin Graf~$G$ s~ohodnocením~$w$.
+\:Zvolíme libovolný vrchol $v_0\in V(G)$.
+\:$T\leftarrow(\left\{v_0\right\},\emptyset)$ (zatím jednovrcholový strom)
+\:Dokud $\vert V(T) \vert \neq n$:
+\::Vybereme hranu $uv\in E(G) : u\in V(T), v\notin V(T)$ tak, aby $w(uv)$ byla minimálni.
+\::$T\leftarrow T+uv$.
+\algout Minimální kostra~$T$.
+\endalgo
+
+\s{Vìta:} Jarníkùv algoritmus se zastaví po maximálnì $n$ iteracích a vydá minimální kostru grafu $G$.
+
+\proof
+Pøi ka¾dé iteraci algoritmus pøidá jeden vrchol do~$T$, a~proto se po~maximálnì $n$ iteracích zastaví.
+Vydaný graf je strom, proto¾e se stále pøidává list k ji¾ existujícímu stromu, a~jeliko¾ má $n$~vrcholù,
+je to kostra. Zbývá nám u¾ jen dokázat, ¾e nalezená kostra je minimální. K~tomu pomu¾e následující lemma:
+
+{\narrower
+
+\s{Definice:} {\I Øez} v~grafu $G=(V,E)$ je mno¾ina hran $F\subseteq E$ taková, ¾e $\exists U\subset V$ :
+$F=\left\{uv\in E:u\in U, v\notin U \right\}$.
+
+\s{Lemma:} Pokud $G$ je graf, $w$ jeho prosté ohodnocení, $F$ je øez v grafu $G$ a $f$ je nejlehèí hrana v øezu
+$F$, pak pro ka¾dou minimální kostru $T$ grafu $G$ je $f\in E(T)$.
+
+\proof
+Buï $T$ kostra a $f=uv\notin E(T)$. Pak existuje cesta $P\subseteq T$ spojující $u$ a $v$.
+Cesta musí øez alespoò jednou pøekroèit. Proto existuje $e\in P \cap F$ a navíc víme, ¾e $w(e) > w(f)$. Uva¾me $T'=T-e+f$.
+Tento graf je rovnì¾ kostra grafu $G$, proto¾e odebraním hrany $e$ se graf rozpadne na dvì komponenty a pøidáním
+hrany $f$ se tyto komponenty opìt spojí. Navíc $w(T')=w(T)-w(e)+w(f)<w(T)$.
+\qed
+
+}
+
+\>V~dùkazu korektnosti Jarníkova algoritmu toto lemma vyu¾ijeme tak, ¾e si v¹imneme, ¾e hrany mezi
+vrcholy stromu~$T$ a zbytkem grafu tvoøí øez a algoritmus nejlehèí hranu tohoto øezu pøidá
+do~$T$. Podle lemmatu tedy v¹echny hrany~$T$ musí být souèástí ka¾dé minimální kostry a jeliko¾~$T$ je strom,
+musí být minimální kostrou.
+
+\qed
+
+
+\s{Dùsledky:} Graf $G$ s prostým ohodnocením má pravì jednu minimální kostru. Minimální kostra je
+jednoznaènì urèená lineárním uspoøádáním hran.
+
+\s{Implementace:}
+\itemize\ibull
+\:Pøímoèará: pamatujeme si, které vrcholy a hrany jsou v kostøe $T$ a které ne. Èasová slo¾itost je $\O(nm)$.
+\:Chytøej¹í: Pro $v\notin V(T)$ si pamatujeme $D(v)=\min\left\{w(uv):u\in T\right\}$. Pøi ka¾dém
+prùchodu hlavním cyklem pak procházíme v¹echna~$D(v)$ (to v¾dy trvá $\O(n)$) a pøi pøidání vrcholu do~$T$ kontrolujeme
+okolní~$D(w)$ pro $vw\in E$ a pøípadnì je sni¾ujeme (za~ka¾dou hranu~$\O(1)$). Èasovou slo¾itost tím celkovì
+zlep¹íme na~$\O(n^2+m)=\O(n^2)$.
+\:Také se dá pou¾ít halda pro uchovávání hran nebo hodnot~$D(v)$.
+\endlist
+
+\h{Borùvkùv algoritmus}
+
+\s{Algoritmus:}
+
+\algo
+\algin Graf~$G$ s~ohodnocením~$w$.
+\:$F\leftarrow(V(G),\emptyset)$
+\:Dokud $F$ má alespoò dvì komponenty:
+\::Pro ka¾dou komponentu $T_i$ grafu $F$ vybereme nejlehèí incidentní hranu $t_i$.
+\::V¹echny hrany $t_i$ pøidáme do $F$.
+\algout Minimální kostra~$F$.
+\endalgo
+
+\s{Vìta:} Borùvkùv algoritmus se zastaví po $\left\lceil \log_2 n\right\rceil$ iteracích a vydá minimální kostru grafu $G$.
+
+\proof
+V¹imnìme si nejprve, ¾e po~$k$ iteracích mají v¹echny komponenty grafu~$F$ minimálnì $2^k$ vrcholù
+(na~poèátku jsou v¹echny komponenty jednovrcholové, v~ka¾dé dal¹í iteraci se komponenty sluèují do~vìt¹ích,
+ka¾dá s~alespoò jednou sousední, tak¾e se velikosti komponent minimálnì zdvojnásobí).
+Proto nejpozdìji po~$\left\lceil \log_2 n\right\rceil$ iteracích u¾~velikost komponenty dosáhne poètu v¹ech vrcholù a algoritmus
+se zastaví.
+
+Hrany mezi ka¾dou komponentou a~zbytkem grafu tvoøí øez, tak¾e podle pøedchozího lemmatu v¹echny
+hrany pøidané do~$F$ musí být souèástí (jednoznaènì urèené) minimální kostry. Graf $F\subseteq G$
+je tedy v¾dy les a a¾ se algoritmus zastaví, bude roven minimální kostøe.
+\qed
+
+\s{Implementace:}
+\itemize\ibull
+\:Inicializace pøímoèará.
+\:Pomocí DFS rozlo¾íme les na komponenty. Pro ka¾dý vrchol si pamatujeme èíslo komponenty.
+\:Pro ka¾dou hranu zjistíme, do které komponenty patøí, a pro ka¾dou komponentu
+si uchováme nejlehèí hranu.
+\endlist
+
+\>Takto doká¾eme ka¾dou iteraci provést v~èase $\O(m)$ a celý algoritmus dobìhne v~$\O(m\log n)$.
+
+\h{Kruskalùv neboli hladový algoritmus}
+
+\s{Algoritmus:}
+
+\algo
+\algin Graf~$G$ s~ohodnocením~$w$.
+\:Setøídíme v¹echny hrany z $E(G)$ tak, aby: $w(e_1)<...<w(e_m)$.
+\:$F\leftarrow (V(G),\emptyset)$.
+\:Pro $i=1$ do $m$:
+\::Pokud $F+e_i$ je acyklický, provedeme $F\leftarrow F+e_i$.
+\algout Minimální kostra~$F$.
+\endalgo
+
+\s{Vìta:} Kruskalùv algoritmus se zastaví po~$m$ iteracích a vydá minimální kostru.
+
+\proof
+Ka¾dá iterace algoritmu zpracovává jednu hranu, tak¾e iterací je~$m$. Indukcí doká¾eme,
+¾e~$F$ je v¾dy podgrafem minimální kostry: prázdné poèáteèní~$F$ je podgrafem èehokoliv,
+ka¾dá hrana, kterou pak pøidáme, je minimální v~øezu oddìlujícím nìjakou
+komponentu~$F$ od~zbytku grafu (ostatní hrany tohoto øezu je¹tì nebyly
+zpracovány, a~tudí¾ jsou tì¾¹í). Naopak ¾ádná hrana, kterou jsme se rozhodli
+do~$F$ nepøidat, nemù¾e být souèástí minimální kostry, jeliko¾ s~hranami,
+o~kterých ji¾ víme, ¾e v~minimální kostøe le¾í, tvoøí kru¾nici. \qed
+
+\s{Implementace:}
+\itemize\ibull
+\:Setøídìní v èase $\O(m\log m)=\O(m\log n)$.
+\:Pak potøebujeme udr¾ovat komponenty souvislosti grafu~$F$, abychom umìli rychle
+ urèit, jestli právì zpracovávaná hrana vytvoøí kru¾nici. Potøebujeme tedy strukturu
+ pro udr¾ování komponent souvislosti, které se $m$-krát zeptáme, zda dva vrcholy
+ le¾í v~té¾e komponentì (tomu budeme øíkat operace \<Find>), a~$(n-1)$-krát spojíme
+ dvì komponenty do jedné (\<Union>).
+\endlist
+
+\>Kruskalùv algoritmus tedy pobì¾í v~èase $\O(m\log n + mT_f + nT_u)$, kde~$T_u$ je
+èas na~provedení jedné operace \<Union> a $T_f$ na~operaci \<Find>.
+
+\s{Jednoduchá struktura pro komponenty:}
+Budeme pamatovat v~poli èísla komponent, ve~kterých le¾í jednotlivé vrcholy. \<Find> zvládneme v~èase $\O(1)$,
+ale \<Union> bude stát $\O(n)$. Celý algoritmus pak pobì¾í v~èase $\O(m\log n+ m + n^2) = \O(m\log n+n^2)$.
+
+\s{Chytøej¹í struktura:} Ka¾dou komponentou si ulo¾íme jako strom orientovaný smìrem ke koøeni
+-- ka¾dý vrchol si pamatuje svého otce, navíc ka¾dý koøen si pamatuje velikost komponenty.
+Operace \<Find> vystoupá z~obou vrcholù ke~koøeni a koøeny porovná. \<Union> rovnì¾ najde
+koøeny a pøipojí koøen men¹í komponenty pod koøen té vìt¹í. Obojí zvládneme v~èase lineárním
+v~hloubce stromu a jak si uká¾eme, tato hloubka je v¾dy nejvý¹e logaritmická, a proto celý Kruskalùv
+algoritmus pobì¾í v~èase $\O(m\log n + m\log n + n\log n) = \O(m\log n)$.\foot{%
+Drobnou úpravou bychom mohli dosáhnout daleko efektivnìj¹í struktury, ale tu bychom neupotøebili,
+jeliko¾ by nás stejnì brzdilo tøídìní, a analýza slo¾itosti by byla \dots\ inu, slo¾itìj¹í.}
+
+\s{Lemma:} Strom hloubky $h$ má alespoò $2^h$ prvkù.
+
+\proof
+Indukcí: Pokud \<Union> spojí strom s~hloubkou $h$ s jiným s~hloubkou men¹í ne¾ $h$, pak hloubka výsledného
+stromu zùstává~$h$. Pokud spojuje dva stromy stejné hloubky~$h$, pak má výsledný strom hloubku $h+1$.
+Z~indukèního pøedpokladu víme, ¾e strom hloubky $h$ má minimálnì $2^h$ vrcholù,
+a~tedy výsledný strom hloubky $h+1$ má alespoò $2^{h+1}$ vrcholù. \qed
+
+\h{Èerveno-èerné stromy}
+
+Ve~zbytku pøedná¹ky se vrátíme k~vyhledávacím stromùm a uká¾eme si je¹tì jeden zpùsob,
+jak udr¾et logaritmickou hloubku stromu.
+
+\s{Definice:} {\I Èerveno-èerný strom} je binární vyhledávací strom, jeho¾ vrcholy jsou
+obarvené (nìkteré èervenì, zbylé èernì), a ke~kterému byly pøidány externí vrcholy v¹ude
+tam, kde pùvodnímu vrcholu chybí nìkterý ze~synù. Listy è-è stromu jsou tedy v¾dy externí.
+Navíc platí následující axiomy:
+
+\s{Axiomy:}
+\itemize\ibull
+\:Koøen a externí vrcholy jsou èerné.
+\:Otec èerveného vrcholu je èerný.
+\:Na v¹ech cestách z~koøene do~externího vrcholu je stejný poèet èerných vrcholù.
+\endlist
+
+\s{Vìta:} Èerveno-èerný strom o~$n$ vrcholech má hloubku $\O(\log n)$.
+
+\proof
+Zkomprimujeme v¹echny hrany spojující èerný a èervený vrchol, èím¾ v¹echny èervené
+vrcholy \uv{zasuneme} do~jejich èerných otcù. Vznikne $2,4$-strom a ten, jak u¾ víme,
+má hloubku $\O(\log n)$. Pùvodní è-è strom má tedy hloubku nejvý¹e $2\log n$, co¾ je
+asymptoticky taky $\O(\log n)$.
+\qed
+
+\noindent{\sl Alternativní dùkaz:} \foot{\uv{Vy máte protipøíklad? Nevadí, já mám dva dùkazy :)}}
+Pøedpokládejme, ¾e $T$ je èerveno-èerný strom, který má na cestì z koøene do listu
+$k$ èerných vrcholù. Pak pro poèet vrcholù stromu $T$ platí $2^k-1\leq n \leq 2^{2k}-1$. Nejmen¹í takový strom má v¹echny vrcholy obarvené èernì a je to úplný
+pravidelný binární strom o hloubce $k-1$, co¾ dává dolní odhad. Nejvìt¹í takový strom má v¹echny vrcholy v sudých hladinách obarveny èervenì a v lichých hladinách èernì, je to úplný pravidelný binární strom o hloubce $2k-1$ a tím je dán horní odhad. Tedy $k\leq 1+\log n$. Z vlastností èerveno-èerných stromù plyne, ¾e $k\leq \<hloubka>(T) \leq 2k$.
+\qed
+
+Nyní si rozebereme vkládaní do èerveno-èerných stromù.
+
+\s{Insert:}
+Vlo¾íme hodnotu do stromu jako do standardního binárního vyhledávacího stromu, tedy jako list,
+a obarvíme èervenì. První ani tøetí axiom jsme neporu¹ili a pokud je pøedek èerný, tak ani
+axiom druhý, èili jsme hotovi:
+\fig{01.eps}{2in}
+Jestli¾e je otec~$o$ na¹eho vrcholu~$t$ také èervený, budeme øíkat, ¾e do¹lo k~{\I poru¹e}
+a budeme se ji sna¾it opravit. Situace musí vypadat následovnì:
+
+\fig{02.eps}{1.5in}
+
+Nyní rozli¹íme následující tøi pøípady:
+\itemize\ibull
+\:Pokud strýc~$s$ vrcholu~$t$ je èervený, pak pouze pøebarvíme $o$, $d$ a $s$ podle následujícího obrázku.
+Nyní $d$ mù¾e být porucha, ov¹em posunutá o 2 hladiny vý¹e, tak¾e pøi nejhor¹ím pokraèujeme
+do~koøene a poslední poruchu opravíme pøebarvením koøene na~èerno.
+\fig{03.eps}{3.5in}
+\:Strýc $s$ je èerný. Pak zále¾í na tom, zda hodnota $t$ le¾í mezi hodnotami $o$ a $d$ nebo ne. Jinými slovy, zda cesta $t-o-d$ obsahuje zatáèku.
+ \numlist\nalpha
+ \:Bez zatáèky: Provedeme rotaci a pøebarvíme. Splnìny budou axiomy 1, 2 i 3, tedy máme èervenoèerný strom:
+ \fig{04.eps}{4in}
+ \:Se zatáèkou: Provedeme dvojitou (LR) rotaci a pøebarvíme. Opìt získáme korektní èervenoèerný strom:
+ \fig{05.eps}{4in}
+ \endlist
+\endlist
+
+\bye
--- /dev/null
+P=12-kostry
+
+include ../Makerules
--- /dev/null
+\input lecnotes.tex
+
+\prednaska{13}{Hashování}{(zapsali Martin Francù a Bohdan Maslowski)}
+
+V~pøedchozích kapitolách jsme ukázali, ¾e aèkoliv v~obecném pøípadì
+nelze tøídit rychleji ne¾ v~èase $\Theta(n\log n)$, pro nìkteré speciální
+druhy hodnot (malá celá èísla, øetìzce apod.) existuje efektivnìj¹í lineární
+algoritmus. Nyní si podobným zpùsobem posvítíme na~reprezentaci mno¾in --
+navrhneme datovou strukturu, která bude umìt s~vhodnými typy prvkù provádìt
+základní mno¾inové operace (\<Insert>, \<Delete>, \<Find>) v~{\I prùmìrnì}
+konstantním èase (pomocí vyhledávacích stromù jsme to zatím umìli v~èase $\O(\log n)$
+na operaci).
+
+Jaké máme mo¾nosti? Zaènìme od~tìch nejjednodu¹¹ích a oznaème si~$n$ velikost
+mno¾iny a $U$ velikost univerza, z~nìj¾ prvky mno¾iny vybíráme.
+
+\s{1. Pole indexované od $0$ do $U-1$}
+
+\indent \<Insert>, \<Delete> i \<Find> pracují v~èase $\O(1)$, ale platíme za~to
+pamì»ovou slo¾itostí $\Theta(U)$.
+
+\s{2. Èíslicový strom (radix tree)}
+
+Zvolíme základ soustavy $b$, prvky universa zapisujeme jako $k$-tice èíslic. Èili:
+$$x\in U \mapsto [b]^{k}, k=\lceil\log_b U\rceil.$$
+Tím nám vznikne strom s~hloubkou $k$, na $k$-té úrovni se rozhodujeme podle $k$-té èíslice.
+
+Operace \<Find> bude trvat $O(\log_b U)$, operace \<Insert> a \<Delete> pak
+$O(b \log_b U)$. Strom zabere prostor velikosti $O(nb\log_b U)$.
+
+Èím bude základ $b$ men¹í, tím lep¹í pamì»ová a hor¹í èasová slo¾itost,
+a obrácenì. Èíslicový strom tedy vy¾aduje peèlivé nastavení parametrù.
+
+\h{3. Písmenkový strom }
+
+Pou¾ívá se pro øezìzce, za základ soustavy $b$ zvolíme abecedu.
+
+\medskip
+
+$\vert x\vert $...délka øetìzce
+
+Find, Insert, Delete: $O(\vert x\vert )$
+
+Pamì»: $O(\sum_{i} \vert x\vert )$
+
+\bigskip
+
+\h{4. Hashování }
+
+
+$h$ ... hashovací funkce $[U] \rightarrow [m]$
+
+$m$ ... poèet pøihrádek, v ka¾dé pøihrádce spojový seznam.
+
+\medskip
+
+Perfektní hashovací funkce pøiøadí do ka¾dé z $m$ pøihrádek jediný prvek z $[U]$.
+
+\s{Pøíklad:} Jaká je pravdìpodobnost, ¾e 2 z k lidí mají narozeniny ve stejný den? Staèí 23 lidí, aby nastala kolize s pravdìpodobností vìt¹í ne¾ jedna polovina. Aby nenastala, potøebovali bychom vìt¹í poèet pøihrádek: $p = cm^2$
+
+
+
+
+\h{Hashování se separovými øetezci}
+
+\medskip
+Insert, Delete i Find pracují v èase $O(\vert x\vert )$, pamì»ová slo¾itost je $O(m+n)$
+\medskip
+
+vlo¾ili jsme $x_1 \ldots x_n \in [U]$
+
+nyní pracujeme s $x \in [U]$
+
+pøedpokládejme, ¾e $h(x_i)$ a $h(x)$ jsou náhodné
+
+$p:=h(x)$
+
+$N_p:=\sharp$ : $h(x_1)=h(x)$
+
+$N_{i,p} := 0$ pokud $h(x_i)\neq p$
+
+$N_{i,p} := 1$ pokud $h(x_i)= p$
+
+$N_p = \sum_{i=1}^n N_{i,p}$
+
+$E N_{i,p} = Pr[h(x_i)=p] = {1 \over m} $
+
+$E N_p = n {1\over m} = {n \over m} $
+
+\medskip
+
+Za pøedpokladu, ¾e hash pøidìluje náhodnou pøihrádku, je v ka¾dé pøihrádce ${n \over m}$ prvkù.
+
+\bigskip
+
+
+\h{Druhá pùlka}
+
+
+\noindent {\sc Vìta:} {\it Pokud jsme do hashovací tabulky vlo¾ili
+hodnoty $x_1,\dots,x_m$ tak, ¾e $h_{x_i}$ jsou v¹echny stejnì
+pravdìpodobné, pak následující Insert, Delete, Find pracují v èase
+${\cal O}\,({n \over m}+1)$. (Za pøedpokladu, ¾e výpoèet $h$ trvá
+${\cal O}\,(1)$.)}
+
+\medskip
+
+{\it Insert}: Spoèítá si $h(x)$ a vlo¾í jej do spojového seznamu v patøièné pøihrádce.
+
+{\it Delete}: Spoèítá si $h(x)$ a najde jej v spojovém seznamu v patøièné pøihrádce a
+odstraní jej.
+
+{\it Find}: Spoèítá si $h(x)$ a najde jej v spojovém seznamu v patøièné pøihrádce.
+
+\medskip
+
+\noindent Tato metoda má dva háèky:
+
+1. vysoká koliznost
+
+2. musíme dopøedu vìdìt, kolik prvkù budeme vkládat
+
+\bigskip
+
+{\large \bf Nafukovací hashovací tabulka}
+
+na poèátku: $n=0;m=4$
+
+v ka¾dém okam¾iku budeme udr¾ovat pomìr ${n\over m}\in \langle
+{1\over 2},1)$
+
+tedy: (pokud nastane)
+$$n=4 \Rightarrow m:=8 $$
+$$n=8 \Rightarrow m:=16$$
+Pøi ka¾dém zvìt¹ování tabulky musíme zvìt¹it obor hodnot hashovací funkce a v¹echny prvky ji¾
+v tabulce ulo¾ené znovu "zhashovat".
+
+Nafukování trvá ${\cal O}\, (n)$, ale mezi dvìma nafouknutími je
+minimálnì $ {n\over 2} $ Insertù.
+
+Pokud tedy ka¾dý Insert pøispìje konstantou, tak to máme zadarmo!
+
+\bigskip
+
+PROBLÉM: Jak udìlat funkci Delete? Jednoduché zmen¹ování tabulky pøi pomìru
+${n\over m}={1\over 2}$ není dostaèující, nebo» by se mohlo stát, ¾e bychom nafukovali a
+zfukovali hned po sobì.
+
+\bigskip
+
+{\bf Verze s Insert a Delete}
+
+Udr¾ujeme ${n\over m}\in \langle {1\over 4},1)$
+
+pøi zfukování tabulku pùlíme, pøi nafukování zdvojnásobujeme
+
+\medskip
+
+Po pøehashování je pomìr ${n\over m}={1\over 2} \Rightarrow$mezi dvìma pøehashováními
+nastane minimálnì ${m\over 2}$ Insertù èi ${m\over 4}$ Deletù, tedy alespoò ${m\over 4}$
+operací.
+
+Zároveò zabereme v¾dy lineárnì prostoru vzhledem k $n$.
+
+\bigskip
+
+\noindent {\sc Vìta:} {\it Nafukovací hashování (se separovanými øetìzci) za pøedpokladu
+rovnomìrné pravdìpodobnosti pracuje v prùmìrném èase ${\cal O}\,(1)$ na operaci a v prostoru
+${\cal O}\,(n)$ (kde $n$ je poèet pøítomných prvkù).}
+
+\medskip
+
+Kde pøijít k rovnomìrné hashovací funkci?
+
+\medskip
+
+\noindent Dìdeèkovo vyprávìní:
+
+a) $h(x)= x$ $mod (m)$ -rychlá, snadná kolize, sudé x v¾dy v sudých pøihrádkách. Na náhodných
+èíslech funguje v pohodì.
+
+\bigskip
+
+\noindent POZOROVÁNÍ: $\alpha \in(0,1)$ iracionální;
+
+$x \in{\sc N}$ pak $\alpha x $ $ mod (1)$ se chová "náhodnì".
+
+\medskip
+
+ b) $h(x)=\lfloor m(\alpha x $ $mod(1))\rfloor$
+ $$ {A\over W} \sim \alpha \Rightarrow h(x)= \lfloor m(Ax \ mod(W)) \rfloor $$
+
+ \medskip
+
+ Jako velmi výhodné se jeví pou¾ívat $\alpha = {{\sqrt{5} -1}\over{2}} $ - pøevrácená hodnota zlatého øezu.
+
+\medskip
+
+c) funkce k hashování øetìzcù
+$$ h(\emptyset)=0 $$
+$$ h(x_1, \dots , x_n)=c(h(x_1, \dots ,x_{n-1})+x_n)$$
+$$ \Rightarrow h(x_1, \dots , x_n)=x_n + cx_{n-1} + c^2 x_{n-2} +
+\dots $$ (polynom)
+
+\medskip
+
+d)Pokud $x$ je $k$-tice (celých) èísel. Pak mù¾eme jako hashovací funkci pou¾ít skalární
+souèín s náhodnì vybraným prvkem z vektorového prostoru $Z_p^k$ (nad koneèným tìlesem):
+$$ h(x)= \overrightarrow{a} * \overrightarrow{x} mod (p) $$
+
+\medskip
+
+Pravdìpodobnost pøes v¹echny volby $\overrightarrow{a}$, ¾e $x,y$ se zahashují stejnì je
+pak:
+$$ p[h(x)=h(y)] = p[\overrightarrow{a}*\overrightarrow{x} =
+\overrightarrow{a}*\overrightarrow{y}]= p[\overrightarrow{a}*
+(\overrightarrow{x}-\overrightarrow{y})=0)] $$
+
+Pokud platí $p[\overrightarrow{a}* (\overrightarrow{x}-\overrightarrow{y}=0)]$, pak
+$\overrightarrow{a}$ je prvkem ortogonálního doplòku k $(x-y)$, ten má $ dim (W^\bot) =
+(k-1)$
+
+$$ p= {{W^\bot}\over{Z_p^k}}= {{p^{k-1}}\over{p^k}} = {1\over p}={1\over m}$$
+
+kde $m$ je poèet pøihrádek. Co¾ je dostaèující.
+
+\bye
+
--- /dev/null
+P=13-hash
+
+include ../Makerules
--- /dev/null
+\input ../lecnotes.tex
+
+\prednaska{2}{Rozdìl a panuj}{(zapsali J. Záloha a P. Ba¹ista)}
+
+Na~minulé pøedná¹ce jsme si zavedli výpoèetní model a èasovou slo¾itost.
+Nyní se pustíme do~analýzy slo¾itosti rùzných algoritmù, zejména rekurzivních
+zalo¾ených na metodì Rozdìl a panuj {\sl (Divide et Impera)}.
+
+\h{Asymptotická notace}
+
+Jak u¾ víme, pøi porovnávání algoritmù je obvykle rozhodující asymptotické
+chování a multiplikativní konstanty mù¾eme zanedbávat (beztak závisí na~konkrétním
+poèítaèi). Proto si zavedeme takzvanou asymptotickou notaci, která slou¾í
+pro právì takové porovnávání rùstu funkcí:
+
+\s{Definice:} Pro funkce $f,g: {\bb N} \rightarrow {\bb R}^+$ øekneme,
+¾e $f$ je $\O(g)$ právì tehdy, kdy¾ $\exists c>0: \forall ^{*} n \in {\bb N}: f(n) \leq c \cdot g(n)$.
+Zde $\forall^* n \in {\bb N}$ je zkratka za \uv{$\exists n_0 \in {\bb N}: \forall n \geq n_0$}, tedy
+\uv{pro v¹echna~$n$ a¾ na~koneènì mnoho výjimek.}
+
+\s{Poznámka:} $\O$-notace tedy vyjadøuje, ¾e funkce~$f$ je skoro v¹ude men¹í nebo nejvý¹e rovná
+nìjakému reálnému násobku funkce~$g$. Aèkoliv zápis vypadá jako rovnost, rozhodnì
+není symetrický: napøíklad platí $\log n=\O(n)$, ale neplatí $n=\O(\log n)$. Formálnì by bylo lep¹í pova¾ovat $\O(g)$
+za tøídu funkcí, pro které platí, ¾e se dají shora odhadnout kladným násobkem funkce~$g$, a~psát tedy~$f\in\O(g)$,
+ale zvyk je bohu¾el ¾elezná ko¹ile.
+
+\s{Pøíklady:} $2.5n^{2} = \O(n^{2})$, $2.5n^{2}+30n = \O(n^{2})$.
+
+\>Také platí:
+$$
+\O(f)+\O(g) \in \O(f+g),
+$$
+èím¾ myslíme, ¾e pokud vezmeme libovolnou $f'=O(f)$ a $g'=O(g)$, bude $f'+g'=O(f+g)$.
+To platí, jeliko¾ skoro v¹ude je $f' \leq cf$, $g'\leq dg$, a~tedy $f'+g' \le cf+dg \le (c+d)(f+g)$.
+
+\s{Cvièení:} Uka¾te, ¾e:
+\itemize\ibull
+\:$\O(f) \cdot \O(g)=\O(f \cdot g)$,
+\:$\O(f+g)=\O(\max(f,g))$,
+\:$\O(n^{2})+\O(n)=\O(n^{2}+n)=\O(n^{2})$.
+\endlist
+
+$\O$-notace popisuje horní odhad asymptotického chování funkce. Mnohdy v¹ak
+potøebujeme také odhandout funkci zespodu (chceme-li øíci, ¾e algoritmus potøebuje
+{\I alespoò} nìjaké mno¾ství èasu nebo pamìti), pøípadnì z~obou stran:
+
+\s{Definice:}
+
+\itemize\ibull
+\:$f=\Omega(g) \equiv \exists c>0:\forall^* n \in {\bb N}: f(n) \geq c\cdot g(n)$.
+
+$\Omega$-notace tedy øíká, ¾e hodnota funkce $f$ je v¾dy stejná nebo vy¹¹í ne¾ nìjaký $c$-násobek funkce $g$, a tedy $g=\O(f)$.
+\:$f=\Theta(g) \equiv f=O(g) \wedge f=\Omega(g)$
+
+nebo výøeènìji:
+
+$f=\Theta(g) \equiv \exists$ $c_{1},c_{2} > 0: c_{1}\cdot g(n) \leq f(n) \leq c_{2}\cdot g(n)$ To znamená, ¾e existují nezáporné reálne konstanty $c_{1},c_{2}$ takové, ¾e se funkce $f(n)$ dá ohranièit $c_{1}$- a $c_{2}$-násobkem funkce $g(n)$.
+\endlist
+
+\s{Porovnání rùstu funkcí:} (aneb jak moc máme algoritmy rádi podle jejich chování od~nejlep¹ích k~nejhor¹ím)
+
+\itemize\ibull
+\:$\Theta(1) \ldots$ funkce zespoda i shora ohranièené konstantami
+\:$\Theta(\log{( \log{n} )})$
+\:$\Theta(\log{n})$ \dots\ logaritmická
+\:$\Theta(n^{\varepsilon}), \varepsilon \in (0,1)$ \dots\ sublineární
+\:$\Theta(n)$ \dots\ lineární
+\:$\Theta(n^{2})$ \dots\ kvadratická
+\:$\Theta(n^{k})$ \dots\ polynomiální
+\:$\Theta(2^{n})$ \dots\ exponenciální pøi základu $2$
+\:$\Theta(3^{n})$ \dots\ exponenciální pøi základu $3$
+\:$\Theta(k^{n})$ \dots\ exponenciální pøi základu $k>1$
+\:$\Theta(n!)$ \dots\ faktoriálová
+\:$\Theta(n^{n})$
+\:\dots\ nekoneènì mnoho dal¹ích tøíd (i mezi tìmi vý¹e uvedenými)
+\endlist
+
+\>{\sl Poznámka:} Pokud se v~odhadu slo¾itosti vyskytne logaritmus (jinde ne¾
+v~exponentu), nezále¾í na tom, jaký má základ, proto¾e platí:
+$$
+\log_k{n}={{\log_c{n}}\over{\log_c{k}}}={{1}\over{\log_c{k}}}\cdot \log_c{n},
+$$
+kde $1/\log_c{k}$ je jen konstanta, tak¾e ji mù¾eme \uv{schovat do~$\O$.}
+
+\s{Pøíklad:} Eukleidùv algoritmus:
+Pokud jej pustíme na $2$ èísla o $n$ bitech, poèet iterací bude $\O(n)$, ka¾dá
+iterace trvá $\O(n^{2})$ krokù. Tak¾e celkovì má tento algoritmus èasovou
+slo¾itost $\O(n^{3})$.
+
+\h{Metoda Rozdìl a panuj}
+
+Nyní pøestaòme chodit kolem horké ka¹e a øekòeme si, co to ono vý¹e zmiòované
+\uv {rozdìl a panuj} znamená. Mìjme nìjaký problém, který má tu vlastnost, ¾e
+kdy¾ jej rozdìlíme na nìjaké podproblémy, které mají stejný charakter, a ty
+vyøe¹íme, slo¾ením jejich øe¹ení mù¾eme získat øe¹ení pùvodního problému.
+Algoritmus tedy bude rekurzivnì volat sám sebe, ne¾ se dostane k~podproblému
+nìjaké konstantní velikosti, který u¾ umí vyøe¹it triviálnì, a pak se zaène
+z~rekurze vracet a skládat jednotlivá dílèí øe¹ení.
+
+\s{Pøíklad 1 -- MergeSort:} Algoritmus {\I MergeSort} pro tøídìní posloupnosti vypadá takto:
+vstup rozdìlíme na~dvì (skoro) stejné èásti, ty rekurzivním voláním setøídíme,
+a~nakonec výsledné dvì posloupnosti slijeme do~jedné. Rozdìlování a slévání nám
+trvá lineárnì dlouho, tak¾e pro èasovou slo¾itost MergeSortu platí tato
+rekurentní rovnice: $$T(n)=2 \cdot T({{n}/{2}})+\O(n).$$ Jak ji vyøe¹íme? Tøeba
+tak, ¾e si pøedstavíme strom rekurzivních volání. Ka¾dý vrchol má dva syny
+(dìlíme vstup na~dvì èásti), v~nich¾ jsou vstupy polovièní velikosti. V~ka¾dém
+vrcholu trávíme èas lineární s~velikostí jeho vstupu, souèet velikostí vstupù
+pøes ka¾dou hladinu je~$n$ a hloubka stromu musí být $\O(\log n)$. Vyjde nám
+tedy, ¾e $T(n)=\O(n\log n)$.
+
+\s{Pøiklad 2 -- násobení èísel:} Pokud násobíme dvì èísla $X$ a $Y$ zpùsobem,
+který nás uèili na základní ¹kole, dostaneme se na èasovou slo¾itost $\Theta(n^2)$.
+Proto¾e se jedná o~dost èastou operaci, zamysleme se, jestli by ne¹la zrychlit.
+Nasmìrujme na¹e úvahy na postup \uv{rozdìl a panuj}. Rozdìlíme ka¾dého èinitele
+na dvì stejnì dlouhé èásti a pro jednoduchost pøedpokládejme, ¾e toto
+roz¹tìpení èinitele probìhne v¾dy bez zbytku:
+$$
+X=A \cdot 10^{{n}/2}+B, \qquad Y=C \cdot10^{{n}/{2}}+D.
+$$
+Zde $A, B, C, D$ jsou u¾ jen $n/2$-ciferná èísla. Pùvodní souèin získáme jako:
+$$
+XY=(A\cdot 10^{{n}/{2}}+B) (C\cdot 10^{{n}/{2}}+D)=AC \cdot 10^{n}+(AD+BC)\cdot 10^{{n}/{2}}+BD.
+$$
+Nyní, jak vidíme, staèí spoèítat souèin ètyø $n/2$-ciferných èísel. Uva¾me,
+jakou bude mít tento algoritmus èasovou slo¾itost:
+$$T(n) = 4T(n/2)+\Theta(n).$$
+Jak takovou rekurenci vyøe¹íme?
+
+\>{\sl 1. zpùsob: Øe¹ení pomocí rozepisování:}
+$$\eqalign{
+T(n)&= 4T(n/2)+\Theta(n) = \cr
+ &= 4T(n/2)+cn = \cr
+ &= 4\cdot (4T(n/4)+cn/2)+cn = 4^2T(n/4)+2cn+cn = 4^2T(n/4)+3cn = \cr
+ &= 4^2\cdot (2T(n/8)+cn/4)+3cn = 4^3T(n/8)+4cn+3cn = 4^3T(n/8)+7cn = \cr
+ &\ldots\cr
+}$$
+Odtud snadno vypozorujeme, ¾e jednotlivé vztahy se vyvíjí podle vzorce
+$T(n)=4^kT(n/2^k) + (2^k-1)cn.$ Pro $k=\lceil\log_2 n\rceil$ je ov¹em
+$2^k\le 1$, tak¾e $T(n/2^k)=\Theta(1)$ a dostaneme (horní celou èást zanedbáme,
+ta ovlivní jen konstanty):
+$$
+T(n) = 4^{\log_2 n}\Theta(1) + (2^{\log_2 n}-1)cn = n^2\Theta(1) + (n-1)cn = \Theta(n^2).
+$$
+
+\>{\sl 2. zpùsob: Úvaha o~stromu:} Nakreslíme si strom rekurzivních volání
+na¹eho algoritmu:
+\fig{figure.eps}{4in}
+Na~$i$-té hladinì stromu máme $4^i$ vrcholù, v~nich jsou vstupy velikosti
+$n/2^i$, tak¾e na~celé hladinì trávíme èas celkem $\Theta(4^i\cdot n/2^i)
+= \Theta(2^in)$. Velikosti vstupù klesají exponenciálnì, tak¾e celý strom
+je hluboký $k=\log_2 n$ (opìt si dovolíme zapomenout na~horní celou èást).
+Celkem tedy trávíme èas $\sum_{i=0}^k \Theta(2^in) = \Theta(n\cdot\sum_{i=0}^k 2^i) = \Theta(n^2)$.
+
+Oba zpùsoby analýzy se tedy shodují, ¾e ná¹ algoritmus má kvadratickou èasovou
+slo¾itost a ¾e jsme si oproti klasickému algoritmu nikterak nepomohli.
+Podívejme se je¹tì jednou na~to, jak se ná¹ algoritmus vìtví:
+$$\vbox{\halign{\hfil#\hfil \quad & \hfil#\hfil \quad &\hfil#\hfil\cr
+hloubka & poèet úloh & velikost podúlohy\cr
+\noalign{\smallskip\hrule\medskip}
+0 & $4^{0}$ & ${n}/{2^{0}}$\cr
+1 & $4^{1}$ & ${n}/{2^{1}}$\cr
+2 & $4^{2}$ & ${n}/{2^{2}}$\cr
+3 & $4^{3}$ & ${n}/{2^{3}}$\cr
+\vdots & \vdots & \vdots\cr
+$k$ & $4^{k}$ & ${n}/{2^{k}}$\cr}}$$
+Naskýtá se otázka, jestli bychom nemohli èasovou slo¾itost zlep¹it. Toho bychom
+mohli dosáhnout buïto zlep¹ením èlenu $cn$ v~na¹í rekurenci, èili zefektivnìním
+spojování podúloh. To ov¹em není pøíli¹ nadìjné (pokud ètenáø nevìøí, mù¾e si to dokázat),
+tak¾e místo toho vyu¾ijeme druhou ¹anci, a~to omezení vìtvení ze~ètyø vìtví na~tøi.
+Pøipomeòme si, ¾e potøebujeme spoèítat:
+$$
+XY=AC\cdot 10^{n}+(AD+BC)\cdot 10^{n/2}+BD.
+$$
+Pøitom ale nepotøebujeme znát souèiny $AD$ ani $BC$ samostatnì, nebo» nám staèí
+zjistit celý èlen $AD+BC$. Kdybychom poèítali $AC$, $BD$
+a potom $(A+B)(C+D)=AC+AD+BC+BD$, tak odèítáním $(AC+BD)$ od $AC+AD+BC+BD$ dostaneme
+hledaný prostøední èlen $AD+BC$. Nyní nám ji¾ staèí jen tøi
+násobení, ale potøebujeme tøi sèítání a jedno odèítání navíc.
+Uká¾eme, ¾e tato komplikace je zanedbatelná oproti práci u¹etøené
+men¹ím vìtvením. Podívejme se opìt na~tabulku:
+$$\vbox{\halign{\hfil#\hfil \quad & \hfil#\hfil \quad &\hfil#\hfil\cr
+hloubka & poèet úloh & velikost podúlohy\cr
+\noalign{\smallskip\hrule\medskip}
+0 & $3^{0}$ & ${n}/{2^{0}}$\cr
+1 & $3^{1}$ & ${n}/{2^{1}}$\cr
+2& $3^{2}$ & ${n}/{2^{2}}$\cr
+3 & $3^{3}$ & ${n}/{2^{3}}$\cr
+\vdots & \vdots & \vdots\cr
+$k$ & $3^{k}$ & ${n}/{2^{k}}$\cr}}$$
+Opìt uva¾me, kolik práce spotøebujeme v~souètu pøes v¹echny hladiny (hloubka stromu
+$k$ je opìt $\lceil\log_2 n\rceil$ a horní celou èást zanedbáme):
+$$\sum_{i=0}^{k}3^{i}\cdot {{n}\over{2^{i}}}=\sum_{i=0}^{k} \left( {{3}\over{2}} \right) ^{i}\cdot n=n\cdot \sum_{i=0}^{k} \left( {{3}\over{2}} \right) ^{i}=n\cdot {{ \left( {{3}\over{2}} \right) ^{k+1}-1}\over{{{3}\over{2}}-1}}=
+$$
+$$
+=n\cdot {{ \left( {3}\over{2} \right) ^{k+1}-1}\over{{{1}\over{2}}}}=2\cdot n\cdot \left[ \left( {{3}\over{2}} \right) ^{k+1}-1 \right] = \O \left( n\cdot \left( {{3}\over{2}} \right) ^{\log_2{n}} \right) =
+$$
+$$
+=\O \left( n\cdot {{3^{\log_2{n}}}\over{2^{\log_2{n}}}} \right)=\O \left( n\cdot {{3^{\log_2{n}}}\over{n}} \right)=\O \left( 3^{\log_2{n}} \right)=\O \left( (2^{\log_2{3}})^{\log_2{n}} \right)=
+$$
+$$
+=\O \left( 2^{(\log_2{n}) \cdot \log_2{3}} \right)=\O \left( (2^{\log_2{n}})^{\log_2{3}} \right)=\O \left( n^{\log_2{3}} \right) =\O \left( n^{1.585} \right).
+$$
+Upravený algoritmus u¾ tedy má lep¹í èasovou slo¾itost, konkrétnì $\O(n^{1.585})$.
+V~praxi bychom samozøejmì pro èinitele ne¹tìpili a¾ na jednociferná èísla,
+ale zastavili se u~nìjaké dostateènì malé délky (øeknìme 50~cifer) a tam
+pøepnuli na~kvadratický algoritmus, který má men¹í re¾ii.
+
+(Mimochodem, pro násobení èísel existuje je¹tì efektívnìj¹í algoritmus, který
+dosahuje èasové slo¾itosti $\O(n \log{n})$, av¹ak na~to u¾ jsou potøeba trochu
+pokroèilej¹í techniky, jako je diskrétní Fourierova transformace, tak¾e
+si jej necháme na~pøí¹tí semestr.)
+
+\h{Master Theorem}
+
+Metody øe¹ení rekurentních rovnic pro slo¾itosti rozdìlovacích a panovacích algoritmù
+by asi fungovaly i na~jiné algoritmy ne¾ jen na¹e dva pøíklady, ale proè se poka¾dé
+moøit upravováním výrazù? Radìji si doká¾eme obecnou vìtu, která pùjde na~vìt¹inu
+takových rekurencí nasadit. Øíká se jí Master Theorem nebo také (vzhledem k~tomu,
+jak se pou¾ívá) Kuchaøková vìta.
+
+\s{Vìta:} \>{\sl (Master Theorem)} Pøedpokládejme, ¾e $T(1)=\O(1)$ a $T(n)=a\cdot T(\lceil {{n}\over{b}} \rceil)+\O(n^d)$, kde $a \geq 1$, $b>1$ a $d \geq 0$. Potom $T(n)$ je:
+
+\smallskip
+
+\halign{#&#&#\cr
+\indent & $\O(n^d)$ & kdy¾ $a<b^d$,\cr
+& $\O(n^d\cdot \log{n})$ & kdy¾ $a=b^d$,\cr
+& $\O(n^{\log_b{a}})$ & kdy¾ $a>b^d$.\cr}
+
+\proof Pøedpokládejme nejdøíve, ¾e $n=b^m, m \in \bb{N}$, aby platilo $\lceil
+{{n}\over{b}} \rceil = {{n}\over{b}}$. Pou¾ijeme opìt \uv{dùkaz stromem}.
+Strom rekurzivních volání se v¾dy vìtví na stejný poèet vìtví, konkrétnì~$a$,
+a~velikosti vstupù klesají $b$-krát. Podívejme se na~tabulku:
+$$\vbox{\halign{\hfil#\hfil \quad & \hfil#\hfil \quad & \hfil#\hfil & \hfil#\hfil \cr
+hloubka & poèet podúloh & velikost podúlohy & èas na hladinì \cr
+\noalign{\medskip\hrule\medskip}
+0 & $1$ & $n$ & $\O(n^d)$\cr
+1 & $a$ & $n/{b^1}$ & ${\O(({n/{b^1}})^d) \cdot a^1}$\cr
+2 & $a^2$ & $n/{b^2}$ & ${\O(({n/{b^2}})^d) \cdot a^2}$\cr
+3 & $a^3$ & $n/{b^3}$ & ${\O(({n/{b^3}})^d) \cdot a^3}$\cr
+\vdots & \vdots & \vdots\cr
+$k$ & $a^k$ & $n/{b^k}$ & ${\O(({n/{b^k}})^d) \cdot a^k}$\cr}}$$
+
+\noindent
+Zkoumejme èas potøebný na vyøe¹ení v¹ech podúloh na jedné hladinì:
+$$
+\O \left( \left( {{n}\over{b^k}} \right) ^d \right) \cdot a^k=\O \left( a^k\cdot n^d \left( {{1}\over{b^k}} \right) ^d \right)=\O \left( a^k \cdot n^d \left( {{1}\over{b^d}} \right) ^k \right)=\O \left( n^d \left( {{a}\over{b^d}} \right) ^k \right).
+$$
+Celkem je tedy èas potøebný na vyøe¹ení v¹ech podúloh na v¹ech hladinách (to
+znamená celé úlohy):
+$$
+T(n)=\sum_{k=0}^{\log_b{n}}\O \left( n^d \left( {{a}\over{b^d}} \right) ^k \right)=\O \left( n^d \sum_{k=0}^{\log_b{n}} \left( {{a}\over{b^d}} \right) ^k \right)
+$$
+V¹imnìme si èlenu ${a}\over{b^d}$. Na jeho hodnotì závisí hodnota výsledné sumy, proto¾e se vlastnì jedná o kvocient geometrické øady. Proto rozli¹me následující pøípady:
+
+\>{\I 1.} ${{a}\over{b^d}}<1$ -- jinými slovy, práce na jednotlivých hladinách exponenciálnì ubývá a souèet sumy, i kdyby
+byla nekoneèná, se dá omezit nìjakou konstantou, která se schová do $\O$, a tak je $T(n)=\O(n^d)$.
+
+\>{\I 2.} ${{a}\over{b^d}}=1$ -- práce na jednotlivých hladinách je stejnì, to znamená, ¾e souèet sumy je právì $\log_b n$, a tedy $T(n) = \O(n^d \cdot \log_b(n))$.
+
+\>{\I 3.} ${{a}\over{b^d}}>1$ -- to znamená, ¾e práce na jednotlivých hladinách
+pøibývá, tak¾e musíme geometrickou øadu seèíst poctivì: $T(n) = \O(n^d \cdot
+({{a}\over{b^d}})^{\log_b{n}})$. Tento výraz vypadá ponìkud o¹klivì,
+ale je¹tì ho trochu (alespoò kosmeticky) upravíme:
+$$
+\O\left(n^d \cdot \left({{a}\over{b^d}}\right)^{\log_b{n}}\right)=\O\left({ a^{\log_b{n}} \cdot n^d \over (b^d)^{\log_b{n}}}\right)=\O\left({\left(b^{\log_b{a}}\right)^{\log_b{n}} \cdot n^d \over{\left(b^d\right)^{\log_b{n}}}}\right)=
+$$
+$$
+=\O\left({\left(b^{\log_b{n}}\right)^{\log_b{a}} \cdot n^d \over{\left(b^{\log_b{n}}\right)^d}}\right)
+=\O\left({n^{\log_b{a}} \cdot n^d \over{n^d}}\right)
+=\O\left(n^{\log_b{a}}\right).
+$$
+Tyto tøi pøípady pøesnì odpovídají rozdìlení pøípadu v~tvrzení vìty.
+
+Vra»me se nyní k~mo¾nosti, kdy $n$ není mocnina~$b$.
+Tehdy platí $b^l<n<b^{l+1}$ pro nìjaké $l \in \bb{N}$. V tomto pøípadì
+zaokrouhleme $n$ a~poèítejme s $n^{\prime}=b^{l+1} < bn$. Jeliko¾ funkce
+$T(n)$ je neklesající, dostaneme $T(n)\le T(n')$ a ve~v¹ech tøech pøípadech
+je výsledek pro~$n'$ maximálnì konstanta-krát vìt¹í ne¾ pro~$n$, co¾ se
+\uv{schová do~$\O$.} Vìta tedy platí i v~tomto pøípadì.
+\qed
+
+\s{Pøíklad:}
+Porovnejme si nìkteré známé algoritmy a jejich èasovou slo¾itost pomocí \>{\sl Master Theoremu}:
+$$\vbox{\halign{# \quad \quad & # \quad \quad & # \quad \quad & # \quad \quad & #\cr
+algoritmus & $a$ & $b$ & $d$ & èasová slo¾itost\cr
+\noalign{\smallskip\hrule\medskip}
+Mergesort & 2 & 2 & 1 & $\O({n \cdot \log{n}})$\cr
+Násobení I. & 4 & 2 & 1 & $\O(n^2)$\cr
+Násobení II. & 3 & 2 & 1 & $\O(n^{\log_2{3}})$\cr
+Binární vyhledávání & 1 & 2 & 0 & $\O(\log{n})$\cr}}$$
+
+\s{Domácí úkol nakonec:} Vymyslete algoritmus, který by z $n$ zadaných bodù v rovinì (prostoru) na¹el takové dva, které~jsou od sebe nejménì vzdálené (zde existuje takový algoritmus s èasovou slo¾itostí $\O(n \cdot \log{n})$).
+
+\bye
--- /dev/null
+P=2-rozdel
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--- /dev/null
+\input ../lecnotes.tex
+\input epsf.tex
+
+\prednaska{3}{Rozdìl a panuj II}{(zapsali Luká¹ Hermann, Vincent Krí¾, Oto Petøík)}
+
+\h{Násobení matic n$\times$n a Strassenùv algoritmus}
+
+Nejdøíve si pøipomeneme definici násobení dvou ètvercových matic typu $n \times n$. Platí, ¾e prvek v $i$-tém øádku a $j$-tém sloupci výsledné matice $Z$ se rovná standardnímu skalárnímu souèinu $i$-tého øádku první matice $X$ a $j$-tého sloupce druhé matice~$Y$. Formálnì zapsáno:
+
+$$ Z_{ij} = \sum_{k=1}^n X_{ik} \cdot Y_{kj}. $$
+
+\figure{nasobeni-matic.eps}{Násobení matic}{\epsfxsize}
+
+Algoritmus, který by násobil matice podle této definice, by mìl èasovou slo¾itost $ \Theta(n^3) $, proto¾e poèet prvkù ve výsledné matici je $n^2$ a jeden skalární souèin vektorù dimenze $n$ vy¾aduje lineární poèet operací.
+
+My se s touto èasovou slo¾itostí ov¹em nespokojíme a budeme postupovat podobnì jako pøi vylep¹ování algoritmu na násobení velkých èísel. Bez újmy na obecnosti pøedpokládejme, ¾e budeme násobit dvì matice typu $n \times n$, kde $n=2^k, k \in \bb N$. Obì tyto matice rozdìlíme na ètvrtiny a tyto èásti postupnì oznaèíme u matice $X$ písmeny $A$, $B$, $C$ a $D$, u matice $Y$ písmeny $P$, $Q$, $R$ a $S$. Z definice násobení matic zjistíme, ¾e ètvrtiny výsledné matice $Z$ mù¾eme zapsat pomocí souèinù èástí násobených matic. Levá horní ètvrtina bude odpovídat výsledku operací $AP+BR$, pravá horní ètvrtina bude $AQ+BS$, levá dolní $CP+DR$ a zbylá $CQ+DS$ (viz obrázek).
+
+\figure{nasobeni-matic-2.eps}{Násobení rozètvrcených matic}{\epsfxsize}
+
+\break
+
+Pøevedli jsme tedy problém násobení ètvercových matic øádu $n$ na násobení ètvercových matic øádu ${n/2}$. Tímto rozdìlováním bychom mohli pokraèovat, dokud bychom se nedostali na matice øádu 1, jejich¾ vynásobení je triviální. Dostali jsme tedy klasický algoritmus typu {\it rozdìl a panuj}. Pomohli jsme si ale nìjak ? V ka¾dém kroku provádíme 8 násobení matic polovièního øádu a navíc konstantní poèet operací na $n^2$ prvcích. Dostáváme tedy rekurentní zápis èasové slo¾itosti:
+
+$$ T(n) = 8T\left({n \over 2}\right) + \O(n^2). $$
+
+Pou¾itím Master Theoremu lehce dojdeme k závìru, ¾e slo¾itost je stále $\Theta(n^3)$, tedy stejná jako pøi násobení matic z definice. Zdánlivì jsme si tedy nepomohli, ale stejnì jako tomu bylo u násobení velkých èísel, i teï mù¾eme zredukovat poèet násobení matic polovièního øádu, které nejvíce ovlivòuje èasovou slo¾itost algoritmu. Není to bohu¾el nic triviálního, a proto si radìji rovnou øekneme správné øe¹ení. Jedná se o Strassenùv algoritmus, který redukuje potøebný poèet násobení na~7, a je¹tì pøed tím, ne¾ si uká¾eme, jak funguje, doká¾eme si, jak nám to s èasovou slo¾itostí vlastnì pomù¾e:
+
+$$ T(n) = 7T\left({n \over 2}\right) + \O(n^2) \Longrightarrow \O(n^{\log_2 7}) = \O(n^{2.808}). $$
+
+Výsledná slo¾itost Strassenova algoritmu je tedy $\O(n^{2.808})$, co¾ je sice malé, ale pro velké matice znatelné zlep¹ení oproti algoritmu vycházejícímu pøímo z~definice.
+
+\s{Lemma:} (vzorce pro násobení blokových matic ve~Strassenovì algoritmu)
+
+\def\\{\noalign{\vskip 7pt}}
+
+$$
+\pmatrix{A & B \cr\\ C & D \cr}
+\cdot
+\pmatrix{P & Q \cr\\ R & S \cr}
+=
+\pmatrix{
+T_1 + T_4 - T_5 + T_7 &
+T_3 + T_5 \cr\\
+T_2 + T_4 &
+T_1 - T_2 + T_3 + T_6 \cr
+},$$
+
+kde:
+
+$$\vbox{\halign{$#$\hfil\qquad &$#$\hfil\qquad \cr
+T_1 = (A+D)\cdot(P+S) & T_5 = (A+B)\cdot S \cr\\
+T_2 = (C+D)\cdot P & T_6 = (C-A)\cdot (P+Q) \cr\\
+T_3 = A\cdot(Q-S) & T_7 = (B-D)\cdot (R+S) \cr\\
+T_4 = D\cdot(R-P) \cr
+}}$$
+
+\medskip
+
+\proof Do ètvercù $4 \times 4$ si napí¹eme znaky $+$ nebo $-$ podle toho, jestli se pøi výpoètu dané matice pøièítá nebo odeèítá pøíslu¹ný souèin dvou matic. Øádky znamenají matice $A$, $B$, $C$ a $D$ a sloupce znaèí matice $P$, $Q$, $R$ a $S$. Pokud se tedy v prvním øádku a prvním sloupci vyskytuje znak $+$, znamená to ¾e pøièteme souèin matic $A$~a~$P$. Nejdøíve si spoèítáme pomocné matice $T_1$ a¾ $T_7$ a z nich pak dopoèítáme, co bude na pøíslu¹ných místech ve výsledné matici.
+
+\def\bbb#1{\vbox to 10pt{\vss\hbox to 10pt{\hss\tenrm #1\hss}\vss}}
+\def\bb#1{\ifx#1.\bbb{$\cdot$}\else\bbb#1\fi}
+\def\zz#1#2#3#4{\bb#1\bb#2\bb#3\bb#4}
+\def\qq#1#2#3#4{{\offinterlineskip\vcenter{\halign{\vrule ##\vrule \cr\noalign{\hrule}\zz#1\cr\zz#2\cr\zz#3\cr\zz#4\cr\noalign{\hrule}}}}}
+
+$$
+T_1 = \qq{+..+}{....}{....}{+..+} \qquad
+T_2 = \qq{....}{....}{+...}{+...} \qquad
+T_3 = \qq{.+.-}{....}{....}{....} \qquad
+T_4 = \qq{....}{....}{....}{-.+.}
+$$
+$$
+T_5 = \qq{...+}{...+}{....}{....} \qquad
+T_6 = \qq{--..}{....}{++..}{....} \qquad
+T_7 = \qq{....}{..++}{....}{..--}
+$$
+
+\medskip
+
+\def\\{\noalign{\vskip 7pt}}
+$$
+\eqalign{
+T_1 + T_4 - T_5 + T_7 &= \qq{+...}{..+.}{....}{....} = AP + BR \cr\\
+T_3 + T_5 &= \qq{.+..}{...+}{....}{....} = AQ + BS \cr\\
+T_2 + T_4 &= \qq{....}{....}{+...}{..+.} = CP + DR \cr\\
+T_1 - T_2 + T_3 + T_6 &= \qq{....}{....}{.+..}{...+} = CQ + DS \cr
+}
+$$
+
+% konec slidu z prednasky
+
+Jak je vidìt, výsledná matice je tvoøena stejnými èástmi jako pøi obyèejném násobení. Touto kapitolou jsme tedy dokázali následující vìtu:
+
+\s{Vìta:}
+Strassenùv algoritmus pro násobení matic $n \times n$ má èasovou slo¾itost
+v nejhor¹ím pøípadì $\O(n^{2.808})$. \qed
+
+\s{Poznámka:}
+Zatím nejlep¹í dokázaný algoritmus má èasovou slo¾itost $\O(n^{2.376})$, $ $ ov¹em s velkou multiplikativní konstantou.
+
+\h{Hledání $k$-tého nejmen¹ího prvku (mediánu)}
+
+V tomto oddílu se budeme zabývat tím, jak co nejrychleji najít v jakékoli posloupnosti $n$ èísel $k$-tý nejmen¹í prvek popøípadì medián. Pro ty, kdo medián neznají, tu máme definici:
+
+\s{Definice:}
+Medián posloupnosti $a_1, a_2,\ldots , a_n$ je takové $m=a_i$, kde nejvý¹e $n/2$ prvkù je men¹ích ne¾ $m$ a nejvý¹e $n/2$ prvkù je vìt¹ích ne¾ $m$.
+
+Nejjednodu¹¹ím øe¹ením by urèitì bylo celou posloupnost nejdøíve setøídit a pak u¾ jednodu¹e vyhmátnout po¾adovaný prvek. To bychom dokázali v celkem slu¹ném èase $\O(n\log n)$, ale u¾ teï mù¾eme prozradit, ¾e to jde v èase $\O(n)$. Jak?
+
+Pou¾ívat budeme metodu {\it rozdìl a panuj}. Nìjakým zpùsobem si zvolíme jeden prvek posloupnosti, který nazveme {\it pivot}. Poté rozdìlíme zadanou posloupnost na~tøi disjunktní mno¾iny. Do první dáme v¹echny prvky men¹í ne¾ pivot, do druhé stejné jako pivot a do tøetí vìt¹í ne¾ pivot. Tímto máme zaji¹tìno, ¾e prvky z první mno¾iny jsou urèitì men¹í ne¾ prvky z druhé a ty ne¾ prvky z tøetí.
+O tom, jak jsou prvky uspoøádány uvnitø tìchto mno¾in, ale nic nevíme.
+
+V posledním kroku na¹eho algoritmu se pak rozhodneme, na kterou mno¾inu svùj algoritmus rekurzivnì zavoláme. Pokud je $k$ men¹í ne¾ velikost první mno¾iny, pokraèujeme v první mno¾inì, pokud je $k$ men¹í ne¾ souèet velikostí první a druhé mno¾iny, pak hledaným prvkem je právì vybraný pivot a algoritmus skonèí, a nakonec pokud ani jedna podmínka splnìna nebyla, pustíme se do hledání ve tøetí mno¾inì, ov¹em u¾ nehledáme $k$-tý nejmen¹í prvek, ale $l$-tý, kde $l$ se rovná $k$ minus velikost prvních dvou mno¾in. Pro vìt¹í názornost zapí¹eme tento algoritmus formálnìji:
+
+{\bo Select($k,X$):} (Hledání $k$-tého nejmen¹ího prvku v mno¾ine $X$)
+\algo
+\:Jestli¾e $\vert X\vert \le 1$, vyøe¹íme triviálnì.
+\:Zvolíme pivota $p \in X$.
+\:Rozdìlíme mno¾inu X na tøi podmno¾iny: $M = \{x \in X; x < p\},$ $ P = \{x \in X; x = p\}, V = \{x \in X; x > p\}$.
+\:Jestli¾e $k \le \vert M\vert$, vrátím výsledek funkce \<Select>($k$, $M$).
+\:Jestli¾e $k \le \vert M\vert + \vert P\vert$, vrátíme pivota $p$.
+\:Jinak vrátíme výsledek funkce \<Select>($k - \vert M\vert - \vert P\vert, V$).
+\endalgo
+
+Na první pohled je vidìt, ¾e se algoritmus zastaví (vstup se v¾dy zmen¹í alespoò o 1) a ¾e vydá v¾dy správný výsledek. Jak je to ov¹em s èasovou slo¾itostí? Rozdìlení do mno¾in a podmínky v druhém a tøetím kroku mají lineární slo¾itost, èemu¾ se nevyhneme. Pøi ne¹»astné volbì pivota se nám mù¾e stát, ¾e poèet rekurencí mù¾e být a¾ $n$, tedy celková slo¾itost v nejhor¹ím pøípadì je $\Theta(n^2)$, èím¾ jsme si oproti prostému setøídìní je¹tì pohor¹ili. Co s tím? Jak je vidìt, velmi dùle¾itá je volba pivota. Tu mù¾eme provést nìkolika zpùsoby:
+
+a) Pivot by se v setøídìné posloupnosti vyskytoval uprostøed, vstup by se tedy stále pùlil. Èasovou slo¾itost vypoèteme z rekurentního zápisu:
+
+$$ T(n) = T\left({n \over 2}\right) + \O(n) = \O\left(n + {n \over 2} + {n \over 4} + \dots\right) = \O(n). $$
+
+To by bylo sice skvìlé, ale nalezení takového pivota je vlastnì vyøe¹ení úlohy hledání mediánu, o co¾ se sna¾íme. Tedy jsme si vùbec nepomohli.
+
+\break
+
+b) Pivot by se v setøídìné posloupnosti náchazel v prostøedních dvou ètvrtinách (nazvìme tento prvek "l¾imedián"). Tím bychom v ka¾dém kroku urèitì odstranili mno¾inu velikosti ètvrtiny vstupu. Èasová slo¾itost tohoto øe¹ení by byla:
+
+$$ T(n) = T\left({3 \over 4}n\right) + \O(n) = \O\left(n + {3 \over 4}n + {9 \over 16}n + \dots\right) = \O(n). $$
+
+Tímto bychom tedy také dosáhli lineární èasové slo¾itosti. Ale jak vybrat pivota tak, aby se nacházel v prostøedních dvou ètvrtinách, aby nám nám to nepokazilo lineární slo¾itost?
+
+c) Pivot bude v¾dy prostøední prvek zadané posloupnosti. V tomto pøípadì je zøejmé, ¾e existují urèité vstupy, pro které bude èasová slo¾itost $\Omega(n^2)$, ale v~prùmìrném pøípadì, bude èasová slo¾itost $\O(n)$, co¾ doká¾eme v pøí¹tí kapitole.
+
+d) Pivot bude náhodnì zvolený prvek zadané posloupnosti. Tímto dosáhneme nìèeho podobného jako v pøedchozím pøípadì, ale u¾ nebude existovat vstup, pro který by èasová slo¾itost byla $\Omega(n^2)$, jen nìkteré prùbìhy tohoto algoritmu na rùzných vstupech budou trvat takto dlouho. Prùmìrná èasová slo¾itost je tedy zase $O(n)$.
+
+My se ale nespokojíme pouze s lineární prùmìrnou èasovou slo¾itostí. Chceme dosáhnout lineární slo¾itosti i v nejhor¹ím pøípadì. Ne¾ si ale prozradíme takové øe¹ení, rozmyslíme si, ¾e u¾ teï známe 3 èasové slo¾itosti, i kdy¾ ne v¹echny je¹tì umíme analyzovat. Je to èasová slo¾itost v nejhor¹ím pøídadì, v prùmìrném pøípadì pøes v¹echny vstupy a v prùmìrném pøípadì pøes v¹echny bìhy programu. V¹imnìte si hlavnì rozdílu posledních dvou.
+
+A nyní u¾ slibované øe¹ení s lineární nejhor¹í èasovou slo¾itostí. Vyjdeme z~mo¾nosti b), tedy pokusíme se najít takového pivota, který by na pøí¹tí krok zaruèenì omezil velikost analyzované mno¾iny. Dosáhneme toho tímto algoritmem:
+
+\s{Volba pivota:}
+\algo
+\:Rozdìlíme vstup na pìtice \dots $\O(n)$
+\:Spoèteme medián ka¾dé pìtice \dots $\O(n)$
+\:Spoèteme medián mediánù pìtic tak, ¾e rekurzivnì zavoláme tentý¾ algoritmus \<Select>($n / 10$, {mediány pìtic}) a to je pivot.
+\endalgo
+
+Abychom dokázali, ¾e tento algoritmus bude mít opravdu lineární èasovou slo¾itost, musíme si nejdøíve dokázat následující lemma:
+
+\s{Lemma:} V ka¾dém kroku vypadne alespoò ${3/10}\cdot n$ prvkù.
+
+\proof Ten provedeme obrázkem. Pøedstavme si vybrané pìtice seøazené podle velikosti od nejvìt¹ího prvku a zakresleme je do sloupcù. Jejich mediány tedy vyplòují prostøední øadu. Tyto pìtice pak seøaïme podle velikosti jejich mediánù (nejmen¹í vlevo). Hledaný pivot se tedy nachází (pokud pøedpokládáme pro jednoduchost lichý poèet pìtic) pøesnì uprostøed. O prvcích nad pivotem a napravo od nìj mù¾eme urèitì øíct, ¾e jsou vìt¹í nebo rovny pivotu, prvky pod ním a nalevo od nìj jsou zase urèitì men¹í nebo rovny pivotu.
+
+Podle konstrukce algoritmu tedy zaruèenì vypadne jedna nebo druhá skupina prvkù. Obì tyto skupiny pøitom obsahují, jak je vidìt z obrázku, alespoò $3/10 \cdot n$ prvkù.
+\qed
+
+\figure{petice.eps}{Pìtice}{125mm}
+
+Nyní u¾ se tedy mù¾eme pustit do výpoètu èasové slo¾itosti. V ka¾dém kroku funkce zavolá sama sebe nejdøíve na vstup velikosti ${n \over 5}$ a poté na vstup velikosti nejvý¹e ${7 \over 10}n$. Ostatní operace zùstávají lineární. Èasovou slo¾itost v nejhor¹ím pøípadì tedy mù¾eme zapsat rekurentním vzorcem:
+
+$$ T(n) = \O(n) + T\left({n \over 5}\right) + T\left({7 \over 10}n\right). $$
+
+Tento rekurentní vzorec ale zatím neumíme obecnì øe¹it. Mohli bychom ho postupnì rozepsat, ale trvalo by dlouho, ne¾ bychom z toho nìco vykoukali. Lep¹í bude rovnou dokázat, ¾e tento rekurentní vzorec odpovídá lineární slo¾itosti, zkusíme tedy dosadit $T(n) = cn$:
+
+$$ cn = n + {1 \over 5}cn + {7 \over 10}cn \quad\Longleftrightarrow\quad c = 10. $$
+
+Tímto jsme tedy dokázali, ¾e èasová slo¾itost tohoto algoritmu v nejhor¹ím pøípadì je $\O(n)$.
+Jeliko¾ to rychleji urèitì není mo¾né, mù¾eme na závìr této kapitoly zformulovat tuto vìtu:
+
+\s{Vìta:} Hledání k-tého nejmen¹ího prvku v posloupnosti délky n má èasovou slo¾itost v nejhor¹ím pøípadì $\Theta(n)$. \qed
+
+\bye
--- /dev/null
+P=3-strassen
+
+include ../Makerules
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+sub neg/$K xd _ 1 0 0 0/$C xd/$M xd/$Y xd/$T xd/$N xd}bd/k{1 index type\r
+/stringtype eq{/$t xd/$n xd}{/$t 0 def/$n _ def}ifelse/$k xd/$y xd/$m xd/$c xd\r
+/$fil 0 def}bd/K{1 index type/stringtype eq{/$T xd/$N xd}{/$T 0 def/$N _ def}\r
+ifelse/$K xd/$Y xd/$M xd/$C xd}bd/x/k ld/X/K ld/sf{1 index type/stringtype eq{\r
+/s1t xd/s1n xd}{/s1t 0 def/s1n _ def}ifelse/s1k xd/s1y xd/s1m xd/s1c xd}bd/i{\r
+dup 0 ne{setflat}{pop}ifelse}bd/v{4 -2 roll 2 copy 6 -2 roll c}bd/V/v ld/y{2\r
+copy c}bd/Y/y ld/@w{matrix rotate/$ptm xd matrix scale $ptm dup concatmatrix\r
+/$ptm xd 1 eq{$ptm exch dup concatmatrix/$ptm xd}if 1 w}bd/@g{1 eq dup/$sdf xd\r
+{/$scp xd/$sca xd/$scf xd}if}bd/@G{1 eq dup/$SDF xd{/$SCP xd/$SCA xd/$SCF xd}\r
+if}bd/@D{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if 3 copy exch $Scra add exch\r
+load SetScr/$dsp xd/$dsa xd/$dsf xd}bd/$ngx{$SDF{$SCF SepMode_5 0 eq{$SCA}\r
+{$dsa}ifelse $SCP @ss}if}bd/@MN{2 copy le{pop}{exch pop}ifelse}bd/@MX{2 copy ge\r
+{pop}{exch pop}ifelse}bd/InRange{3 -1 roll @MN @MX}bd/@sqr{dup 0 rl dup 0 exch\r
+rl neg 0 rl @cp}bd/currentscale{1 0 dtransform matrix defaultmatrix idtransform\r
+dup mul exch dup mul add sqrt 0 1 dtransform matrix defaultmatrix idtransform\r
+dup mul exch dup mul add sqrt}bd/@unscale{}bd/wDstChck{2 1 roll dup 3 -1 roll\r
+eq{1 add}if}bd/@dot{dup mul exch dup mul add 1 exch sub}bd/@lin{exch pop abs 1\r
+exch sub}bd/cmyk2rgb{3{dup 5 -1 roll add 1 exch sub dup 0 lt{pop 0}if exch}\r
+repeat pop}bd/rgb2cmyk{3{1 exch sub 3 1 roll}repeat 3 copy @MN @MN 3{dup 5 -1\r
+roll sub neg exch}repeat}bd/rgb2g{2 index .299 mul 2 index .587 mul add 1 index\r
+.114 mul add 4 1 roll pop pop pop}bd/WaldoColor_5 where{pop}{/SetRgb\r
+/setrgbcolor ld/GetRgb/currentrgbcolor ld/SetGry/setgray ld/GetGry/currentgray\r
+ld/SetRgb2 systemdict/setrgbcolor get def/GetRgb2 systemdict/currentrgbcolor\r
+get def/SetHsb systemdict/sethsbcolor get def/GetHsb systemdict\r
+/currenthsbcolor get def/rgb2hsb{SetRgb2 GetHsb}bd/hsb2rgb{3 -1 roll dup floor\r
+sub 3 1 roll SetHsb GetRgb2}bd/setcmykcolor where{pop/LumSepsDict where{pop\r
+/SetCmyk_5{LumSepsDict/setcmykcolor get exec}def}{/AldusSepsDict where{pop\r
+/SetCmyk_5{AldusSepsDict/setcmykcolor get exec}def}{/SetCmyk_5/setcmykcolor ld\r
+}ifelse}ifelse}{/SetCmyk_5{cmyk2rgb SetRgb}bd}ifelse/currentcmykcolor where{\r
+pop/GetCmyk/currentcmykcolor ld}{/GetCmyk{GetRgb rgb2cmyk}bd}ifelse\r
+/setoverprint where{pop}{/setoverprint{/$op xd}bd}ifelse/currentoverprint where\r
+{pop}{/currentoverprint{$op}bd}ifelse/@tc_5{5 -1 roll dup 1 ge{pop}{4{dup 6 -1\r
+roll mul exch}repeat pop}ifelse}bd/@trp{exch pop 5 1 roll @tc_5}bd\r
+/setprocesscolor_5{SepMode_5 0 eq{SetCmyk_5}{0 4 $ink_5 sub index exch pop 5 1\r
+roll pop pop pop pop SepsColor true eq{$ink_5 3 gt{1 sub neg SetGry}{0 0 0 4\r
+$ink_5 roll SetCmyk_5}ifelse}{1 sub neg SetGry}ifelse}ifelse}bd\r
+/findcmykcustomcolor where{pop}{/findcmykcustomcolor{5 array astore}bd}ifelse\r
+/Corelsetcustomcolor_exists false def/setcustomcolor where{pop\r
+/Corelsetcustomcolor_exists true def}if CorelIsSeps true eq CorelIsInRIPSeps\r
+false eq and{/Corelsetcustomcolor_exists false def}if\r
+Corelsetcustomcolor_exists false eq{/setcustomcolor{exch aload pop SepMode_5 0\r
+eq{pop @tc_5 setprocesscolor_5}{CurrentInkName_5 eq{4 index}{0}ifelse 6 1 roll\r
+5 rp 1 sub neg SetGry}ifelse}bd}if/@scc_5{dup type/booleantype eq{dup\r
+currentoverprint ne{setoverprint}{pop}ifelse}{1 eq setoverprint}ifelse dup _ eq\r
+{pop setprocesscolor_5 pop}{findcmykcustomcolor exch setcustomcolor}ifelse\r
+SepMode_5 0 eq{true}{GetGry 1 eq currentoverprint and not}ifelse}bd/colorimage\r
+where{pop/ColorImage{colorimage}def}{/ColorImage{/ncolors xd/$multi xd $multi\r
+true eq{ncolors 3 eq{/daqB xd/daqG xd/daqR xd pop pop exch pop abs{daqR pop\r
+daqG pop daqB pop}repeat}{/daqK xd/daqY xd/daqM xd/daqC xd pop pop exch pop abs\r
+{daqC pop daqM pop daqY pop daqK pop}repeat}ifelse}{/dataaq xd{dataaq ncolors\r
+dup 3 eq{/$dat xd 0 1 $dat length 3 div 1 sub{dup 3 mul $dat 1 index get 255\r
+div $dat 2 index 1 add get 255 div $dat 3 index 2 add get 255 div rgb2g 255 mul\r
+cvi exch pop $dat 3 1 roll put}for $dat 0 $dat length 3 idiv getinterval pop}{\r
+4 eq{/$dat xd 0 1 $dat length 4 div 1 sub{dup 4 mul $dat 1 index get 255 div\r
+$dat 2 index 1 add get 255 div $dat 3 index 2 add get 255 div $dat 4 index 3\r
+add get 255 div cmyk2rgb rgb2g 255 mul cvi exch pop $dat 3 1 roll put}for $dat\r
+0 $dat length ncolors idiv getinterval}if}ifelse}image}ifelse}bd}ifelse\r
+/setcmykcolor{1 5 1 roll _ currentoverprint @scc_5/$ffpnt xd}bd\r
+/currentcmykcolor{GetCmyk}bd/setrgbcolor{rgb2cmyk setcmykcolor}bd\r
+/currentrgbcolor{currentcmykcolor cmyk2rgb}bd/sethsbcolor{hsb2rgb setrgbcolor}\r
+bd/currenthsbcolor{currentrgbcolor rgb2hsb}bd/setgray{dup dup setrgbcolor}bd\r
+/currentgray{currentrgbcolor rgb2g}bd/InsideDCS false def/IMAGE/image ld/image\r
+{InsideDCS{IMAGE}{/EPSDict where{pop SepMode_5 0 eq{IMAGE}{dup type/dicttype eq\r
+{dup/ImageType get 1 ne{IMAGE}{dup dup/BitsPerComponent get 8 eq exch\r
+/BitsPerComponent get 1 eq or currentcolorspace 0 get/DeviceGray eq and{\r
+CurrentInkName_5(Black)eq{IMAGE}{dup/DataSource get/TCC xd/Height get abs{TCC\r
+pop}repeat}ifelse}{IMAGE}ifelse}ifelse}{2 index 1 ne{CurrentInkName_5(Black)eq\r
+{IMAGE}{/TCC xd pop pop exch pop abs{TCC pop}repeat}ifelse}{IMAGE}ifelse}\r
+ifelse}ifelse}{IMAGE}ifelse}ifelse}bd}ifelse/WaldoColor_5 true def/$fm 0 def\r
+/wfill{1 $fm eq{fill}{eofill}ifelse}bd/@Pf{@sv SepMode_5 0 eq $Psc 0 ne or\r
+$ink_5 3 eq or{0 J 0 j[]0 d $t $c $m $y $k $n $o @scc_5 pop $ctm setmatrix 72\r
+1000 div dup matrix scale dup concat dup Bburx exch Bbury exch itransform\r
+ceiling cvi/Bbury xd ceiling cvi/Bburx xd Bbllx exch Bblly exch itransform\r
+floor cvi/Bblly xd floor cvi/Bbllx xd $Prm aload pop $Psn load exec}{1 SetGry\r
+wfill}ifelse @rs @np}bd/F{matrix currentmatrix $sdf{$scf $sca $scp @ss}if $fil\r
+1 eq{CorelPtrnDoFill}{$fil 2 eq{@ff}{$fil 3 eq{@Pf}{$fil 4 eq\r
+{CorelShfillDoFill}{$t $c $m $y $k $n $o @scc_5{wfill}{@np}ifelse}ifelse}\r
+ifelse}ifelse}ifelse $sdf{$dsf $dsa $dsp @ss}if setmatrix}bd/f{@cp F}bd/S{\r
+matrix currentmatrix $ctm setmatrix $SDF{$SCF $SCA $SCP @ss}if $T $C $M $Y $K\r
+$N $O @scc_5{matrix currentmatrix $ptm concat stroke setmatrix}{@np}ifelse $SDF\r
+{$dsf $dsa $dsp @ss}if setmatrix}bd/s{@cp S}bd/B{@gs F @gr S}bd/b{@cp B}bd/_E{\r
+5 array astore exch cvlit xd}bd/@cc{currentfile $dat readhexstring pop}bd/@sm{\r
+/$ctm $ctm currentmatrix def}bd/@E{/Bbury xd/Bburx xd/Bblly xd/Bbllx xd}bd/@c{\r
+@cp}bd/@P{/$fil 3 def/$Psn xd/$Psc xd array astore/$Prm xd}bd/tcc{@cc}def/@B{\r
+@gs S @gr F}bd/@b{@cp @B}bd/@sep{CurrentInkName_5(Composite)eq{/$ink_5 -1 def}\r
+{CurrentInkName_5(Cyan)eq{/$ink_5 0 def}{CurrentInkName_5(Magenta)eq{/$ink_5 1\r
+def}{CurrentInkName_5(Yellow)eq{/$ink_5 2 def}{CurrentInkName_5(Black)eq\r
+{/$ink_5 3 def}{/$ink_5 4 def}ifelse}ifelse}ifelse}ifelse}ifelse}bd/@whi{@gs\r
+-72000 dup m -72000 72000 l 72000 dup l 72000 -72000 l @cp 1 SetGry fill @gr}\r
+bd/@neg{[{1 exch sub}/exec cvx currenttransfer/exec cvx]cvx settransfer @whi}\r
+bd/deflevel 0 def/@sax{/deflevel deflevel 1 add def}bd/@eax{/deflevel deflevel\r
+dup 0 gt{1 sub}if def deflevel 0 gt{/eax load}{eax}ifelse}bd/eax{{exec}forall}\r
+bd/@rax{deflevel 0 eq{@rs @sv}if}bd systemdict/pdfmark known not{/pdfmark\r
+/cleartomark ld}if/wclip{1 $fm eq{clip}{eoclip}ifelse}bd\r
+% Copyright (c)1992-2003 Corel Corporation\r
+% All rights reserved. v12 r0.0\r
+/z{exch findfont exch scalefont setfont}bd/ZB{9 dict dup begin 4 1 roll\r
+/FontType 3 def/FontMatrix xd/FontBBox xd/Encoding 256 array def 0 1 255{\r
+Encoding exch/.notdef put}for/CharStrings 256 dict def CharStrings/.notdef{}\r
+put/Metrics 256 dict def Metrics/.notdef 3 -1 roll put/BuildChar{exch dup\r
+/$char exch/Encoding get 3 index get def dup/Metrics get $char get aload pop\r
+setcachedevice begin Encoding exch get CharStrings exch get end exec}def end\r
+definefont pop}bd/ZBAddChar{findfont begin dup 4 1 roll dup 6 1 roll Encoding 3\r
+1 roll put CharStrings 3 1 roll put Metrics 3 1 roll put end}bd/Z{findfont dup\r
+maxlength 2 add dict exch dup{1 index/FID ne{3 index 3 1 roll put}{pop pop}\r
+ifelse}forall pop dup dup/Encoding get 256 array copy dup/$fe xd/Encoding exch\r
+put dup/Fontname 3 index put 3 -1 roll dup length 0 ne{0 exch{dup type 0 type\r
+eq{exch pop}{$fe exch 2 index exch put 1 add}ifelse}forall pop}if dup 256 dict\r
+dup/$met xd/Metrics exch put dup/FontMatrix get 0 get 1000 mul 1 exch div 3\r
+index length 256 eq{0 1 255{dup $fe exch get dup/.notdef eq{pop pop}{5 index 3\r
+-1 roll get 2 index mul $met 3 1 roll put}ifelse}for}if pop definefont pop pop\r
+}bd/CorelIsValidCharpath{pathbbox 3 -1 roll sub abs 0.5 ge 3 1 roll sub abs 0.5\r
+ge and}bd/@ftx{{currentpoint 3 -1 roll(0)dup 3 -1 roll 0 exch put dup @gs true\r
+charpath $ctm setmatrix CorelIsValidCharpath{@@txt}if @gr @np stringwidth pop 3\r
+-1 roll add exch m}forall}bd/@ft{matrix currentmatrix exch $sdf{$scf $sca $scp\r
+@ss}if $fil 1 eq{/@@txt/@pf ld @ftx}{$fil 2 eq{/@@txt/@ff ld @ftx}{$fil 3 eq\r
+{/@@txt/@Pf ld @ftx}{$fil 4 eq{/@@txt/CorelShfillDoFill ld @ftx}{$t $c $m $y $k\r
+$n $o @scc_5{show}{pop}ifelse}ifelse}ifelse}ifelse}ifelse $sdf{$dsf $dsa $dsp\r
+@ss}if setmatrix}bd/@st{matrix currentmatrix exch $SDF{$SCF $SCA $SCP @ss}if $T\r
+$C $M $Y $K $N $O @scc_5{{currentpoint 3 -1 roll(0)dup 3 -1 roll 0 exch put dup\r
+@gs true charpath $ctm setmatrix $ptm concat stroke @gr @np stringwidth pop 3\r
+-1 roll add exch m}forall}{pop}ifelse $SDF{$dsf $dsa $dsp @ss}if setmatrix}bd\r
+/@te{@ft}bd/@tr{@st}bd/@ta{dup @gs @ft @gr @st}bd/@t@a{dup @gs @st @gr @ft}bd\r
+/@tm{@sm concat}bd/e{/t{@te}def}bd/r{/t{@tr}def}bd/o{/t{pop}def}bd/a{/t{@ta}\r
+def}bd/@a{/t{@t@a}def}bd/t{@te}def/T{@np $ctm setmatrix/$ttm matrix def}bd/ddt\r
+{t}def/@t{/$stm $stm currentmatrix def 3 1 roll m $ttm concat ddt $stm\r
+setmatrix}bd/@n{/$ttm exch matrix rotate def}bd/@s{}bd/@l{}bd/_lineorientation\r
+0 def/_bitfont null def/_bitlobyte 0 def/_bitkey null def/_bithibyte 0 def\r
+% Copyright (c)1992-2003 Corel Corporation\r
+% All rights reserved. v12 r0.0\r
+/@ii{concat 3 index 3 index m 3 index 1 index l 2 copy l 1 index 3 index l 3\r
+index 3 index l clip pop pop pop pop}bd/@i{@sm @gs @ii 6 index 1 ne{/$frg true\r
+def pop pop}{1 eq{s1t s1c s1m s1y s1k s1n $O @scc_5/$frg xd}{/$frg false def}\r
+ifelse 1 eq{@gs $ctm setmatrix F @gr}if}ifelse @np/$ury xd/$urx xd/$lly xd\r
+/$llx xd/$bts xd/$hei xd/$wid xd/$dat $wid $bts mul 8 div ceiling cvi string\r
+def $bkg $frg or{$SDF{$SCF $SCA $SCP @ss}if $llx $lly Tl $urx $llx sub $ury\r
+$lly sub scale $bkg{$t $c $m $y $k $n $o @scc_5 pop}if $wid $hei abs $bts 1 eq\r
+{$bkg}{$bts}ifelse[$wid 0 0 $hei neg 0 $hei 0 gt{$hei}{0}ifelse]/tcc load $bts\r
+1 eq{imagemask}{image}ifelse $SDF{$dsf $dsa $dsp @ss}if}{$hei abs{tcc pop}\r
+repeat}ifelse @gr $ctm setmatrix}bd/@I{@sm @gs @ii @np/$ury xd/$urx xd/$lly xd\r
+/$llx xd/$ncl xd/$bts xd/$hei xd/$wid xd $ngx $llx $lly Tl $urx $llx sub $ury\r
+$lly sub scale $wid $hei abs $bts[$wid 0 0 $hei neg 0 $hei 0 gt{$hei}{0}ifelse\r
+]$msimage false eq $ncl 1 eq or{/$dat $wid $bts mul $ncl mul 8 div ceiling cvi\r
+string def/@cc load false $ncl ColorImage}{$wid $bts mul 8 div ceiling cvi $ncl\r
+3 eq{dup dup/$dat1 exch string def/$dat2 exch string def/$dat3 exch string def\r
+/@cc1 load/@cc2 load/@cc3 load}{dup dup dup/$dat1 exch string def/$dat2 exch\r
+string def/$dat3 exch string def/$dat4 exch string def/@cc1 load/@cc2 load\r
+/@cc3 load/@cc4 load}ifelse true $ncl ColorImage}ifelse $SDF{$dsf $dsa $dsp\r
+@ss}if @gr $ctm setmatrix}bd/@cc1{currentfile $dat1 readhexstring pop}bd/@cc2{\r
+currentfile $dat2 readhexstring pop}bd/@cc3{currentfile $dat3 readhexstring pop\r
+}bd/@cc4{currentfile $dat4 readhexstring pop}bd/$msimage false def/COMP 0 def\r
+/MaskedImage false def L2?{/@I_2{@sm @gs @ii @np/$ury xd/$urx xd/$lly xd/$llx\r
+xd/$ncl xd/$bts xd/$hei xd/$wid xd/$dat $wid $bts mul $ncl mul 8 div ceiling\r
+cvi string def $ngx $ncl 1 eq{/DeviceGray}{$ncl 3 eq{/DeviceRGB}{/DeviceCMYK}\r
+ifelse}ifelse setcolorspace $llx $lly Tl $urx $llx sub $ury $lly sub scale 8\r
+dict begin/ImageType 1 def/Width $wid def/Height $hei abs def/BitsPerComponent\r
+$bts def/Decode $ncl 1 eq{[0 1]}{$ncl 3 eq{[0 1 0 1 0 1]}{[0 1 0 1 0 1 0 1]}\r
+ifelse}ifelse def/ImageMatrix[$wid 0 0 $hei neg 0 $hei 0 gt{$hei}{0}ifelse]def\r
+/DataSource currentfile/ASCII85Decode filter COMP 1 eq{/DCTDecode filter}{COMP\r
+2 eq{/RunLengthDecode filter}if}ifelse def currentdict end image $SDF{$dsf $dsa\r
+$dsp @ss}if @gr $ctm setmatrix}bd}{/@I_2{}bd}ifelse/@I_3{@sm @gs @ii @np/$ury\r
+xd/$urx xd/$lly xd/$llx xd/$ncl xd/$bts xd/$hei xd/$wid xd/$dat $wid $bts mul\r
+$ncl mul 8 div ceiling cvi string def $ngx $ncl 1 eq{/DeviceGray}{$ncl 3 eq\r
+{/DeviceRGB}{/DeviceCMYK}ifelse}ifelse setcolorspace $llx $lly Tl $urx $llx sub\r
+$ury $lly sub scale/ImageDataDict 8 dict def ImageDataDict begin/ImageType 1\r
+def/Width $wid def/Height $hei abs def/BitsPerComponent $bts def/Decode $ncl 1\r
+eq{[0 1]}{$ncl 3 eq{[0 1 0 1 0 1]}{[0 1 0 1 0 1 0 1]}ifelse}ifelse def\r
+/ImageMatrix[$wid 0 0 $hei neg 0 $hei 0 gt{$hei}{0}ifelse]def/DataSource\r
+currentfile/ASCII85Decode filter COMP 1 eq{/DCTDecode filter}{COMP 2 eq{\r
+/RunLengthDecode filter}if}ifelse def end/MaskedImageDict 7 dict def\r
+MaskedImageDict begin/ImageType 3 def/InterleaveType 3 def/MaskDict\r
+ImageMaskDict def/DataDict ImageDataDict def end MaskedImageDict image $SDF\r
+{$dsf $dsa $dsp @ss}if @gr $ctm setmatrix}bd/@SetMask{/$mbts xd/$mhei xd/$mwid\r
+xd/ImageMaskDict 8 dict def ImageMaskDict begin/ImageType 1 def/Width $mwid def\r
+/Height $mhei abs def/BitsPerComponent $mbts def/DataSource maskstream def\r
+/ImageMatrix[$mwid 0 0 $mhei neg 0 $mhei 0 gt{$mhei}{0}ifelse]def/Decode[1 0]\r
+def end}bd/@daq{dup type/arraytype eq{{}forall}if}bd/@BMP{/@cc xd UseLevel 3 eq\r
+MaskedImage true eq and{7 -2 roll pop pop @I_3}{12 index 1 gt UseLevel 2 eq\r
+UseLevel 3 eq or and{7 -2 roll pop pop @I_2}{11 index 1 eq{12 -1 roll pop @i}{\r
+7 -2 roll pop pop @I}ifelse}ifelse}ifelse}bd\r
+end
+%%EndResource\r
+%%EndProlog\r
+%%BeginSetup\r
+wCorel12Dict begin\r
+@BeginSysCorelDict\r
+2.6131 setmiterlimit\r
+1.00 setflat\r
+/$fst 128 def\r
+%%BeginResource: font TimesNewRomanPSMT-NormalItalic\r
+%!FontType1-1.0: TimesNewRomanPSMT-NormalItalic 001.003\r%%Creator: Corel PostScript Engine\r10 dict begin\r/FontName /TimesNewRomanPSMT-NormalItalic def\r/PaintType 0 def\r/FontType 1 def\r/FontMatrix [0.001 0 0 0.001 0 0] readonly def\r/Encoding 256 array 0 1 255 {1 index exch /.notdef put} for\rdup 77 /M put\rdup 101 /e put\rdup 100 /d put\rdup 105 /i put\rdup 225 /aacute put\rdup 110 /n put\rdup 121 /y put\rdup 112 /p put\rdup 116 /t put\rdup 99 /c put\rdup 72 /H put\rdup 108 /l put\rdup 97 /a put\rdup 253 /yacute put\rdup 118 /v put\rdup 111 /o put\rreadonly def\r/FontBBox {0 0 0 0} readonly def\rcurrentdict end\rcurrentfile eexec\r\r
+A22DD33CB9A1B84FC323D538B9AE6C6014672C02872FAD31037218C4EC2B7124C58AFC4A0E2584B50A778936CFE1053450FEC35486F87A4DA48EF5124EE42DE6\r
+9DDB8A5C33C2868DDADC1C9B4725483A678DFD1BEF77D7BDC50D39DB17FF02031F39030455C675716EB1B292EA6078E9937BB936698E457C365396CC5708EAAB\r
+921AD0271E16D4A5F1689C7D8DEDA69051F9EA8B689EDEA8949F2C93CE777268FD3CE5D1713388D0E33A640C77DFB1D300C88E302BEFDF0083AF060D407FD007\r
+23D3F76465C679461FC0471E7F6EFFFCB5A8E513C1661D98B93E8667005CB8B30093BCB089336AFAB7D61973B6F27AC993F52C52238826E221A63575C2C867DB\r
+E9C0264C99B65994DB79F83B4627E129923C7C8B2B1AAA93324C834C75D1F5601DFFE6B14A5A85B0ACAD953462B5AC8C800841BA8EFB424E9A608C7245B55BBD\r
+BAA431986E188F126C069DFD8AFA988AE03C02B04B4C8AEB5E66664FC066DACEAFDBA51EC23B3497A280F0CD52CECEADFA90F2E9D334C0883ED7936FF52AB11E\r
+9CAA4A29E503DCE59D40A92836D9231EFD0388538350B3BBC2B5D06F1F3E36439DAC9ED1785FEE5B5BD5A9D52D5A5DB0918E8687E45D4E1140BDE9BDF2FD7DC7\r
+B5D09D36455EEC816C190B49AE7C5AC067455643E4FC4F9BA32B0295CC7184B7B1DA5C5C2D055FC9CA0D1D0ADE65750036EA8031AC20D89DA23EF765DC3CFFF8\r
+68D5AEB4CCE654439976E1DE697CC3ABA7F216B347C174B7B54775779670965FA04FAEA4B4427C7D36A49BB15F06F514C40EBA0EA9DA47A3D67F1F6800E179D7\r
+24E5BD03AE75BD68EF48B2C7E1ED09DFD7990BC9A1CF25DAB084E5FE7F0EC25767DD8FED594B680F8065A2D1CFC3500901A23BB3656ECEB83964AE367AA96953\r
+84574040B90B4B4F30EE57F7FAC901F1BACEDF4F17FFDA48AC7982F19F3657C6DCD6C56605D1BB5A0DDE2EDD505AF6B69D9E6FD6B92C9B04C07856E3B11E05D1\r
+ED463A49371CD5D4D3167314E97D74BA2CD0F8DA75A01BF876784C72226D5C8947F3EC864A3338924DD1E13E3B1C97472C7E46DACDD9FFDEBC4251519743F272\r
+ACC820CEB82EEFCBD7E8B36212A7614E5635D5803157D6D417143DEF0EA7C6626C744355B882231B7509DAB611D4856EE63531C33CDA3737A8662F4B2EE5936D\r
+DEA195A6BDBB9625457F0B6EA17B690CAA12017C1CB0980398CD30C5A6D17BA278C65D664293441DE6119436508C8DF6B3DA7595A077D6F596573E323A3B8E13\r
+81C6240F390E9C657471F5BD0FCFA2CDF9A1591AC9B592F228AC9A6E2EEF48D12C3B83221A4EC5B048555F958A83D6E3CEC669B0013D966FD04C11690A0822D9\r
+9C21A1B92B76A40EA5EA3EDF77C4498F0C6D7A2634BEC793700FE35ED4E8BEDAF98FB0DC3AA246CF067B963DED084C12C88EC9E573921F80BBB143F5A0101A44\r
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+\input ../lecnotes.tex
+
+\def\E{{\bb E}}
+
+\prednaska{4}{Prùmìrná èasová slo¾itost}{(R. Cinkais, R. Barczi a R. Vansa)}
+
+\h{První pomoc z~teorie pravdìpodobnosti}
+
+\s{Definice:} {\I Diskrétní pravdìpodobnostní prostor} je uspoøádaná dvojice
+$(\Omega,P)$, kde $\Omega$ je nejvý¹e spoèetná mno¾ina {\I elementárních jevù}
+a $P:\Omega\rightarrow[0,1]$ je funkce, pro kterou platí:
+$$\sum_{x\in\Omega}P(x)=1.$$
+Hodnotám $P(x)$ budeme øíkat {\I pravdìpodobnosti} jednotlivých elementárních jevù.
+
+\s{Pøíklad:} Mno¾ina elementárních jevù pøi vrhu kostkou je $\Omega=\{1,2,3,4,5,6\}$ a pokud je kostka spravedlivá, platí $\forall x\in\Omega:P(x)=1/6$.
+
+\s{Definice:} {\I Jevem} $A$ rozumíme libovolnou mno¾inu $A\subseteq\Omega$. {\I Pravdìpodobnost}
+jevu $A$ poèítáme jako: $$\Pr[A]=\sum_{a\in A}P(a).$$
+
+\s{Pøíklad:}
+Jaká je pravdìpodobnost slo¾eného jevu \uv{na kostce padne sudé èíslo}? Oznaèíme
+$A=\{2,4,6\}\subseteq\Omega$. Hledanou pravdìpodobností je pak
+$P(A)=1/6+1/6+1/6=1/2$.
+
+\s{Definice:} {\I Náhodná promìnná} $X$ je funkce $X:\Omega\rightarrow{\bb
+R}$. {\I Støední hodnotou} náhodné promìnné $X$ rozumíme výraz
+$$\E{}X=\sum_{x\in\Omega}P(x)\cdot X(x).$$
+
+\s{Pøíklad:} Nech» $H$ je náhodná promìnná pøiøazující vrhu kostkou dosa¾enou
+hodnotu, tedy $\forall x\in\Omega:H(x)=x$. Støední (oèekávanou) hodnotou této
+promìnné je $\E{}H= 1/6\cdot1 +1/6\cdot2 +1/6\cdot3 +1/6\cdot4 +1/6\cdot5
++1/6\cdot6=7/2$.
+
+\s{Vìta:} ({\I Linearita støední hodnoty})
+Nech» $A,B:\Omega\rightarrow\bb R$ jsou náhodné promìnné a $\alpha\in\bb R$ je libovolná konstanta. Potom:
+\numlist\ndotted
+\:$\E{}(A+B)=\E{}(A)+\E{}(B)$,
+\:$\E{}(\alpha\cdot A)=\alpha\cdot\E{}(A)$.
+\endlist
+
+\proof
+\numlist\ndotted
+\:Podle definice støední hodnoty je:
+$$\eqalign{
+\E{}(A+B)&=\sum_{x\in\Omega}(A(x)+B(x))\cdot P(x)=\sum_{x\in\Omega}(A(x)\cdot P(x)+B(x)\cdot P(x))=\cr
+ &=\sum_{x\in\Omega}A(x)\cdot P(x)+\sum_{x\in\Omega}B(x)\cdot P(x)=\E{}A+ \E{}B.
+}$$
+\:Opìt pøímo z definice støední hodnoty:
+$$\E{}(\alpha\cdot A)=\sum_{x\in\Omega}\alpha\cdot A(x)\cdot P(x)=\alpha\cdot \sum_{x\in\Omega}A(x)\cdot P(x)=\alpha\cdot\E{}A.$$
+\endlist
+\qed
+
+\s{Pøíklad:}
+Házíme dvìmi odli¹nými hracími kostkami. Uva¾ovaný prostor elementárních jevù
+je proto $\Omega=\{(1,1),(1,2),\dots, (6,5),(6,6)\}$ a $\forall
+(a,b)\in\Omega:P((a,b))=1/36$. Mìjme náhodnou promìnnou $S$ pøiøazující vrhu
+kostkami souèet dosa¾ených hodnot, tedy $\forall (a,b)\in\Omega:S((a,b))=a+b$.
+Chceme urèit $\E{}S$.
+
+Oznaème $A$ náhodnou promìnnou, která pøi vrhu kostkami vrací hodnotu na první
+kostce, tedy $\forall (a,b)\in\Omega:A((a,b))=a$. Podobnì oznaème $B$ náhodnou
+promìnnou, která vrací hodnotu na druhé kostce, tedy $\forall
+(a,b)\in\Omega:B((a,b))=b$. V~pøedcházejícím pøíkladì jsme si ukázali, ¾e
+$\E{}A=\E{}B=7/2$. Proto¾e $S((a,b))=a+b=A((a,b))+B((a,b))$, pou¾itím vìty
+o linearitì støední hodnoty dostáváme $\E{}S=\E{}(A+B)=\E{}A+\E{}B=7/2+7/2=7$.
+
+\s{Poznámka:}
+Pro ka¾dou náhodnou promìnnou $Y$ platí: $$\E{}Y=\sum_{x\in{\bb R}}x\cdot\Pr[Y=x].$$
+\proof
+Uvedený vztah plyne okam¾itì z definice støední hodnoty, slouèíme-li dohromady v¹echny èleny $x\in\Omega$ se stejnou hodnotou $Y(x)$.
+\qed
+
+\s{Prùmìrná slo¾itost algoritmu}
+
+Prùmìrnou slo¾itost algoritmu mù¾eme poèítat pøes v¹echny vstupy nebo pøes náhodné
+volby vstupù. V prvním pøípadì je algoritmus pravdìpodobnostní (\uv{hází mincí}) a slo¾itost poèítáme jako prùmìr pøes v¹echny vstupy.
+Ve druhém je zase algoritmus deterministický a zajímá nás pouze slo¾itost v nejhor¹ím pøípadì.
+Pokud jde o koneèný výsledek, jsou oba uvedené zpùsoby ekvivalentní.
+
+Na¹im cílem bude odvodit prùmìrnou slo¾itost algoritmu $\<Select>(k,X)$ z~minulé pøedná¹ky.
+K~tomu si uká¾eme algoritmus slou¾ící k výbìru pivota pro $\<Select>(k,X)$ ($\<Select>(k,X)$ jej pøi svém bìhu
+bude spou¹tìt jako podprogram). Bez újmy na~obecnosti pøedpokládejme,
+¾e v¹echny prvky mno¾iny $X$ jsou navzájem rùzné.
+
+\s{Algoritmus:} ({\I Hledání l¾imediánu v mno¾inì $X$})
+\algo
+\:Zvol náhodnì $x\in X$ (v¹echna~$x$ se stejnou pravdìpodobností).
+\:Spoèítej $A=\vert \{y\in X:y<x\} \vert$ a $B=\vert\{y\in X: y>x\}\vert$.
+\:Pokud $A\geq \vert X\vert/4$ a~zároveò $B\geq \vert X\vert/4$, vra» $x$, jinak skoè na~1.
+\endalgo
+
+\s{Pozorování:}
+Uva¾ujme, ¾e házíme spravedlivou mincí. Aby nedo¹lo k ¾ádnému mezinárodnímu sporu, oznaème strany mince jako tuèòák ($T$) a ryba ($R$). Tedy pravdìpodobnost pádu na nìjakou stranu je $1/2$.
+Budeme zkoumat, jaká je støední hodnota èekání na pád tuèòáka $\E{}T$. Mno¾inu v¹ech jevù (posloupností vrhù), které mohou nastat, oznaème jako $\Omega=\{T,RT,RRT,\dots,RRR\dots\}$. Platí:
+$$\Pr[T]={1\over 2}, \Pr[RT]={1\over 4}, \Pr[R^{k}T]={1\over 2^{k+1}}, \Pr[RRR\dots]=\lim_{k\rightarrow\infty}{1\over 2^k}=0.$$
+Podle pozorování máme: $$\E{}T=\sum_{k=1}^{\infty}k\cdot\underbrace{\Pr[T=k]}_{1\over 2^k}=\sum_{k=1}^{\infty}{k\over 2^k}.$$
+Tuto sumu je v¹ak trochu obtí¾né spoèítat (kdo chce si to mù¾e zkusit), proto provedeme následující trik:
+$$\E{}T=1+{1\over 2}\cdot0+{1\over 2}\cdot\E{}T,$$
+to znamená, ¾e po jednom tahu jsme s polovièní pravdìpodobností vyhráli
+a s polovièní pravdìpodobností se situace nezmìnila. Jednoduchou úpravou tedy
+zjistíme, ¾e: $$\E{}T=2.$$
+
+\s{Lemma:}
+Èekání na jev, který nastane s pravdìpodobností $p$, trvá prùmìrnì $1/p$.
+
+\proof
+$\E{}T=1+(1-p)\E{}T$, tedy $\E{}T=1/p$.
+\qed
+
+Pøi hledání l¾imediánu máme polovièní ¹anci, ¾e se trefíme pøi náhodném výbìru
+do prostøedních dvou ètvrtin -- prùmìrnì tedy vykonáme dva prùchody cyklem, pøièem¾
+jeden prùchod trvá $\Theta(n)$. Celkem bude tedy hledání l¾imediánu trvat prùmìrnì
+$\Theta(n)$.
+
+\s{Vìta:}
+\<Select> s~náhodnou volbou pivota má v prùmìru èasovou slo¾itost $\Theta(n)$.
+
+\proof
+Rozdìlíme algoritmus na {\I fáze} $\equiv$ posloupnosti iterací konèící dobrou
+volbou pivota (jako l¾imedián). V¹imneme si, ¾e jedna fáze zmen¹í vstup na nejvý¹e ${3\over 4}\cdot n$
+a~trvá v~prùmìru $\Theta(n)$ (prùmìrnì 2~iterace po $\Theta(n)$).
+Tedy $\E{}T=\E{}T_1+\E{}T_2+{\dots}=\Theta(n)$, kde $\E{}T_i$ oznaèuje $i$-tou fázi.
+\qed
+
+\s{Vìta:}
+\<Select> s~pivotem rovným prostøednímu prvku má v~prùmìru pøes permutace na vstupu èasovou slo¾itost $\Theta(n)$.
+
+\proof
+Uká¾eme, ¾e algoritmus s~pevnou volbou pivota se na~náhodném vstupu chová
+stejnì, jako algoritmus s~náhodnou volbou pivota na~pevném vstupu, jeho¾
+prùmìrnou slo¾itost odhaduje pøedchozí vìta.
+
+Pokud je na~vstupu náhodná permutace~$\pi$, jsou v¹echny hodnoty pivota~$p$ stejnì pravdìpodobné.
+Zbývá ukázat, ¾e dal¹í iterace algoritmu dostane na vstupu prvky vìt¹í nebo men¹í ne¾~$p$
+opìt v~náhodném poøadí (s~rovnomìrným rozdìlením pravdìpodobností).
+K~tomu sestrojíme bijekci mezi mno¾inou v¹ech permutací na~$\{1,\ldots,n\}$ a mno¾inou ètveøic $(p, L, \pi_M, \pi_V)$, kde:
+
+\itemize\ibull
+\:$p\in\{1,\ldots,n\}$ je prostøední prvek permutace~$\pi$ (pivot), tedy $\pi[l]$ le¾ící na~pozici $l=\lfloor {(1+n)/2} \rfloor$,
+\:$L\in{{\{1,{\dots},l-1,l+1,{\dots},n\}} \choose {p-1}}$ je mno¾ina pozic, na~nich¾ jsou
+ umístìny prvky men¹í ne¾~$p$ (ty, které ná¹ algoritmus umístí do~mno¾iny~$M$),
+\:$\pi_M$ permutace na~$\{1,\ldots, p-1\}$ popisující poøadí men¹ích prvkù a
+\:$\pi_V$ permutace na~$\{p+1,\ldots, n\}$, tedy poøadí vìt¹ích prvkù.
+\endlist
+
+Jakmile zvolíme~$p$, jsou volby~$L$, $\pi_M$ a $\pi_V$ navzájem nezávislé. Ka¾dé volbì~$\pi_M$
+tedy odpovídá stejný poèet permutací~$\pi$, a~proto jsou v¹echny~$\pi_M$ stejnì pravdìpodobné.
+Analogicky pro~$\pi_V$.
+
+\qed
+
+\bye
--- /dev/null
+P=4-avg
+
+include ../Makerules
--- /dev/null
+\input ../lecnotes.tex
+
+\prednaska{5}{QuickSort a slo¾itost tøídìní}{(Michal Sta¹a, Jan Návrat)}
+
+Dostaneme posloupnost, její¾ prvky dovedeme porovnávat, a sna¾íme se co
+nejefektivnìji posloupnost setøídit. Uká¾eme si tøídící algoritmus QuickSort
+(pro pøátele QSort nebo QS) zalo¾ený na metodì Rozdìl a panuj:
+
+\s{Algoritmus:} (QuickSort)
+\def\concat{\mathop{\hbox{.}}}
+
+\algo
+\:Pokud $\vert X\vert \leq 1$, vrátíme~$X$.
+\:Vybereme pivota $p \in X$.\foot{pozdìji upøesníme, jak}
+\:$M \leftarrow \{x \in X ; x \le p\}$.
+\:$P \leftarrow \{x \in X ; x = p\}$.
+\:$V \leftarrow \{x \in X ; x \ge p\}$.
+\:Vrátíme $\<Quicksort>(M) \concat P \concat \<Quicksort>(V)$. \foot{Operátor \uv{$\concat$} znaèí zøetìzení seznamù}
+\endalgo
+
+\s{Rozbor:}
+V¹imnìme si, ¾e QS se urèitì zastaví a také ¾e vydá
+správný výsledek. To mù¾eme ovìøit napøíklad indukcí podle $\vert X\vert$.
+
+Podobnì jako u vybírání $k$-tého nejmen¹ího prvku v minulých pøedná¹kách
+i zde èasová slo¾itost závisí hlavnì na volbì pivota. Kdybychom za pivota
+zvolili medián, vy¹la by èasová slo¾itost stejnì jako u MergeSortu:
+ $$ T(n) = 2T(n/2) + O(n) = O(n\log n). $$
+Pokud naopak budeme volit pivota ne¹ikovnì, dostaneme:
+ $$ T(n) = T(n - 1) + \Theta(n) = \Theta(n^2). $$
+V ideálním pøípadì bychom tedy chtìli za pivota zvolit medián, av¹ak jeho pøímým výpoètem
+bychom algoritmus pøíli¹ zpomalili.
+Pou¾ívá se proto mnoho zpùsobù, jak vybrat rychle pivota blízkého mediánu.
+Èasto pou¾ívanou metodou je náhodný výbìr, v praxi realizovaný nìjakým pseudonáhodným
+generátorem. Uká¾eme, ¾e v tomto pøípadì je QS v prùmìrném pøípadì rychlý:
+
+\s{Vìta:} QS s náhodnou volbou pivota má èasovou slo¾itost v prùmìru $\O(n\log n)$.
+
+\s{Poznámka:} Stejnì jako u výbìru $k$-tého nejmen¹ího prvku bychom také
+mohli ukázat, ¾e QS s pevnou volbou pivota spu¹tìný na náhodnou permutaci
+má tuté¾ èasovou slo¾itost. Detaily nicménì vynecháme.
+
+\noindent {\sl Dùkaz vìty:}
+
+Dùkaz provádíme rozdìlením algoritmu na fáze.
+Rekurzivní volání QSortu zle zobrazit jako¾to strom, pøièem¾ fází rozumíme cestu ve stromu volání,
+která sleduje vìt¹í díl a konèí, kdy¾ se povede vybrat za pivota l¾imedián.
+
+\figure{strom-dukaz.eps}{Dùkaz rozdìlením na fáze}{0.3\hsize}
+
+\figure{Faze.eps}{Fáze}{0.3\hsize}
+
+Ka¾dá fáze pøitom rozdìlí vstup na disjunktní èásti $X_1, \ldots, X_k$ ($k\ge 2$)
+a pivoty, kteøí je oddìlují.
+Oznaème si $n$ za velikost vstupu (poèet prvkù vstupní posloupnosti) a $n_i$ za velikost $i$-té èásti.
+Nahlédneme, ¾e platí $\sum_i n_i \leq n$.
+Velikost ka¾dé èásti je navíc nejvý¹e $3/4 \cdot n$ (na konci fáze to platí proto, ¾e jsme
+zvolili l¾imedián, pøedtím jsme v¾dy oddìlili men¹í z èástí, èili nejvý¹e $n/2$).
+
+Jedna iterace algoritmu trvá $\O(n)$ a jeliko¾ l¾imedián vybereme s pravdìpodobností alespoò $1/2$,
+je v jedné fázi v prùmìru $\O(1)$ iterací a celá fáze proto v prùmìru trvá èas $\O(n)$.
+Z toho dostaneme následující rekurenci pro prùmìrnou èasovou slo¾itost celého algoritmu:
+$${\bb E}T(n) = \sum_i {\bb E}T (n_i) + \O(n).$$
+
+Tento typ rekurence jsme je¹tì nepotkali a Kuchaøková vìta\foot{MasterTheorem} na ni nezabere,
+ov¹em mù¾eme si pomoci jednoduchou úvahou. Pøedstavme si, ¾e v na¹em stromu
+rekurzivních volání zkomprimujeme ka¾dou fázi do jednoho vrcholu. Tím vznikne
+strom, který odpovídá algoritmu, jen¾ v jedné iteraci v prùmìrnì lineárním
+èase rozdìlí vstup na nìkolik èástí a rekurzivnì se na nì zavolá.
+
+\figure{KomprimovanyStrom.eps}{Komprimovaný Strom}{0.3\hsize}
+
+\figure{Komprimace.eps}{Zpùsob vytvoøení komprimovaného stromu}{0.3\hsize}
+
+Nový strom má logaritmickou hloubku, proto¾e na ka¾dé dal¹í hladinì jsou
+délky posloupností nejvý¹e $3/4$ délek z pøedchozí hladiny. Navíc souèet
+délek pøes ka¾dou hladinu je maximálnì $n$. Proto na ka¾dé hladinì
+trávíme èas v prùmìru $\O(n)$ a v celém stromu tedy $\O(n\log n)$.
+\qed
+
+\s{Pozorování:} Na¹e první verze QS spotøebuje lineární mno¾ství pamìti
+na pomocné pole a na zásobník. Na pøedná¹ce jsme ukazovali rùzná jeho
+praktická vylep¹ení, které staèí pomocná pamì» o velikosti $\O(\log n)$.
+Detaily viz webové stránky pøedná¹ky.
+
+Známe u¾ nìkolik tøídících algoritmù s èasovou slo¾itostí $\O(n\log n)$.
+Následující vìta ukazuje, ¾e efektivnìj¹í algoritmus v obecném pøípadì
+nese¾eneme.
+
+\s{Vìta:}
+Ka¾dý tøídící algoritmus zalo¾ený na porovnávání a prohazování prvkù
+potøebuje na vstup délky $n$ v nejhor¹ím pøípadì $\Omega (n \log n)$ porovnání.
+
+\proof
+Bez újmy na obecnosti budeme nejdøíve pøedpokládat o algoritmu dvì vìci:
+Jednak to, ¾e algoritmus nejprve porovnává, a teprve potom prohazuje.\foot{Algoritmus
+mù¾eme upravit tak, aby si pamatoval aktuální permutaci prvkù a podle ní prohazoval a¾ na konci.}
+Také pøedpokládáme, ¾e vstup algoritmu je permutace na mno¾inì $\{1, \ldots, n\}$.
+
+Chování tohoto algoritmu popí¹eme rozhodovacím stromem. V rozhodovacím stromu vnitøní vrcholy
+urèují jednotlivá porovnání prvkù a listy odpovídají okam¾ikùm, kdy algoritmus pøestal porovnávat a zaèal prohazovat.
+
+\figure{RozhodovaciStrom.eps}{Rozhodovací Strom}{0.3\hsize}
+
+Vstup je tedy permutace $n$ prvkù, a víme ¾e poèet rùzných permutací je $n!$. Existuje tedy právì $n!$ rùzných vstupù.
+Dále si v¹imneme, ¾e nemohou existovat dvì vstupní permutace, pro které by algoritmus skonèil v tém¾e listu rozhodovacího stromu.
+Listù stromu je tedy alespoò tolik, kolik je rùzných vstupù, tedy $n!$.
+
+{\narrower
+ \s{Lemmátko:} Binární strom hloubky $k$ má nejvý¹e $2^k$ listù.
+ \par\noindent {\sl Dùkazík:} Uva¾me binární strom hloubky $k$ s maximálním poètem
+ listù. V takovém stromu budou v¹echny listy urèitì le¾et na poslední hladinì
+ (kdyby nele¾ely, mù¾eme pod nìkterý list na vy¹¹í hladinì pøidat dal¹í dva vrcholy a získat
+ tak \uv{listnatìj¹í} strom stejné hloubky). Jeliko¾ na $i$-té hladinì je nejvý¹e $2^i$
+ vrcholù, v¹ech listù je nejvý¹e $2^k$.
+ \qed
+}
+
+\>Z~tohoto lemmátka plyne, ¾e rozhodovací strom musí být hluboký alespoò $\log n!$.
+
+\>Zbytek je u¾ snadné cvièení z diskrétní matematiky:
+
+{\narrower
+ \s{Lemmátko:} $ n! \ge n^{n / 2}$.
+ \par\noindent {\sl Dùkazík:} $n! = \sqrt{(n!)^2} = \sqrt{1(n-1)\cdot 2(n-2) \cdot \ldots \cdot n\cdot 1}$,
+ co¾ mù¾eme také zapsat jako $\sqrt{1(n-1)}\cdot \sqrt{2(n-2)} \cdot \ldots \cdot \sqrt{n\cdot 1}$.
+ Pøitom pro ka¾dé $1\le k\le n$ je $k(n+1-k) = kn + k - k^2 = n + (k-1)n + k(1-k) = n + (k-1)(n-k) \ge n$.
+ Proto je ka¾dá z~odmocnin vìt¹í nebo rovna $n^{1/2}$ a $n!\ge (n^{1/2})^n = n^{n/2}$.
+ \qed
+}
+
+\>Hloubka stromu tedy èiní minimálnì $\log n! \ge \log(n^{n/2}) = n/2 \cdot \log n = \Omega(n\log n)$,
+co¾ také zdola odhaduje poèet porovnání, který algoritmus provede v nejhor¹ím pøípadì.
+\qed
+
+\bye
--- /dev/null
+P=5-qs
+
+include ../Makerules
--- /dev/null
+\input ../lecnotes.tex
+
+\prednaska{6}{Tøídìní v lineárním èase}{(zapsal M.~Kupec, J.~Volec, J.~Ivánek)}
+
+Na minulých pøedna¹kách jsme si ukázali tøídìní v èase $ \O(N\log{N}) $ a také dokázali, ¾e líp to v obecném pøípadì nejde. Jak je tedy mo¾né tøídit v lineárním èase?
+
+V¹echny dosud pøedvedené alrogitmy tøídily pouze pomocí porovnávní hodnot na vstupu. Ale co kdy¾ o vstupu víme víc, tøeba rozsah hodnot na vstup, nedá se toho vyu¾ít?
+
+Následující algoritmy vyu¾ívají znalosti rozsahu hodnot na vstupu a toho, ¾e tento rozsah je malý.
+
+\h{Counting sort}
+
+Counting sort je algoritmus pro tøídìní $N$ celých èísel s maximálním rozsahem hodnot $R$. Tøídí v èase $ \O(N + R) $ s~pamì»ovou nároèností $ \O(R) $.
+
+Algoritmus postupnì prochází vstup a poèítá si ke ka¾dému prvku z~rozsahu kolikrát jej vidìl. Poté a¾ projde celý vstup, projde poèítadla a indexy u~tech, kde nìco napoèítal vydá jako výstup.
+
+\s{Algoritmus:} (tøídìní posloupnosti $ \left ( x_i \right )_0^N $ pomocí {\sc Counting sort}u)
+
+\algo
+\:Pro $ i \leftarrow 1 $ do $R$ opakuj:
+\::$ p_i \leftarrow 0 $
+\:Pro $ i \leftarrow 1 $ do $N$ opakuj:
+\::$ p_{x_i} \leftarrow p_{x_i} + 1 $
+\:$ j \leftarrow 1 $
+\:Pro $ i \leftarrow 1 $ do $R$ opakuj:
+\::Pokud $ p_i \neq 0 $
+\:::$ v_j \leftarrow i $
+\:::$ j \leftarrow j + 1 $
+\:Vra» $v$
+\endalgo
+
+\h{Pøihrádkové tøídìní}
+
+Counting sort nám moc nepomù¾e, pokud chceme tøídit neco jiného ne¾ celá èísla, ale i to se dá øe¹it. Pøihrádkové, popøípadì kbelíkové tøídìní, neboli bucket-sort umí tøídit mno¾inu prvkù oèíslovaných èísly $ 1 \ldots R $.
+
+\>Potøebuje k tomu èas $ \O(N + R) $ a pamì» $ \O(N + R) $.
+
+Bucket sort ma je¹tì jednu pøíjemnou vlastnost: jedná se o stabilní tøídìní.
+
+\s{Definice:} {\it Stabilní tøídìní} je takové, ¾e ka¾dé 2 prvky vstupu se stejnou hodnotou klíèe jsou na výstupu ve stejnìm poøadí jako na vstupu.
+
+\s{Algoritmus:} (tøídìní mno¾iny $ x_1,x_2,\ldots,x_n $ oèíslované $ c_1,c_2,\ldots,c_n $ pomocí {\sc bucket-sort}u)
+\algo
+\:Pro $ k \leftarrow 1 $ do $R$ opakuj:
+\::$ p_k \leftarrow 0 $
+\:Pro $ i \leftarrow 1 $ do $N$ opakuj:
+\::$ p_{c_i} \leftarrow p_{c_i} + 1 $
+\:$ b_1 \leftarrow 1 $
+\:Pro $ k \leftarrow 2 $ do $R$ opakuj:
+\::$ b_k \leftarrow b_{k - 1} + p_{b - 1} $
+\:Pro $ i \leftarrow 1 $ do $N$ opakuj:
+\::$ t \leftarrow c_i $
+\::$ v_{b_t} \leftarrow x_i $
+\::$ b_t \leftarrow b_t + 1 $
+\:Vra» $v$
+\endalgo
+
+\h{Lexikografické tøídìní k-tic}
+Mìjme $n$ uspoøádaných $k$-tic prvkù z mno¾iny $ \{1 \ldots R \}^k $. Úkol zní seøadit \hbox{$k$-tice} slovníkovì (lexikograficky).
+Mù¾eme pou¾ít metodu rozdìl a panuj, tak¾e prvky setøídíme nejprve podle první souøadnice $k$-tic a pak se rekurzivnì zavoláme na ka¾dou pøihrádku a tøídíme podle následující souøadnice.
+% a dal??
+Nebo mù¾eme vyu¾ít toho, ¾e bucket-sort je stabilní a tøídit takto:
+
+\s{Algoritmus: } (tøídìní $k$-tic $ x_1, x_2, \ldots ,x_n $ lexikograficky pomocí {\sc Bucket-sort}u)
+\algo
+\:$ S \leftarrow \{x_1, x_2, \dots, x_n\} $
+\:Pro $ i \leftarrow k $ do $ 1 $ opakuj:
+\::$ S \leftarrow $ bucket-sort $ S $ podle $i$-té souøadnice
+\:vysledek $ S $
+\endalgo
+
+Pro pøehlednost v následujícím pozorování oznaème $ l = k - i + 1 $, co¾ pøesnì odpovídá tomu, v kolikátém prùchodu cyklu jsme.
+
+\s{Pozorování:}
+Po $l$-tém prùchodu cyklem jsou prvky uspoøádány lexikograficky podle $i$-té a¾ $k$-té souøadnice.
+
+\proof
+Indukcí podle $l$
+% FIXME nevim jak presne by mel byt list
+\itemize
+\next Pro $ l = 1 $ jsou prvky uspoøádány podle poslední souøadnice.
+\next Po $l$ prùchodech ji¾ máme prvky setøídìny lexikograficky podle $i$-té a¾ $k$-té souøadnice a spou¹tíme $(l + 1)$-ní prùchod, tj. budeme tøídít podle $(i - 1)$-ní souøadnice.
+Proto¾e bucket-sort tøídí stabilnì, zùstanou prvky se stejnou $(i - 1)$-ní souøadnicí vùèi sobì seøazeny tak, jak byly seøazeny na vstupu.
+Z IP tam v¹ak byly seøazeny lexikograficky podle $i$-té a¾ $k$-té souøadnice. Tudí¾ po $(l + 1)$-ním prùchodu jsou prvky seøazeny podle $(i - 1)$-ní a¾ $k$-té souøadnice.
+\endlist
+\qed
+
+Èasová slo¾itost je $\O( k \cdot (n + R))$, co¾ je lineární s délkou vstupu $(k \cdot n)$ pro pevné $k$ a pamì»ová slo¾itost je $\O(n + R)$.
+
+\h{Tøídìní èísel $ 1 \ldots R $ podruhé (Radix sort)}
+Zvolíme základ $Z$ a èísla zapí¹eme v soustavì o základu $Z$, èím¾ získáme $ ( \lfloor \log_z{R} \rfloor +1) $-tice, na které spustíme pøedcházející algoritmus.
+Díky tomu budeme tøídít v èase $\O({\log{R} \over \log{Z}} \cdot (n + Z))$. A jak zvolit vhodnì $Z$?
+
+Pokud bychom zvolili $ Z = konstanta $, èasová slo¾itost bude $\O(\log{R} \cdot n) $, co¾ mù¾e být $ n * \log{n} $ nebo i víc.
+Zvolíme-li $ Z = n $, dostáváme $ \O({\log{R} \over \log{n}} \cdot n) $, co¾ pro $ R \leq n^\alpha $ znamená $ \O(\alpha \cdot n) $.
+
+\h{Tøídìní øetìzcù}
+Mìjme $n$ øetìzcù $ r_1, r_2 \dots r_n $ dlouhých $ l_1, l_2 \dots l_n $
+Oznaème si $ L = \max_{1 \leq i \leq n} l_i $ délku nejdel¹ího øetìzce a $R$ poèet znakù abecedy. Problém je, ¾e øetìzce nemusí být stejnì dlouhé (pokud by byly, lze se na øetìzce dívat jako na $k$-tice, které u¾ tøídít umíme). S tím se mù¾eme pokusit vypoøádat doplnìním øetìzcù mezerami tak, aby mìly v¹echny stejnou délku, a spustit na nìj algoritmus pro $k$-tice. Tím dostaneme algoritmus, který bude mít èasovou slo¾itost $ \O(L * n) $, co¾ bohu¾el mù¾e být a¾ kvadratické vzhledem k velikosti vstupu.
+Pøíklad: na vstupu mìjme k øetìzcù, kde prvních $k-1$ z nich bude mít délku $1$ a poslední øetìzec bude dlouhý pøesnì $k$. Vstup má tedy celkovou délku $2k-1$ a my teï doplníme prvních $k-1$ øetìzcù mezerami. Vidíme, ¾e algoritmus teï bude pracovat v èase $ \O(k^2) $.
+To, co nám nejvíce zpùsobovalo problémy u pøedchozího pøíkladu, bylo velké mno¾ství èasu zabraného porovnáváním doplnìných mezer. Zkusíme proto øe¹it ná¹ problem trochu chytøeji a koncové mezery do øetìzù vùbec pøidávat nebudeme.
+Nejprve roztøídíme bucket-sortem øetìzce do pøihrádek (mno¾in) $ P_i $ podle jejich délek, kde $i$ znaèí délku øetìzcù v dané pøihrádce, neboli definujme $ P_i = \left\{ r_j \vert l_j = i \right\} $. Dále si zavedeme seznam setøídìných øetìzcù $S$ takový, ¾e v nìm po $k$-tém prùchodu tøídícím cyklem budou øetìzce s délkou alespoò $ L - k + 1 $ (oznaème $l$) a zároveò v nìm tyto øetìzce budou setøídìny lexikograficky od $l$-tého znaku po $L$-tý. Z definice tohoto seznamu je patrné, ¾e po $L$ krocích tøídícího cyklu bude tento seznam obsahovat v¹echny øetìzce a tyto øetìzce v nìm budou lexikograficky seøazeny. Zbývá u¾ jen popsat, jak tento cyklus pracuje. Nejprve vezme $l$-tou mno¾inu $ P_l $ a její øetìzce roztøídí do pøihrádek $ Q_j $ (kde index $j$ znaèí j-tý znak abecedy) podle jejich $l$-tého (neboli posledního) znaku. Dále vezme seznam $S$ a jeho øetìzce pøidá opìt podle jejich $l$-tého znaku do stejných pøihrádek $ Q_j $ za ji¾ døíve pøidané øetìzce z $ P_l $. Na závìr postupnì projde v¹echny pøihrádky $ Q_j $ a øetìzce v nich pøesune do seznamu $S$. Proto¾e øetìzce z pøihrádek $ Q_j $ bude brát ve stejném poøadí, v jakém do nich byly umístìny, a proto¾e ze seznamu $S$, který je setøízený podle $ (l + 1) $-ního znaku po $L$-tý, bude také brát øetìzce postupnì, bude seznam $S$ po $k$-tém prùchodu pøesnì takový, jaký jsme chtìli (indukcí bychom dokázali, ¾e cyklus pracuje skuteènì správnì). Zároveò z popisu algoritmu je jasné, ¾e bìhem tøídìní ka¾dý znak ka¾dého øetìzce pou¾ijeme právì jednou, tudí¾ algoritmus bude lineární s délkou vstupu (pro úplnost dodejme, ¾e popsaný algoritmus funguje v pøípadech, kdy abeceda má pevnou velikost).
+
+Algoritmus: (tøídìní øetìzcù)
+\algo
+\:$L \leftarrow \max(l_1, l_2,\ldots,l_n)$
+\:Pro $ i \leftarrow 1 $ do $L$ opakuj:
+\::$ P_i \leftarrow \emptyset $
+\:Pro $ i \leftarrow 1 $ do $ n $ opakuj:
+\::$ \<pridej>(P_{l_i}, r_i) $
+
+\:$ S \leftarrow \emptyset $
+\:Pro $ i \leftarrow L $ do $ 1 $ opakuj:
+\::Pro $ j \leftarrow 1 $ do $ R $ opakuj:
+\:::$ Q_j \leftarrow \emptyset $
+
+\::Pro $ j \leftarrow 1 $ do velikost($P_i$) opakuj:
+\:::$ \<vezmi>(P_i, r) $
+\:::$ \<pridej>(Q_{r[i]}, r) $
+\::Pro $ j \leftarrow 1 $ do velikost($S$) opakuj:
+\:::$ \<vezmi>(S, r) $
+\:::$ \<pridej>(Q_{r[i]}, r) $
+
+\::$ S \leftarrow \emptyset $
+\::Pro $ j \leftarrow 1 $ do $ R $ opakuj:
+\:::Pro $ k \leftarrow 1 $ do velikost($Q_j$) opakuj:
+\::::$ \<vezmi>(Q_j, r) $
+\::::$ \<pridej>(S, r) $
+
+\:výsledek $S$
+\endalgo
+
+Èasová slo¾itost tohoto algoritmu tedy bude $ \O(RN) $, kde $N$ je délka vstupu a $R$ poèet znakù abecedy.
+
+\h{Sorting Zoo}
+
+% FIXME pridat svisle cary..mozna i vodorovne
+
+$$
+\vbox {
+\halign{$#$ \hfil & \quad \hfil $#$ & \quad \hfil $#$ & \quad \hfil $#$ \cr
+& \O(1) & \O(\log{N}) & \O(N) \cr
+\noalign{\medskip\hrule\bigskip}
+\O(N^2) & Buble & & \cr
+\O(N\log{N}) & Heap & Quick & Merge \cr
+\O(N) & & & Bucket \cr
+}
+}
+$$
+
+\h{K èemu je vlastnì tøídìní dobré?}
+Díky nìmu mù¾eme rychle vyhledávat prvky v mno¾inì, konkrétnì v èase $ \O(\log(n)) $ napø. pomocí pùlení intervalù.
+Dal¹ím problémem, na který se hodí pou¾ít tøídìní, je zji¹tìní, zda se v posloupnosti nìjaké její hodnoty opakují. Dá se dokázat, ¾e tuto úlohu nelze vyøe¹it lépe (rychleji), ne¾ tak, ¾e hodnoty nejprve setøídíme a poté setøidìnou posloupnost projdeme.
+
+Ukázali jsme si tedy, ¾e v se setøízené mno¾inì jsme schopni vyhledávat prvky v èase $ \O(\log(n)) $. Bohu¾el do ní v èase $ \O(\log(n)) $ neumíme prvky pøidávat.
+Jedním ze zpùsobu, jak zrychlit pøidávání nových prvkù a zároveò nepøijít o logaritmické vyhledávání, je pou¾ít dal¹í datouvou strukturu, vyhledávací stromy.
+
+\h{Vyhledávací stromy}
+Problém: chceme udr¾ovat nìjaká data setøízená. To znamená, ¾e chceme udr¾ovat mno¾inu $X$ prvkù z lineárnì uspoøádaného universa $U$ s operacemi Find (najdi prvek), Insert (vlo¾ prvek), Delete (sma¾ prvek). K tomu se nám bude hodit reprezentace dat pomocí stromu.
+
+\s{Definice:} {\sc Binární strom} je zakoøenìný strom; ka¾dý vrchol má max. 2 syny, pravého a levého.
+
+Dále budeme pou¾ívat toto znaèení:
+$v$ ... vrchol
+$l(v)$ ... levý syn, $p(v)$ ... pravý syn
+$L(v)$ ... levý podstrom, $p(v)$ ... pravý podstrom
+$T(v)$ ... podstrom s koøenem $v$.
+
+\s{Definice:} {\sc Binární vyhledávací strom} je takový binární strom, ¾e ka¾dý vrchol obsahuje prvek z $U$
+a pro $ \forall v: \forall x \in L(v): x < v, \forall x \in P(v): x > v $.
+
+\s{Find($T(v)$,$x$)}
+Procházíme strom od koøene, na ka¾dém vrcholu se podle hodnoty $x$ rozhodneme jak budeme pokraèovat. Buï se hodnota vrcholu rovná $x$, pak konèíme (nalezli jsme hledaný prvek). V opaèném pøípadì pokraèujeme v hledání v levém podstromu ($x$ je men¹í ne¾ hodnota vrcholu), nebo v pravém podstromu ($x$ je vìt¹í ne¾ hodnota vrcholu). Pokud dojdeme do listu a prvek jsme nena¹li, znamená to ¾e tento prvek ve stromì není.
+Toto nám zabere $\O($Hloubka stromu$)$.
+
+\s{Insert($T(v)$,$x$)}
+Nejprve najdeme místo kam je potøeba prvek vlo¾it, k tomu pou¾ijeme operaci Find. Pokud Find najde $x$ neprovedeme nic (máme strom s jedineènými hodnotami). Jinak vlo¾íme prvek na místo, kde by mìl být (jako syna vrcholu ve kterém skonèil Find).
+Nejslo¾itìj¹í je operace Find, která trvá $\O($Hloubka stromu$)$.
+
+\s{Min($T(v)$)}
+Minimum se nachází v nejlevìj¹ím listu stromu. Jdeme tedy z koøene poøád doleva, dokud nenarazíme na list.
+Èasová slo¾itost je $\O($hloubka stromu$)$.
+
+\s{Delete($T(v)$,$x$)}
+Opìt pou¾ijeme Find, zjistíme zda prvek vùbec existuje a kde se nachází. Pokud existuje mù¾ou nastat tyto pøípady:
+% FIXME nevim jak presne by mel byt list
+\itemize\ibull
+\next $x$ je list: Prostì ho sma¾eme.
+\next $x$ má jednoho syna: Zru¹íme $x$ a syna pøipojíme na uzel, který ukazoval na $x$.
+\next $x$ má dva syny: Najdeme minimum $P(v)$, toto minimum pøehodíme s $x$ a $x$ sma¾eme.
+\endlist
+Èas je tentokrát malièko slo¾itìj¹í, Find nám trvá $\O($Hloubka stromu$)$ a v posledním pøípadì nalezení Min($P(v)$) trvá taky a¾ $\O($Hloubka stromu$)$, co¾ nám dohromady dává pìkných $\O($Hloubka stromu$)$.
+
+Strom se mù¾e pou¾íváním operací "kazit", napø. opakováním Insert na max. hodnotu co¾ vede a¾ k hloubce $\sim n$. Mù¾eme
+% FIXME nevim jak presne by mel byt list
+\itemize\ibull
+\next vyka¹lat se na to $\rightarrow$ potøeba dokázat prùmìrný pøípad
+\next vylep¹it Insert a Delete $\rightarrow$ vyva¾ování stromù
+\endlist
+
+\bye
--- /dev/null
+P=6-trideni
+
+include ../Makerules
--- /dev/null
+\input ../lecnotes.tex
+
+% Vkladani obrazku
+\input ../mjipe.tex
+\def\treepic#1{
+\medskip
+\IpeInput{treepic/t#1.ipe}
+\medskip
+}
+\def\abpic#1{
+\medskip
+\centerline{\epsfxsize=0.7\hsize\epsfbox{abpic/#1.eps}}
+\medskip
+}
+\def\abpics#1#2{
+\medskip
+\centerline{\epsfxsize=0.4\hsize\epsfbox{abpic/#1.eps}\qquad\epsfxsize=0.4\hsize\epsfbox{abpic/#2.eps}}
+\medskip
+}
+
+\prednaska{7}{Vyhledávací stromy}{zapsali M. Øezáè, ©. Masojídek, B. Urbancová}
+
+\h{Pár obrázkù, které by stály za pøesun do~pøedchozí kapitoly:}
+
+\treepic{1}
+\treepic{2}
+\treepic{3}
+
+\h{Vyvá¾ené binární vyhledávací stromy}
+
+V minulé kapitole jsme se zabývali problematikou pøidávání a ubírání prvkù
+binárního vyhledávacího stromu a jeho slo¾itostí a zjistili, ¾e v¹e zále¾í
+na~hloubce stromu. Víme, ¾e chceme hloubku logaritmickou, ale jak ji mù¾eme
+udr¾et pøi~operacích? Øe¹ením je následující definice:
+
+\s{Definice:} {\I Dokonale vyvá¾ení} je takové vyvá¾ení, ve~kterém pro v¹echny vrcholy~$v$ platí $\left\vert \vert L(v)\vert - \vert P(v)\vert \right \vert \leq 1 $.
+
+Toto nám jistì zaji¹»uje logaritmickou hloubku, ale je velmi pracné na udr¾ování.
+Zkusíme proto slab¹í podmínku:
+
+\s{Definice:} {\I Hloubkové vyvá¾ení} je takové vyvá¾ení, ve~kterém pro v¹echny vrcholy~$v$ platí $\left \vert h(L(v)) - h(P(v)) \right \vert \leq 1 $.
+Stromùm s hloubkovým vyvá¾ením se øíká {\I AVL stromy} a platí o nich následující lemma.
+
+\s{Lemma:} AVL strom o $n$ vrcholech má hloubku $ \O(\log{n}) $.
+
+\proof
+Uva¾me $a_k = $ minimální poèet vrcholù AVL stromu o~hloubce $k$.
+Lehce spoèteme:
+$$\eqalign{
+a_0 &= 0 \cr
+a_1 &= 1 \cr
+a_2 &= 2 \cr
+&\vdots \cr
+a_k &= 1 + a_{k - 1} + a_{k - 2}. \cr
+}$$
+
+Rekurentní vzorec jsme dostali rekurzivním stavìním stromu hloubky $k$: nový koøen a 2 podstromy o hloubce $k - 1$ a $k - 2$.
+
+Indukcí doká¾eme, ¾e $ a_k \geq 2^{k \over 2} $.
+První indukèní krok jsme si u¾ ukázali, teï pro $ k \geq 2 $ platí:
+$ a_k = 1 + a_{k - 1} + a_{k - 2} > 2^{{k - 1} \over 2} + 2^{{k - 2} \over 2} = 2^{k \over 2} \cdot (2^{-{1 \over 2}} + 2^{-1}) \cong 2^{k \over 2} \cdot 1.21 > 2^{k \over 2} $
+
+Tímto jsme dokázali, ¾e na ka¾dé hladinì je minimálnì exponenciálnì vrcholù, co¾ nám zaruèuje hloubku $ \O(\log{n})$
+\qed
+
+\s{Operace s AVL stromy}
+
+\<Find> se neli¹í od~operace find v~binárních stromech.
+
+Dùraz klademe na operace \<Insert> a \<Delete>, proto¾e pøi~nich musíme o¹etøit udr¾ení struktury AVL~stromù.
+
+První nutnou podmínkou je, ¾e si musíme {\I pamatovat stav} v~ka¾dém vrcholu tohoto stromu. A~to {\I vyvá¾ení} hloubky jeho podstromù.
+
+Umluvíme~se napø. na~tomto oznaèení:
+
+\>Dostaneme tøi typy vrcholù, které se mohou v~AVL~stromu vyskytnout:
+\itemize\ibull
+\:{\I Vrchol typu~$\oplus$}, pokud je pravý podstrom hlub¹í
+\:{\I Vrchol typu~$\ominus$}, pokud je levý podstrom hlub¹í a
+\:{\I Vrchol typu~$\odot$ (nulou)}, který má oba syny schodné hloubky.
+\endlist
+
+\s {Sestavení} \rm AVL stromu:
+
+Postupujeme po~struktuøe binárního stromu od~listù ke~koøeni a~kontrolujeme, zda jsou vrcholy v~jednom ze~tøí uvedených stavù. Pokud ne, opravíme ho operací jménem rotace.
+
+\s {Rotace}
+\treepic{4}
+\treepic{5}
+
+Jde o~pøevrácení hrany mezi pùvodním otcem (koøenem podstromu) a nevyvá¾eným vrcholem tak, aby byli i po pøeskupení synové vzhledem k~otcùm správnì uspoøádáni.
+
+
+\s {Insert} \rm - vlo¾ení vrcholu do~AVL~stromu.
+
+Vlo¾íme jej jako list. Nový list má v¾dy \uv{znaménko} nula $\odot$. Pøedpokládáme, ¾e patøí nalevo od posledního otce. Podíváme~se na~znaménko jeho otce:
+\itemize\ibull
+\:{\I mìl~$\odot$ (nemìl syna) $\rightarrow$ teï má~$\ominus$}, po struktuøe stromu nahoru posíláme informaci, ¾e se podstrom prohloubil o~1, co¾ mù¾e mít samozøejmì vliv na~znaménka vrcholù na~cestì ke~koøeni.
+\:{\I mìl~$\oplus$ (mìl pravého syna, který je listem) $\rightarrow$ teï má~$\odot$}, hloubka podstromu se~nemìní
+\:{\I mìl $\ominus$} -- nenastane, proto¾e v binární struktuøe nemohou být dva leví synové
+\endlist
+\>Pøipadne-li pøidaný list napravo, øe¹íme zrcadlovì.
+
+\treepic{6}
+\treepic{7}
+
+\>{\I Prohloubil-li se strom} vlo¾ením nového listu, musíme pracovat s vyvá¾ením:
+\itemize\ibull
+\:Informace o~prohloubení pøi¹la zleva {\I do~vrcholu typu~$\odot$} $\rightarrow$ mìní jej na~vrchol se~znaménkem~$\ominus$ a informace o~prohloubení je tøeba poslat o~úroveò vý¹.
+\:Informace o~prohloubení pøi¹la zleva {\I do~vrcholu typu~$\oplus$} $\rightarrow$ mìní jej na~vrchol se~znaménkem~$\odot$, hloubka je vyrovnána, dál nic neposíláme.
+\:Informace o~prohloubení pøi¹la zleva {\I do vrcholu s~$\ominus$} $\rightarrow$
+
+\>rozebereme na~tøi pøípady podle znaménka vrcholu, ze~kterého pøi¹la informace o~prohloubení:
+\itemize\ibull
+\:Informace pøi¹la {\I z~vrcholu typu~$\ominus$} $\rightarrow$ provedeme rotaci doprava tak, ¾e novým koøenem se~stane vrchol~$y$, ze~kterého pøi¹la informace o~prohloubení.
+
+\treepic{8}
+
+{\I Pozorování 1:} znaménko vrcholù~$y$ a~$x$ je~$\odot$\
+
+{\I Pozorování 2:} hloubka pøed vkládáním byla $h+1$ a~nyní je také $h+1$, tedy nemusíme dále posílat informaci o~prohloubení a mù¾eme skonèit
+\:Informace pøi¹la {\I z~vrcholu typu~$\oplus$}
+\itemize\ibull
+\:uva¾me je¹te vrchol~$z$ jako pravého syna vrcholu~$y$, ze~kterého pri¹la informace o~prohloubení, a~jeho podstromy~$B$ a~$C$
+\:vrcholy~$B$ a~$C$ mají hloubku~$h$ nebo $h-1$ $\rightarrow$ oznaème~ji tedy $h-$ (to zøejmì proto¾e vrchol~$y$ má znaménko~$\oplus$, tedy jeho pravý podstrom s~koøenem~$z$ má hloubku~$h+1$ )
+\:provedeme dvojrotaci tak, ¾e novým koøenem se stane vrchol~$z$
+\endlist
+\treepic{9}
+
+{\I Pozorování 1:} znaménko vrcholu~$z$ bude~$\odot$\
+
+{\I Pozorování 2:} znaménka vrcholu~$x$ a~$y$ se~dopoèítají v~závislosti na~hloubce~$B$ a~$C$\
+
+{\I Pozorování 3:} rozdíl hloubky pravého a~levého podstromu bude u~tìchto vrcholù $0$ nebo~$1$\
+
+{\I Pozorování 4:} hloubka pøed vkládáním byla $h+2$ a~nyní je také $h+2$, tedy nemusíme dále posílat informaci o~prohloubení a~mù¾eme skonèit
+\:informace pøi¹la {\I z~vrcholu typu~$\odot$} -- to nemù¾e nastat, proto¾e v~tom pøípadì by ne¹lo o~prohloubení
+\endlist
+\endlist
+
+\s {Delete} -- odebrání vrcholu z~AVL~stromu
+\> Buï ma¾eme list nebo ma¾eme vrchol, který mìl nìjaké syny.
+
+\itemize\ibull
+\:pokud ma¾eme list, podíváme~se na~typ otce. Pøedpokládáme mazání levého syna.
+\itemize\ibull
+\:byl typu $\ominus$ (nemìl pravého syna) $\rightarrow$ zmìní~se na~$\odot$ (vrchol teï nemá ¾ádné syny)
+\:byl typu $\odot$ (mìl oba syny) $\rightarrow$ zmìní~se na~$\oplus$
+\endlist
+(ma¾eme-li pravý list, øe¹íme zrcadlovì)
+\:ma¾eme vrchol s~jedním (levým nebo pravým) synem $\rightarrow$ syn nastupuje na~místo otce a~získává typ~$\odot$\
+
+\>V~obou pøípadech posílame informaci o~zmìnì hloubky stromu...
+\:mazaný vrchol mìl oba syny (listy) $\rightarrow$ vybereme jednoho ze~synù na~místo smazaného otce. Hloubka se nemìní.
+\:mazaný vrchol mìl syny podstromy $\rightarrow$ na~jeho místo vezmeme nejvìt¹í prvek levého podstromu (nebo nejmen¹í prvek pravého podstromu) a od~odebraného (nahrazujícího) listu kontrolujeme vyvá¾ení podstromu.
+\endlist
+
+
+
+
+\>{\I Úprava vyvá¾ení} stromu po~odebrání listu z~podstromu
+\itemize\ibull
+\:informace o~zmìnì hloubky pøi¹la z~levého podstromu do~vrcholu typu~$\odot$ $\rightarrow$ vrchol se~zmìní na~$\oplus$ a~dál se hloubka nemìní
+
+\:informace pøi¹la zleva do~vrcholu s~$\ominus$ $\rightarrow$ mìní~se na~$\odot$ a~posíláme informaci o~zmìnì hloubky.
+
+\treepic{10}
+\treepic{11}
+
+\:problémová situace nastává, kdy¾ informace o~zmìnì pøi¹la zleva do~vrcholu se~znaménkem~$\oplus$
+\endlist
+\>Rozebereme na~tøi~pøípady podle znaménka pravého syna nevyvá¾eného vrcholu
+\itemize\ibull
+\:{\I pravý syn je typu~$\oplus$} $\rightarrow$ provedeme rotaci vlevo, novým koøenem se~stává~$y$ (pravý syn), oba vrcholy zmìní typ na~$\odot$ a~posíláme informaci o~zmìnì hloubky.
+
+\treepic{12}
+
+\:{\I pravý syn je typu~$\odot$} $\rightarrow$ provedeme opìt rotaci vlevo, koøenem se~stává~$y$, následnì se u~$y$ zmìní typ na~$\ominus$ , u~vrcholu~$x$ se typ nemìní. Hloubka stromu se~nemìní, tudí¾ není tøeba posílat informaci.
+
+\treepic{12y}
+
+\:{\I pravý syn je typu~$\ominus$} $\rightarrow$ v~tomto pøípadì uva¾ujeme je¹tì vrchol~$z$ jako levého syna vrcholu~$y$, s~podstromy $B$ a~$C$, podstromy $B$ a~$C$ mají hloubku~$h$ nebo~$h-1$. Provedeme dvojrotaci, napøed vpravo rotujeme vrcholy $z$ a~$y$, potom vlevo vrcholy~$x$ a~$z$ tak, ¾e se $z$ stane novým koøenem, typ vecholu~$x$ bude potom~$\ominus$ nebo~$\odot$, typ~$y$~$\oplus$ nebo~$\odot$ (podle toho, jaké znaménko mìl pùvodnì vrchol~$z$), typ~$z$ bude~$\odot$ a~opìt posíláme informaci o~zmìnì hloubky stromu.
+
+\treepic{13}
+\endlist
+
+\h{Obecné vyhledávací stromy}
+
+Pøi ulo¾ení dat na~disku se~sna¾íme, aby~se ètení z~disku provádìlo pokud mo¾no
+co nejménìkrát a~nezále¾í nám tolik na~tom, kolik operací se~vykoná v~jednom
+uzlu. (Èasovì je operace porovnávání zanedbatelná oproti ètení z~disku.)
+
+\s{Definice:} {\I $(a,b)$-strom} pro parametry $a,b$, $a \geq 2$, $b\geq 2a - 1$ je zakoøenìný strom s~uspoøádanými syny a~vnìj¹ími vrcholy, pro který platí
+následující axiomy:
+\numlist\nparen
+\:Data jsou ulo¾ena ve~vnitøních vrcholech a~ka¾dý vrchol obsahuje o~1 ménì klíèù ne¾ má synù.
+\:Platí stromové uspoøádání, tedy ¾e $ A < x_1 < B < x_2 < C < x_3 < D $.
+\:Koøen má $2$ a¾~$b$ synù, ostatní vnitøní vrcholy $a$ a¾ $b$ synù.
+\:V¹echny vnìj¹í vrcholy jsou ve~stejné hloubce (vnìj¹í vrchol$=$list).
+\endlist
+\>{\I Poznámka:} kdekoli~by mohl být syn a~není, pøipojíme vrchol, kterému øíkáme vnìj¹í vrchol)
+
+\abpic{ab-strom11}
+
+\s{Lemma:} $(a,b)$-strom na~$n$~vrcholech má hloubku~$O(\log_a n)$.
+
+\proof
+Zjistíme jeho minimální poèet listù (oznaème jej $m$): ka¾dý vrchol a¾ na~koøen má alespoò $a$ synù, hloubku si oznaèíme~$d$ $\rightarrow$
+$$m\geq~a^{(d -1)}$$
+$$\log_a m \geq d -1$$
+$$d \leq 1+ \log_a m$$
+\centerline{co¾ je øádovì $O(\log_a n)$, kde $n$ je poèet vrcholù.}\
+
+\s{Operace s (a,b)-stromy:}
+
+\s{Find}
+\item{-}V¾dy zjistíme, mezi které 2 klíèe hledaný vrchol patøí a potom se zanoøíme hloubìji.\
+
+\>Èasová slo¾itost nalezení prvku v $(a,b)$-stromu je $O(\log b \cdot \log_a n)$, kde $\log b$ je èas strávený na~jednom vrcholu pro zji¹tení, mezi které 2 vrcholy hledaný patøí, $\log_a n$ je hloubka stromu.
+
+\s{Insert}
+
+\>Jako Find, pøièem¾ jestli¾e nena¹el, skonèí na~posledním patøe a~pøidáme klíè
+\itemize\ibull
+\:pokud pøidáním nepøesáhneme maximální poèet klíèù mù¾eme skonèit
+\abpic{insert1}
+\:pokud pøidáním pøesáhneme maximální poèet klíèù
+\endlist
+\algo
+\:rozdìlíme vrchol na~3 èásti: $L$,$x$,$P$
+\:$L$ a $P$ jsou nové vrcholy
+\:$x$ je hodnota mezi $L$ a $P$, kterou vlo¾íme o patro vý¹ jako klíè oddìlující novì vzniklé vrcholy $L$ a $P$
+\:tím jsme pøevedli problém o patro vý¹ a opakujeme algoritmus
+\endalgo
+
+\abpics{b-klicu1}{b-klicu2}
+
+\s{Poznámka:} Jestli¾e se dostaneme a¾ do koøene, rozdìlí se koøen na dvì èásti, vznikne nám nový koøen se dvìma syny (co¾ je povoleno) a celému stromu vzroste hloubka o jedna.
+
+\s{Korektnost:}
+Potøebujeme, aby
+$$\vert L\vert \geq a-1$$
+$$\vert P\vert \geq a-1$$
+po seètení obou nerovností a~priètení 1 na~obì strany rovnice:
+$$\vert L\vert +\vert P\vert +1\geq 2a-2+1=2a-1$$
+pravá strana je rovna $b$ a~to podle definice $\geq 2a-1$. \par
+\s{Èasová slo¾itost:} vkládání prvku do $(a,b)$-stromu je $O(b\cdot \log_a n)$.
+
+
+\s{Delete}
+\item{-} pøevedeme na~delete z~listu (stejný postup jako u~stromu: jestli¾e to není list, prohodíme tuto hodnotu s~nejmen¹í hodnotou podstromu jeho pravého syna) -- v tomto pøípadì na~klíè posledního vnitøního vrcholu, proto¾e listy jsou vnìj¹í vrcholy bez dat.
+\itemize\ibull
+\:pokud má vrchol, ze~kterého odebíráme stále $a-1$ klíèù, mù¾eme skonèit
+\:pokud má vrchol($V$), ze~kterého odebíráme $a-2$ klíèù a~jeho levý sousední vrchol($L$) alespoò $a$ klíèù (klíè otce oddìlující tyto vrcholy oznaème $x$):
+\endlist
+\algo
+\:sma¾eme nejvìt¹í klíè levého sousedního vrcholu ($L$) a~nahradíme tím klíè otce obou vrcholù (nahradíme $x$ za~tuto hodnotu)
+\:pùvodní klíè otce($x$) pøidáme jako nejmen¹í klíè odebíranému vrcholu ($V$)
+\:tím mají oba tyto vrcholy $a-1$ klíèù a mù¾eme skonèit
+\endalgo
+\abpics{delete21}{delete22}
+\itemize\ibull
+\:pokud má vrchol, z kterého odebíráme($V$) $a-2$ klíèù a jeho levý sousední vrchol($L$) $a-1$ klíèù (klíè otce oddìlující tyto vrcholy oznaème $x$):
+\endlist
+\algo
+\:slouèíme $V$,$x$,$L$ do jednoho vrcholu
+\:tím jsme problém pøevedli o patro vý¹ a opakujeme algoritmus \par
+\endalgo
+\abpics{delete31}{delete32}
+
+\>{\I Poznámka:} Dojdeme-li takto a¾ do koøene, na místo klíèe odebraného z koøene lze pou¾ít nejmen¹í nebo nejvìt¹í klíè novì slouèeného podstromu. Ten odebrat lze, proto¾e po slouèení (které bylo pøíèinou této situace), je v nejni¾¹ím vrcholu $2a-2$ klíèù.
+
+\>{\I Èasová slo¾itost:} $$O(b\cdot \log_a n)$$
+
+\bye
--- /dev/null
+P=7-stromy
+
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+%%PageTrailer
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--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: (ImageMagick)
+%%Title: (b-klicu1.eps)
+%%CreationDate: (Mon Nov 12 12:22:07 2007)
+%%BoundingBox: 0 0 517 375
+%%HiResBoundingBox: 0 0 517 375
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 1
+%%Pages: 1
+%%EndComments
+
+%%BeginDefaults
+%%EndDefaults
+
+%%BeginProlog
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+% Display a color image. The image is displayed in color on
+% Postscript viewers or printers that support color, otherwise
+% it is displayed as grayscale.
+%
+/DirectClassPacket
+{
+ %
+ % Get a DirectClass packet.
+ %
+ % Parameters:
+ % red.
+ % green.
+ % blue.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile color_packet readhexstring pop pop
+ compression 0 eq
+ {
+ /number_pixels 3 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add 3 mul def
+ } ifelse
+ 0 3 number_pixels 1 sub
+ {
+ pixels exch color_packet putinterval
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/DirectClassImage
+{
+ %
+ % Display a DirectClass image.
+ %
+ systemdict /colorimage known
+ {
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { DirectClassPacket } false 3 colorimage
+ }
+ {
+ %
+ % No colorimage operator; convert to grayscale.
+ %
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { GrayDirectClassPacket } image
+ } ifelse
+} bind def
+
+/GrayDirectClassPacket
+{
+ %
+ % Get a DirectClass packet; convert to grayscale.
+ %
+ % Parameters:
+ % red
+ % green
+ % blue
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile color_packet readhexstring pop pop
+ color_packet 0 get 0.299 mul
+ color_packet 1 get 0.587 mul add
+ color_packet 2 get 0.114 mul add
+ cvi
+ /gray_packet exch def
+ compression 0 eq
+ {
+ /number_pixels 1 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add def
+ } ifelse
+ 0 1 number_pixels 1 sub
+ {
+ pixels exch gray_packet put
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/GrayPseudoClassPacket
+{
+ %
+ % Get a PseudoClass packet; convert to grayscale.
+ %
+ % Parameters:
+ % index: index into the colormap.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile byte readhexstring pop 0 get
+ /offset exch 3 mul def
+ /color_packet colormap offset 3 getinterval def
+ color_packet 0 get 0.299 mul
+ color_packet 1 get 0.587 mul add
+ color_packet 2 get 0.114 mul add
+ cvi
+ /gray_packet exch def
+ compression 0 eq
+ {
+ /number_pixels 1 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add def
+ } ifelse
+ 0 1 number_pixels 1 sub
+ {
+ pixels exch gray_packet put
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/PseudoClassPacket
+{
+ %
+ % Get a PseudoClass packet.
+ %
+ % Parameters:
+ % index: index into the colormap.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile byte readhexstring pop 0 get
+ /offset exch 3 mul def
+ /color_packet colormap offset 3 getinterval def
+ compression 0 eq
+ {
+ /number_pixels 3 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add 3 mul def
+ } ifelse
+ 0 3 number_pixels 1 sub
+ {
+ pixels exch color_packet putinterval
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/PseudoClassImage
+{
+ %
+ % Display a PseudoClass image.
+ %
+ % Parameters:
+ % class: 0-PseudoClass or 1-Grayscale.
+ %
+ currentfile buffer readline pop
+ token pop /class exch def pop
+ class 0 gt
+ {
+ currentfile buffer readline pop
+ token pop /depth exch def pop
+ /grays columns 8 add depth sub depth mul 8 idiv string def
+ columns rows depth
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { currentfile grays readhexstring pop } image
+ }
+ {
+ %
+ % Parameters:
+ % colors: number of colors in the colormap.
+ % colormap: red, green, blue color packets.
+ %
+ currentfile buffer readline pop
+ token pop /colors exch def pop
+ /colors colors 3 mul def
+ /colormap colors string def
+ currentfile colormap readhexstring pop pop
+ systemdict /colorimage known
+ {
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { PseudoClassPacket } false 3 colorimage
+ }
+ {
+ %
+ % No colorimage operator; convert to grayscale.
+ %
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { GrayPseudoClassPacket } image
+ } ifelse
+ } ifelse
+} bind def
+
+/DisplayImage
+{
+ %
+ % Display a DirectClass or PseudoClass image.
+ %
+ % Parameters:
+ % x & y translation.
+ % x & y scale.
+ % label pointsize.
+ % image label.
+ % image columns & rows.
+ % class: 0-DirectClass or 1-PseudoClass.
+ % compression: 0-none or 1-RunlengthEncoded.
+ % hex color packets.
+ %
+ gsave
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+ /byte 1 string def
+ /color_packet 3 string def
+ /pixels 768 string def
+
+ currentfile buffer readline pop
+ token pop /x exch def
+ token pop /y exch def pop
+ x y translate
+ currentfile buffer readline pop
+ token pop /x exch def
+ token pop /y exch def pop
+ currentfile buffer readline pop
+ token pop /pointsize exch def pop
+ /Times-Roman findfont pointsize scalefont setfont
+ x y scale
+ currentfile buffer readline pop
+ token pop /columns exch def
+ token pop /rows exch def pop
+ currentfile buffer readline pop
+ token pop /class exch def pop
+ currentfile buffer readline pop
+ token pop /compression exch def pop
+ class 0 gt { PseudoClassImage } { DirectClassImage } ifelse
+ grestore
+} bind def
+%%EndProlog
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+%%PageTrailer
+%%Trailer
+%%EOF
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: (ImageMagick)
+%%Title: (b-klicu2.eps)
+%%CreationDate: (Mon Nov 12 12:22:07 2007)
+%%BoundingBox: 0 0 694 407
+%%HiResBoundingBox: 0 0 694 407
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 1
+%%Pages: 1
+%%EndComments
+
+%%BeginDefaults
+%%EndDefaults
+
+%%BeginProlog
+%
+% Display a color image. The image is displayed in color on
+% Postscript viewers or printers that support color, otherwise
+% it is displayed as grayscale.
+%
+/DirectClassPacket
+{
+ %
+ % Get a DirectClass packet.
+ %
+ % Parameters:
+ % red.
+ % green.
+ % blue.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile color_packet readhexstring pop pop
+ compression 0 eq
+ {
+ /number_pixels 3 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add 3 mul def
+ } ifelse
+ 0 3 number_pixels 1 sub
+ {
+ pixels exch color_packet putinterval
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/DirectClassImage
+{
+ %
+ % Display a DirectClass image.
+ %
+ systemdict /colorimage known
+ {
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { DirectClassPacket } false 3 colorimage
+ }
+ {
+ %
+ % No colorimage operator; convert to grayscale.
+ %
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { GrayDirectClassPacket } image
+ } ifelse
+} bind def
+
+/GrayDirectClassPacket
+{
+ %
+ % Get a DirectClass packet; convert to grayscale.
+ %
+ % Parameters:
+ % red
+ % green
+ % blue
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile color_packet readhexstring pop pop
+ color_packet 0 get 0.299 mul
+ color_packet 1 get 0.587 mul add
+ color_packet 2 get 0.114 mul add
+ cvi
+ /gray_packet exch def
+ compression 0 eq
+ {
+ /number_pixels 1 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add def
+ } ifelse
+ 0 1 number_pixels 1 sub
+ {
+ pixels exch gray_packet put
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/GrayPseudoClassPacket
+{
+ %
+ % Get a PseudoClass packet; convert to grayscale.
+ %
+ % Parameters:
+ % index: index into the colormap.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile byte readhexstring pop 0 get
+ /offset exch 3 mul def
+ /color_packet colormap offset 3 getinterval def
+ color_packet 0 get 0.299 mul
+ color_packet 1 get 0.587 mul add
+ color_packet 2 get 0.114 mul add
+ cvi
+ /gray_packet exch def
+ compression 0 eq
+ {
+ /number_pixels 1 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add def
+ } ifelse
+ 0 1 number_pixels 1 sub
+ {
+ pixels exch gray_packet put
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/PseudoClassPacket
+{
+ %
+ % Get a PseudoClass packet.
+ %
+ % Parameters:
+ % index: index into the colormap.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile byte readhexstring pop 0 get
+ /offset exch 3 mul def
+ /color_packet colormap offset 3 getinterval def
+ compression 0 eq
+ {
+ /number_pixels 3 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add 3 mul def
+ } ifelse
+ 0 3 number_pixels 1 sub
+ {
+ pixels exch color_packet putinterval
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/PseudoClassImage
+{
+ %
+ % Display a PseudoClass image.
+ %
+ % Parameters:
+ % class: 0-PseudoClass or 1-Grayscale.
+ %
+ currentfile buffer readline pop
+ token pop /class exch def pop
+ class 0 gt
+ {
+ currentfile buffer readline pop
+ token pop /depth exch def pop
+ /grays columns 8 add depth sub depth mul 8 idiv string def
+ columns rows depth
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { currentfile grays readhexstring pop } image
+ }
+ {
+ %
+ % Parameters:
+ % colors: number of colors in the colormap.
+ % colormap: red, green, blue color packets.
+ %
+ currentfile buffer readline pop
+ token pop /colors exch def pop
+ /colors colors 3 mul def
+ /colormap colors string def
+ currentfile colormap readhexstring pop pop
+ systemdict /colorimage known
+ {
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { PseudoClassPacket } false 3 colorimage
+ }
+ {
+ %
+ % No colorimage operator; convert to grayscale.
+ %
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { GrayPseudoClassPacket } image
+ } ifelse
+ } ifelse
+} bind def
+
+/DisplayImage
+{
+ %
+ % Display a DirectClass or PseudoClass image.
+ %
+ % Parameters:
+ % x & y translation.
+ % x & y scale.
+ % label pointsize.
+ % image label.
+ % image columns & rows.
+ % class: 0-DirectClass or 1-PseudoClass.
+ % compression: 0-none or 1-RunlengthEncoded.
+ % hex color packets.
+ %
+ gsave
+ /buffer 512 string def
+ /byte 1 string def
+ /color_packet 3 string def
+ /pixels 768 string def
+
+ currentfile buffer readline pop
+ token pop /x exch def
+ token pop /y exch def pop
+ x y translate
+ currentfile buffer readline pop
+ token pop /x exch def
+ token pop /y exch def pop
+ currentfile buffer readline pop
+ token pop /pointsize exch def pop
+ /Times-Roman findfont pointsize scalefont setfont
+ x y scale
+ currentfile buffer readline pop
+ token pop /columns exch def
+ token pop /rows exch def pop
+ currentfile buffer readline pop
+ token pop /class exch def pop
+ currentfile buffer readline pop
+ token pop /compression exch def pop
+ class 0 gt { PseudoClassImage } { DirectClassImage } ifelse
+ grestore
+} bind def
+%%EndProlog
+%%Page: 1 1
+%%PageBoundingBox: 0 0 694 407
+userdict begin
+DisplayImage
+0 0
+694 407
+12.000000
+694 407
+1
+1
+1
+8
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+%%PageTrailer
+%%Trailer
+%%EOF
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: (ImageMagick)
+%%Title: (delete21.eps)
+%%CreationDate: (Mon Nov 12 12:22:07 2007)
+%%BoundingBox: 0 0 694 404
+%%HiResBoundingBox: 0 0 694 404
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 1
+%%Pages: 1
+%%EndComments
+
+%%BeginDefaults
+%%EndDefaults
+
+%%BeginProlog
+%
+% Display a color image. The image is displayed in color on
+% Postscript viewers or printers that support color, otherwise
+% it is displayed as grayscale.
+%
+/DirectClassPacket
+{
+ %
+ % Get a DirectClass packet.
+ %
+ % Parameters:
+ % red.
+ % green.
+ % blue.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile color_packet readhexstring pop pop
+ compression 0 eq
+ {
+ /number_pixels 3 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add 3 mul def
+ } ifelse
+ 0 3 number_pixels 1 sub
+ {
+ pixels exch color_packet putinterval
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/DirectClassImage
+{
+ %
+ % Display a DirectClass image.
+ %
+ systemdict /colorimage known
+ {
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { DirectClassPacket } false 3 colorimage
+ }
+ {
+ %
+ % No colorimage operator; convert to grayscale.
+ %
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { GrayDirectClassPacket } image
+ } ifelse
+} bind def
+
+/GrayDirectClassPacket
+{
+ %
+ % Get a DirectClass packet; convert to grayscale.
+ %
+ % Parameters:
+ % red
+ % green
+ % blue
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile color_packet readhexstring pop pop
+ color_packet 0 get 0.299 mul
+ color_packet 1 get 0.587 mul add
+ color_packet 2 get 0.114 mul add
+ cvi
+ /gray_packet exch def
+ compression 0 eq
+ {
+ /number_pixels 1 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add def
+ } ifelse
+ 0 1 number_pixels 1 sub
+ {
+ pixels exch gray_packet put
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/GrayPseudoClassPacket
+{
+ %
+ % Get a PseudoClass packet; convert to grayscale.
+ %
+ % Parameters:
+ % index: index into the colormap.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile byte readhexstring pop 0 get
+ /offset exch 3 mul def
+ /color_packet colormap offset 3 getinterval def
+ color_packet 0 get 0.299 mul
+ color_packet 1 get 0.587 mul add
+ color_packet 2 get 0.114 mul add
+ cvi
+ /gray_packet exch def
+ compression 0 eq
+ {
+ /number_pixels 1 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add def
+ } ifelse
+ 0 1 number_pixels 1 sub
+ {
+ pixels exch gray_packet put
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/PseudoClassPacket
+{
+ %
+ % Get a PseudoClass packet.
+ %
+ % Parameters:
+ % index: index into the colormap.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile byte readhexstring pop 0 get
+ /offset exch 3 mul def
+ /color_packet colormap offset 3 getinterval def
+ compression 0 eq
+ {
+ /number_pixels 3 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add 3 mul def
+ } ifelse
+ 0 3 number_pixels 1 sub
+ {
+ pixels exch color_packet putinterval
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/PseudoClassImage
+{
+ %
+ % Display a PseudoClass image.
+ %
+ % Parameters:
+ % class: 0-PseudoClass or 1-Grayscale.
+ %
+ currentfile buffer readline pop
+ token pop /class exch def pop
+ class 0 gt
+ {
+ currentfile buffer readline pop
+ token pop /depth exch def pop
+ /grays columns 8 add depth sub depth mul 8 idiv string def
+ columns rows depth
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { currentfile grays readhexstring pop } image
+ }
+ {
+ %
+ % Parameters:
+ % colors: number of colors in the colormap.
+ % colormap: red, green, blue color packets.
+ %
+ currentfile buffer readline pop
+ token pop /colors exch def pop
+ /colors colors 3 mul def
+ /colormap colors string def
+ currentfile colormap readhexstring pop pop
+ systemdict /colorimage known
+ {
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { PseudoClassPacket } false 3 colorimage
+ }
+ {
+ %
+ % No colorimage operator; convert to grayscale.
+ %
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { GrayPseudoClassPacket } image
+ } ifelse
+ } ifelse
+} bind def
+
+/DisplayImage
+{
+ %
+ % Display a DirectClass or PseudoClass image.
+ %
+ % Parameters:
+ % x & y translation.
+ % x & y scale.
+ % label pointsize.
+ % image label.
+ % image columns & rows.
+ % class: 0-DirectClass or 1-PseudoClass.
+ % compression: 0-none or 1-RunlengthEncoded.
+ % hex color packets.
+ %
+ gsave
+ /buffer 512 string def
+ /byte 1 string def
+ /color_packet 3 string def
+ /pixels 768 string def
+
+ currentfile buffer readline pop
+ token pop /x exch def
+ token pop /y exch def pop
+ x y translate
+ currentfile buffer readline pop
+ token pop /x exch def
+ token pop /y exch def pop
+ currentfile buffer readline pop
+ token pop /pointsize exch def pop
+ /Times-Roman findfont pointsize scalefont setfont
+ x y scale
+ currentfile buffer readline pop
+ token pop /columns exch def
+ token pop /rows exch def pop
+ currentfile buffer readline pop
+ token pop /class exch def pop
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--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: (ImageMagick)
+%%Title: (delete22.eps)
+%%CreationDate: (Mon Nov 12 12:22:08 2007)
+%%BoundingBox: 0 0 694 375
+%%HiResBoundingBox: 0 0 694 375
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 1
+%%Pages: 1
+%%EndComments
+
+%%BeginDefaults
+%%EndDefaults
+
+%%BeginProlog
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+% Display a color image. The image is displayed in color on
+% Postscript viewers or printers that support color, otherwise
+% it is displayed as grayscale.
+%
+/DirectClassPacket
+{
+ %
+ % Get a DirectClass packet.
+ %
+ % Parameters:
+ % red.
+ % green.
+ % blue.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile color_packet readhexstring pop pop
+ compression 0 eq
+ {
+ /number_pixels 3 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add 3 mul def
+ } ifelse
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+ {
+ pixels exch color_packet putinterval
+ } for
+ pixels 0 number_pixels getinterval
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+/DirectClassImage
+{
+ %
+ % Display a DirectClass image.
+ %
+ systemdict /colorimage known
+ {
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { DirectClassPacket } false 3 colorimage
+ }
+ {
+ %
+ % No colorimage operator; convert to grayscale.
+ %
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { GrayDirectClassPacket } image
+ } ifelse
+} bind def
+
+/GrayDirectClassPacket
+{
+ %
+ % Get a DirectClass packet; convert to grayscale.
+ %
+ % Parameters:
+ % red
+ % green
+ % blue
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile color_packet readhexstring pop pop
+ color_packet 0 get 0.299 mul
+ color_packet 1 get 0.587 mul add
+ color_packet 2 get 0.114 mul add
+ cvi
+ /gray_packet exch def
+ compression 0 eq
+ {
+ /number_pixels 1 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add def
+ } ifelse
+ 0 1 number_pixels 1 sub
+ {
+ pixels exch gray_packet put
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/GrayPseudoClassPacket
+{
+ %
+ % Get a PseudoClass packet; convert to grayscale.
+ %
+ % Parameters:
+ % index: index into the colormap.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile byte readhexstring pop 0 get
+ /offset exch 3 mul def
+ /color_packet colormap offset 3 getinterval def
+ color_packet 0 get 0.299 mul
+ color_packet 1 get 0.587 mul add
+ color_packet 2 get 0.114 mul add
+ cvi
+ /gray_packet exch def
+ compression 0 eq
+ {
+ /number_pixels 1 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
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+ {
+ pixels exch gray_packet put
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+ pixels 0 number_pixels getinterval
+} bind def
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+/PseudoClassPacket
+{
+ %
+ % Get a PseudoClass packet.
+ %
+ % Parameters:
+ % index: index into the colormap.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile byte readhexstring pop 0 get
+ /offset exch 3 mul def
+ /color_packet colormap offset 3 getinterval def
+ compression 0 eq
+ {
+ /number_pixels 3 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add 3 mul def
+ } ifelse
+ 0 3 number_pixels 1 sub
+ {
+ pixels exch color_packet putinterval
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+ pixels 0 number_pixels getinterval
+} bind def
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+/PseudoClassImage
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+ %
+ % Display a PseudoClass image.
+ %
+ % Parameters:
+ % class: 0-PseudoClass or 1-Grayscale.
+ %
+ currentfile buffer readline pop
+ token pop /class exch def pop
+ class 0 gt
+ {
+ currentfile buffer readline pop
+ token pop /depth exch def pop
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+ columns rows depth
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { currentfile grays readhexstring pop } image
+ }
+ {
+ %
+ % Parameters:
+ % colors: number of colors in the colormap.
+ % colormap: red, green, blue color packets.
+ %
+ currentfile buffer readline pop
+ token pop /colors exch def pop
+ /colors colors 3 mul def
+ /colormap colors string def
+ currentfile colormap readhexstring pop pop
+ systemdict /colorimage known
+ {
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { PseudoClassPacket } false 3 colorimage
+ }
+ {
+ %
+ % No colorimage operator; convert to grayscale.
+ %
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { GrayPseudoClassPacket } image
+ } ifelse
+ } ifelse
+} bind def
+
+/DisplayImage
+{
+ %
+ % Display a DirectClass or PseudoClass image.
+ %
+ % Parameters:
+ % x & y translation.
+ % x & y scale.
+ % label pointsize.
+ % image label.
+ % image columns & rows.
+ % class: 0-DirectClass or 1-PseudoClass.
+ % compression: 0-none or 1-RunlengthEncoded.
+ % hex color packets.
+ %
+ gsave
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+ /byte 1 string def
+ /color_packet 3 string def
+ /pixels 768 string def
+
+ currentfile buffer readline pop
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+ token pop /y exch def pop
+ x y translate
+ currentfile buffer readline pop
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+ token pop /y exch def pop
+ currentfile buffer readline pop
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+ /Times-Roman findfont pointsize scalefont setfont
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+ currentfile buffer readline pop
+ token pop /columns exch def
+ token pop /rows exch def pop
+ currentfile buffer readline pop
+ token pop /class exch def pop
+ currentfile buffer readline pop
+ token pop /compression exch def pop
+ class 0 gt { PseudoClassImage } { DirectClassImage } ifelse
+ grestore
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+%%PageTrailer
+%%Trailer
+%%EOF
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: (ImageMagick)
+%%Title: (delete31.eps)
+%%CreationDate: (Mon Nov 12 12:22:08 2007)
+%%BoundingBox: 0 0 731 404
+%%HiResBoundingBox: 0 0 731 404
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 1
+%%Pages: 1
+%%EndComments
+
+%%BeginDefaults
+%%EndDefaults
+
+%%BeginProlog
+%
+% Display a color image. The image is displayed in color on
+% Postscript viewers or printers that support color, otherwise
+% it is displayed as grayscale.
+%
+/DirectClassPacket
+{
+ %
+ % Get a DirectClass packet.
+ %
+ % Parameters:
+ % red.
+ % green.
+ % blue.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile color_packet readhexstring pop pop
+ compression 0 eq
+ {
+ /number_pixels 3 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add 3 mul def
+ } ifelse
+ 0 3 number_pixels 1 sub
+ {
+ pixels exch color_packet putinterval
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/DirectClassImage
+{
+ %
+ % Display a DirectClass image.
+ %
+ systemdict /colorimage known
+ {
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { DirectClassPacket } false 3 colorimage
+ }
+ {
+ %
+ % No colorimage operator; convert to grayscale.
+ %
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { GrayDirectClassPacket } image
+ } ifelse
+} bind def
+
+/GrayDirectClassPacket
+{
+ %
+ % Get a DirectClass packet; convert to grayscale.
+ %
+ % Parameters:
+ % red
+ % green
+ % blue
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile color_packet readhexstring pop pop
+ color_packet 0 get 0.299 mul
+ color_packet 1 get 0.587 mul add
+ color_packet 2 get 0.114 mul add
+ cvi
+ /gray_packet exch def
+ compression 0 eq
+ {
+ /number_pixels 1 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add def
+ } ifelse
+ 0 1 number_pixels 1 sub
+ {
+ pixels exch gray_packet put
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/GrayPseudoClassPacket
+{
+ %
+ % Get a PseudoClass packet; convert to grayscale.
+ %
+ % Parameters:
+ % index: index into the colormap.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile byte readhexstring pop 0 get
+ /offset exch 3 mul def
+ /color_packet colormap offset 3 getinterval def
+ color_packet 0 get 0.299 mul
+ color_packet 1 get 0.587 mul add
+ color_packet 2 get 0.114 mul add
+ cvi
+ /gray_packet exch def
+ compression 0 eq
+ {
+ /number_pixels 1 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add def
+ } ifelse
+ 0 1 number_pixels 1 sub
+ {
+ pixels exch gray_packet put
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/PseudoClassPacket
+{
+ %
+ % Get a PseudoClass packet.
+ %
+ % Parameters:
+ % index: index into the colormap.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile byte readhexstring pop 0 get
+ /offset exch 3 mul def
+ /color_packet colormap offset 3 getinterval def
+ compression 0 eq
+ {
+ /number_pixels 3 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add 3 mul def
+ } ifelse
+ 0 3 number_pixels 1 sub
+ {
+ pixels exch color_packet putinterval
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/PseudoClassImage
+{
+ %
+ % Display a PseudoClass image.
+ %
+ % Parameters:
+ % class: 0-PseudoClass or 1-Grayscale.
+ %
+ currentfile buffer readline pop
+ token pop /class exch def pop
+ class 0 gt
+ {
+ currentfile buffer readline pop
+ token pop /depth exch def pop
+ /grays columns 8 add depth sub depth mul 8 idiv string def
+ columns rows depth
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { currentfile grays readhexstring pop } image
+ }
+ {
+ %
+ % Parameters:
+ % colors: number of colors in the colormap.
+ % colormap: red, green, blue color packets.
+ %
+ currentfile buffer readline pop
+ token pop /colors exch def pop
+ /colors colors 3 mul def
+ /colormap colors string def
+ currentfile colormap readhexstring pop pop
+ systemdict /colorimage known
+ {
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { PseudoClassPacket } false 3 colorimage
+ }
+ {
+ %
+ % No colorimage operator; convert to grayscale.
+ %
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { GrayPseudoClassPacket } image
+ } ifelse
+ } ifelse
+} bind def
+
+/DisplayImage
+{
+ %
+ % Display a DirectClass or PseudoClass image.
+ %
+ % Parameters:
+ % x & y translation.
+ % x & y scale.
+ % label pointsize.
+ % image label.
+ % image columns & rows.
+ % class: 0-DirectClass or 1-PseudoClass.
+ % compression: 0-none or 1-RunlengthEncoded.
+ % hex color packets.
+ %
+ gsave
+ /buffer 512 string def
+ /byte 1 string def
+ /color_packet 3 string def
+ /pixels 768 string def
+
+ currentfile buffer readline pop
+ token pop /x exch def
+ token pop /y exch def pop
+ x y translate
+ currentfile buffer readline pop
+ token pop /x exch def
+ token pop /y exch def pop
+ currentfile buffer readline pop
+ token pop /pointsize exch def pop
+ /Times-Roman findfont pointsize scalefont setfont
+ x y scale
+ currentfile buffer readline pop
+ token pop /columns exch def
+ token pop /rows exch def pop
+ currentfile buffer readline pop
+ token pop /class exch def pop
+ currentfile buffer readline pop
+ token pop /compression exch def pop
+ class 0 gt { PseudoClassImage } { DirectClassImage } ifelse
+ grestore
+} bind def
+%%EndProlog
+%%Page: 1 1
+%%PageBoundingBox: 0 0 731 404
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+%%PageTrailer
+%%Trailer
+%%EOF
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: (ImageMagick)
+%%Title: (delete32.eps)
+%%CreationDate: (Mon Nov 12 12:22:08 2007)
+%%BoundingBox: 0 0 517 375
+%%HiResBoundingBox: 0 0 517 375
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 1
+%%Pages: 1
+%%EndComments
+
+%%BeginDefaults
+%%EndDefaults
+
+%%BeginProlog
+%
+% Display a color image. The image is displayed in color on
+% Postscript viewers or printers that support color, otherwise
+% it is displayed as grayscale.
+%
+/DirectClassPacket
+{
+ %
+ % Get a DirectClass packet.
+ %
+ % Parameters:
+ % red.
+ % green.
+ % blue.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile color_packet readhexstring pop pop
+ compression 0 eq
+ {
+ /number_pixels 3 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add 3 mul def
+ } ifelse
+ 0 3 number_pixels 1 sub
+ {
+ pixels exch color_packet putinterval
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/DirectClassImage
+{
+ %
+ % Display a DirectClass image.
+ %
+ systemdict /colorimage known
+ {
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { DirectClassPacket } false 3 colorimage
+ }
+ {
+ %
+ % No colorimage operator; convert to grayscale.
+ %
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { GrayDirectClassPacket } image
+ } ifelse
+} bind def
+
+/GrayDirectClassPacket
+{
+ %
+ % Get a DirectClass packet; convert to grayscale.
+ %
+ % Parameters:
+ % red
+ % green
+ % blue
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile color_packet readhexstring pop pop
+ color_packet 0 get 0.299 mul
+ color_packet 1 get 0.587 mul add
+ color_packet 2 get 0.114 mul add
+ cvi
+ /gray_packet exch def
+ compression 0 eq
+ {
+ /number_pixels 1 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add def
+ } ifelse
+ 0 1 number_pixels 1 sub
+ {
+ pixels exch gray_packet put
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/GrayPseudoClassPacket
+{
+ %
+ % Get a PseudoClass packet; convert to grayscale.
+ %
+ % Parameters:
+ % index: index into the colormap.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile byte readhexstring pop 0 get
+ /offset exch 3 mul def
+ /color_packet colormap offset 3 getinterval def
+ color_packet 0 get 0.299 mul
+ color_packet 1 get 0.587 mul add
+ color_packet 2 get 0.114 mul add
+ cvi
+ /gray_packet exch def
+ compression 0 eq
+ {
+ /number_pixels 1 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add def
+ } ifelse
+ 0 1 number_pixels 1 sub
+ {
+ pixels exch gray_packet put
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/PseudoClassPacket
+{
+ %
+ % Get a PseudoClass packet.
+ %
+ % Parameters:
+ % index: index into the colormap.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile byte readhexstring pop 0 get
+ /offset exch 3 mul def
+ /color_packet colormap offset 3 getinterval def
+ compression 0 eq
+ {
+ /number_pixels 3 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add 3 mul def
+ } ifelse
+ 0 3 number_pixels 1 sub
+ {
+ pixels exch color_packet putinterval
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/PseudoClassImage
+{
+ %
+ % Display a PseudoClass image.
+ %
+ % Parameters:
+ % class: 0-PseudoClass or 1-Grayscale.
+ %
+ currentfile buffer readline pop
+ token pop /class exch def pop
+ class 0 gt
+ {
+ currentfile buffer readline pop
+ token pop /depth exch def pop
+ /grays columns 8 add depth sub depth mul 8 idiv string def
+ columns rows depth
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { currentfile grays readhexstring pop } image
+ }
+ {
+ %
+ % Parameters:
+ % colors: number of colors in the colormap.
+ % colormap: red, green, blue color packets.
+ %
+ currentfile buffer readline pop
+ token pop /colors exch def pop
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+%%EOF
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: (ImageMagick)
+%%Title: (Insert1.eps)
+%%CreationDate: (Mon Nov 12 12:22:07 2007)
+%%BoundingBox: 0 0 764 79
+%%HiResBoundingBox: 0 0 764 79
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 1
+%%Pages: 1
+%%EndComments
+
+%%BeginDefaults
+%%EndDefaults
+
+%%BeginProlog
+%
+% Display a color image. The image is displayed in color on
+% Postscript viewers or printers that support color, otherwise
+% it is displayed as grayscale.
+%
+/DirectClassPacket
+{
+ %
+ % Get a DirectClass packet.
+ %
+ % Parameters:
+ % red.
+ % green.
+ % blue.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile color_packet readhexstring pop pop
+ compression 0 eq
+ {
+ /number_pixels 3 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add 3 mul def
+ } ifelse
+ 0 3 number_pixels 1 sub
+ {
+ pixels exch color_packet putinterval
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/DirectClassImage
+{
+ %
+ % Display a DirectClass image.
+ %
+ systemdict /colorimage known
+ {
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { DirectClassPacket } false 3 colorimage
+ }
+ {
+ %
+ % No colorimage operator; convert to grayscale.
+ %
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { GrayDirectClassPacket } image
+ } ifelse
+} bind def
+
+/GrayDirectClassPacket
+{
+ %
+ % Get a DirectClass packet; convert to grayscale.
+ %
+ % Parameters:
+ % red
+ % green
+ % blue
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile color_packet readhexstring pop pop
+ color_packet 0 get 0.299 mul
+ color_packet 1 get 0.587 mul add
+ color_packet 2 get 0.114 mul add
+ cvi
+ /gray_packet exch def
+ compression 0 eq
+ {
+ /number_pixels 1 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add def
+ } ifelse
+ 0 1 number_pixels 1 sub
+ {
+ pixels exch gray_packet put
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/GrayPseudoClassPacket
+{
+ %
+ % Get a PseudoClass packet; convert to grayscale.
+ %
+ % Parameters:
+ % index: index into the colormap.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile byte readhexstring pop 0 get
+ /offset exch 3 mul def
+ /color_packet colormap offset 3 getinterval def
+ color_packet 0 get 0.299 mul
+ color_packet 1 get 0.587 mul add
+ color_packet 2 get 0.114 mul add
+ cvi
+ /gray_packet exch def
+ compression 0 eq
+ {
+ /number_pixels 1 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add def
+ } ifelse
+ 0 1 number_pixels 1 sub
+ {
+ pixels exch gray_packet put
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/PseudoClassPacket
+{
+ %
+ % Get a PseudoClass packet.
+ %
+ % Parameters:
+ % index: index into the colormap.
+ % length: number of pixels minus one of this color (optional).
+ %
+ currentfile byte readhexstring pop 0 get
+ /offset exch 3 mul def
+ /color_packet colormap offset 3 getinterval def
+ compression 0 eq
+ {
+ /number_pixels 3 def
+ }
+ {
+ currentfile byte readhexstring pop 0 get
+ /number_pixels exch 1 add 3 mul def
+ } ifelse
+ 0 3 number_pixels 1 sub
+ {
+ pixels exch color_packet putinterval
+ } for
+ pixels 0 number_pixels getinterval
+} bind def
+
+/PseudoClassImage
+{
+ %
+ % Display a PseudoClass image.
+ %
+ % Parameters:
+ % class: 0-PseudoClass or 1-Grayscale.
+ %
+ currentfile buffer readline pop
+ token pop /class exch def pop
+ class 0 gt
+ {
+ currentfile buffer readline pop
+ token pop /depth exch def pop
+ /grays columns 8 add depth sub depth mul 8 idiv string def
+ columns rows depth
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { currentfile grays readhexstring pop } image
+ }
+ {
+ %
+ % Parameters:
+ % colors: number of colors in the colormap.
+ % colormap: red, green, blue color packets.
+ %
+ currentfile buffer readline pop
+ token pop /colors exch def pop
+ /colors colors 3 mul def
+ /colormap colors string def
+ currentfile colormap readhexstring pop pop
+ systemdict /colorimage known
+ {
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { PseudoClassPacket } false 3 colorimage
+ }
+ {
+ %
+ % No colorimage operator; convert to grayscale.
+ %
+ columns rows 8
+ [
+ columns 0 0
+ rows neg 0 rows
+ ]
+ { GrayPseudoClassPacket } image
+ } ifelse
+ } ifelse
+} bind def
+
+/DisplayImage
+{
+ %
+ % Display a DirectClass or PseudoClass image.
+ %
+ % Parameters:
+ % x & y translation.
+ % x & y scale.
+ % label pointsize.
+ % image label.
+ % image columns & rows.
+ % class: 0-DirectClass or 1-PseudoClass.
+ % compression: 0-none or 1-RunlengthEncoded.
+ % hex color packets.
+ %
+ gsave
+ /buffer 512 string def
+ /byte 1 string def
+ /color_packet 3 string def
+ /pixels 768 string def
+
+ currentfile buffer readline pop
+ token pop /x exch def
+ token pop /y exch def pop
+ x y translate
+ currentfile buffer readline pop
+ token pop /x exch def
+ token pop /y exch def pop
+ currentfile buffer readline pop
+ token pop /pointsize exch def pop
+ /Times-Roman findfont pointsize scalefont setfont
+ x y scale
+ currentfile buffer readline pop
+ token pop /columns exch def
+ token pop /rows exch def pop
+ currentfile buffer readline pop
+ token pop /class exch def pop
+ currentfile buffer readline pop
+ token pop /compression exch def pop
+ class 0 gt { PseudoClassImage } { DirectClassImage } ifelse
+ grestore
+} bind def
+%%EndProlog
+%%Page: 1 1
+%%PageBoundingBox: 0 0 764 79
+userdict begin
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--- /dev/null
+\input ../lecnotes.tex
+
+\prednaska{8}{Základní grafové algoritmy}{(Zapsali M. Petr, O.Tichý, ¥. Magic)}
+
+\h{Reprezentace grafu}
+Oznaème vrcholy grafu na následujícím obrázku písmeny A, B, C, D.
+Pokud bychom chtìli tento graf uchovat v pamìti poèítaèe, máme na výbìr
+hned nìkolik zpùsobù, jak to udìlat.
+\figure{img1_stvorec.eps}{}{\epsfxsize}
+
+\s{1. seznam hran}
+
+Jednodu¹e vypí¹eme seznam v¹ech dvojic vrcholù grafu spojených hranou.
+
+AB, BC, DC, CA, BC
+\medskip
+
+\s{2. matice sousednosti}
+
+Matice sousednosti je pole $A$ o velikosti $n \times n$, jeho¾ prvky na
+souøadnicích $i, j$ jsou dány následujícím pøedpisem:
+
+$$ A_{i,j} = \left\{ \matrix {0 \Leftrightarrow \{i,j\} \notin E \cr
+ 1 \Leftrightarrow \{i,j\} \in E \cr
+ }
+\right.$$
+
+Ná¹ graf z obrázku vý¹e by tedy v maticové reprezentaci vypadal takto:
+
+$$\bordermatrix{
+ & A & B & C & D\cr
+A & 0 & 1 & 1 & 0\cr
+B & 1 & 0 & 1 & 1\cr
+C & 1 & 1 & 0 & 1\cr
+D & 0 & 1 & 1 & 0\cr
+}$$
+
+S touto matici se pracuje velmi snadno (napø. v¹echny sousedy i-tého vrcholu
+zjistíme jednodu¹e tak, ¾e projdeme i-tý øádek matice, co¾ pøedstavuje èasovou
+slo¾itost $O(n)$), ale má i jednu zøejmou nevýhodu: její velikost je v¾dy
+kvadratická bez ohledu na to, jak "øídký" je graf. U grafu s mnoha vrcholy, ale
+s malým poètem hran, tedy budeme zbyteènì plýtvat místem v pamìti.
+
+Nabízí se proto tøetí, úspornìj¹í, zpùsob, a to:
+\medskip
+
+\s{3. seznam sousedù}
+
+%$$\matrix{
+%A&\rightarrow&BC\quad \cr
+%B&\rightarrow&ACD\cr
+%C&\rightarrow&ABD\cr
+%D&\rightarrow&BC\quad \cr
+%}$$
+
+V pamìti poèítaèe by se dal seznam sousedù uchovávat dvìma poli: polem vrcholù
+$V$ grafu, jeho¾ prvky postupnì pro ka¾dý vrchol udávají index na zaèátek
+odpovídajícího úseku v poli $E$, ve kterém by byli ulo¾eni jeho sousedé.
+\figure{img4_susedia.eps}{Znázornìní polí seznamu sousedù.}{\epsfxsize}
+
+Na tuto reprezentaci u¾ staèí prostor $O(n + m)$, co¾ u¾ je, na rozdíl od
+pøedchozího kvadratického prostoru, docela pøíjemné.
+
+Je¹tì dodejme, ¾e aèkoliv jsme jednotlivé reprezentace ukazovali na pøíkladì
+neorientovaného grafu, orientované grafy se dají reprezentovat prakticky stejnì. U
+orientovaných grafù je toti¾ jediný rozdíl: existuje-li hrana z vrcholu $u$ do
+vrcholu $v$, neznamená to, ¾e existuje i hrana z $v$ do $u$ (tedy $(u, v)$ a $(v, u)$
+tvoøí uspoøádané dvojice).
+
+\h{Prùchod grafem}
+
+\s{Definice:} Vrchol je dosa¾itelný, pokud do nìj vede cesta z poèáteèního vrcholu.
+
+\s{Efektivní prùchod by mìl splòovat tyto podmínky:}
+\itemize\ibull
+\:ka¾dou hranu projdu právì jednou (resp. maximálnì dvakrát - 1x tam, 1x zpìt)
+\:hranou se vracím zpìt, a¾ kdy¾ u¾ z vrcholu, na kterém momentálnì jsem, nevede
+ dosud neprozkoumaná hrana
+\:pokud jsem do¹el po hranì do ji¾ døíve nav¹tíveného vrcholu, hned se vrátím
+\endlist
+
+\s{Koneènost}
+
+\proof
+Plyne z prvního bodu.
+\qed
+
+\s{Korektnost} (algoritmus projde celý graf)
+
+\proof
+Sporem: Nech» existuje vrchol, který je dosa¾itelný, ale algoritmus ho nepro¹el.
+Pak jsme ale pøi prùchodu narazili na $\{u, v\}\in E$ takovou, ¾e vrchol $u$ je
+dosa¾itelný a $w$ dosa¾itelný není. Tzn., ¾e $\{u, w\}$ tvoøí neprozkoumanou
+hranu $\Rightarrow$ spor s druhým bodem.
+\qed
+
+\s{Èasová slo¾itost}
+
+Z toho, ¾e ¾ádná hrana ani vrchol se neprojde vícekrát (první podmínka
+efektivního prùchodu), plyne èasová slo¾itost $O(n + m)$.
+
+\>Nyní si nìco povíme o dvou zpùsobech, kterak mù¾eme procházet grafy.
+
+\s{Prùchod do hloubky} (Depth First Search, DFS)
+
+Prùchod do hloubky je prvním základním postupem pøi procházení grafem. Pokud bychom
+chtìli DFS trochu pøiblí¾it, mohli bychom ho pøirovnat k Théseovì postupu
+pøi prùchodu Minotaurovho bludi¹te za pomoci Ariadniny nitì.
+
+\s{Prùchod do ¹íøky} (Breadth First Search, BFS)
+
+Na rozdíl od prohledávání do hloubky vyu¾ívá prohledávání do ¹íøky namísto zásobníku
+frontu. Motivaèní úlohou by mohlo být napø. hledání nejkrat¹í cesty mezi dvìma vrcholy
+v grafu, nebo známý pøevozníkùv problém.
+\figure{img6_BFS.eps}{}{\epsfxsize}
+
+\s{Pø.} Koza, vlk a zelí
+\figure{img8_kvz.eps}{}{\epsfxsize}
+
+Øe¹ení ulohy by se dalo popsat jako hledání cesty grafem stavù od poèáteèního stavu
+(koza, vlk i zelí jsou na jednom bøehu) do koncového (v¹ichni byli bez újmy pøevezeni
+na druhý bøeh øeky).
+
+\h{DFS na neorientovaném grafu}
+
+Pøi prùchodu grafu do hloubky pou¾íváme zásobník. Na zaèátku se v nìm nachází pouze
+vstupní vrchol v. Dokud není zásobník prázdný, tak z nìj odebíráme vrchol; ten oznaèíme
+jako zpracovaný (abychom ho v budoucnu, kdybychom na nìj je¹tì nìkdy narazili,
+nezpracovávali znovu) a do zásobníku pøidáme v¹echny jeho sousedy takové, kteøí
+je¹tì nebyli oznaèeni jako zpracovaní.
+
+\>Vrcholy znaèíme tìmito barvami
+\itemize\ibull
+\:bílá - dosud nenav¹tívìné vrcholy (na zaèátku jsou tedy bílé v¹echny)
+\:¹edá - oznaèuje právì zpracovávaný vrchol
+\:èerná - u¾ zpracované vrcholy
+\endlist
+
+\>U ka¾dého vrcholu si dále také zapamatujeme èas pøíchodu a èas odchodu
+($in[v]$, $out [v]$).
+
+\s{Algoritmus}
+
+\algo
+\:procedure Projdi(v)
+\::color[v] := ¹edá
+\::time := time + 1
+\::in[v] := time
+\::$\forall$ w $\in$ Sousedi(v):
+\:::if color[w] = bílá $\Rightarrow$
+\::::Projdi(w)
+\::color[v] := èerná
+\::time := time + 1
+\::out[v] := time
+\::
+\:procedure DFS()
+\::$\forall$ v: color[v] := bílá
+\::time := 0
+\::$\forall$ v $\in$ V :
+\:::if color[v] = bílá $\Rightarrow$
+\::::Projdi(v)
+\endalgo
+
+Implementaci zásobníku jsme v tomto meta-algoritmu nahradili pou¾itím rekurze
+(která ale funguje na podobném principu, kdy se na zásobník vkládají jednotlivá
+volání rekurzivní funkce).
+
+Na následujících obrázcích je znázornìno, jak probíhá prùchod grafu pomocí DFS:
+
+\figure{img2_dfs.eps}{}{\epsfxsize}
+\figure{img3_dfs.eps}{Znázornìní prùbìhu algoritmu.}{\epsfxsize}
+
+\s{Hrany v tomto znázornìní dìlíme na:}
+\itemize\ibull
+\:stromové - v momentì prùchodu vedou do dosud nenav¹tíveného, bílého, vrcholu
+\:zpìtné - ostatní hrany
+\endlist
+
+\s{Pozorování:} plné hrany tvoøí strom (DFS strom, strom prùchodu do hloubky).
+
+\s{Aplikace:}
+\item{(1)} hledání kostry
+\item{(2)} hledání komponent souvislosti
+
+Èasová slo¾itost tìchto aplikací DFS je stejná jako slo¾itost pùvodního
+meta-algoritmu, tedy $O(n + m)$.
+
+\item{(3)} hledání artikulací
+
+Tuto aplikaci si teï rozebereme tro¹ku podrobnìji. Nejprve v¹ak nìkolik definic.
+
+\s{Definice:} {\I Artikulace} je vrchol v grafu, jeho¾ odebráním se graf rozpadne na alespoò
+dvì komponenty souvislosti.
+
+\s{Definice:} {\I Most} je hrana v grafu, jejím¾ odebráním se graf rozpadne na dvì
+komponenty souvislosti.
+
+\s{Definice:} Graf je {\I 2-souvislý,} jestli¾e po odstranìní libovolného vrcholu
+zùstane souvislý.
+
+\s{Definice:} {\I 2-souvislá komponenta} je maximální 2-souvislý podgraf.
+
+Jak bychom tedy pøi hledání artikulací v grafu mohli postupovat? Jako první
+se nabízí tento algoritmus:
+
+\s{Hloupý algoritmus}
+
+Berme si postupnì v¹echny vrcholy grafu; pro ka¾dý zjistíme, jestli se jeho
+odebráním zvý¹il poèet komponent. K tomu bychom mohli pou¾ít prùchod do hloubky,
+který jsme si u¾ ukázali. Procedura Projdi(v) projde a oznaèí v¹echny vrcholy
+dosa¾itelné z v, tedy v¹echny vrcholy komponenty obsahující vrchol v. Mù¾eme
+tedy udìlat takovou zmìnu, ¾e si budeme nav¹tívené vrcholy oznaèovat èíslem
+komponenty.
+
+Takový algoritmus by mìl slo¾itost $n \times O(n + m))$ (pro ka¾dý vrchol grafu
+bychom n-krát spou¹tìli DFS), co¾ se nám vùbec nelíbí. Pokusme se tedy najít
+nìco lep¹ího. Nejprve uèiòmì toto pozorování:
+
+\s{Pozorování:} v je artikulace $\Leftrightarrow$ v je koøen DFS stromu a má
+alespoò dva potomky nebo v není koøen a má syna w takového, ¾e ze ¾ádného
+jeho potomka nevede hrana do pøedchùdce v.
+
+\s{Definice:} Pøípustná cesta je taková cesta, pokud vede pøes stromové hrany
+DFS stromu (mù¾e konèit i skokem zpìt).
+
+\figure{img7_hrany.eps}{Pøíklady pøípustných cest.}{\epsfxsize}
+
+\>Dále si definujme funkci low pro ka¾dý vrchol v, takto:
+
+\s{low(v)} = min(in[w], z v do w vede pøípustná cesta)
+
+\s{Pozorování:} v je artikulace $\Leftrightarrow$ v je koøen DFS stromu a má
+alespoò dva potomky nebo má syna w takového, ¾e platí $low(w) \geq in[w]$.
+
+\s{Jak ale hodnotu funkce low(w) spoèítat?}
+
+low(v) = min(x, y, z)
+
+x = in[v]
+
+y = min(in[w], $\{w, v\}$ je zpìtná hrana)
+
+z = min(low(w), $\{v, w\}$ je stromová hrana)
+
+\h{DFS na orientovaném grafu}
+
+\figure{img5_dfso.eps}{Graf a znázornení prùbehu:}{\epsfxsize}
+
+\s{Hrany v dìlíme na: }
+\itemize\ibull
+\:stromové $\{A,B\}$, $\{B,C\}$, $\{B,D\}$
+\:zpìtné: vedou do pøedchùdce v DFS stromì (do ¹edého vrcholu)
+\:pøíèné: vedou do vrcholu v jiném podstromì (do èerného vrcholu)
+\:dopøedné: vedou do svého potomka (do èerného vrcholu v tém¾e podstromì)
+\endlist
+\bigskip
+
+\>Na závìr nìkolik \s{pozorování:}
+
+Intervaly (in[v], out[v]) $\forall v \in V $ tvoøí dobré uzávorkování.
+
+v je následník u $\Leftrightarrow (in[v], out[v]) \subseteq (in[u], out[u]) $
+
+$\{u,v\}$ je zpìtná hrana $\Leftrightarrow (in[u], out[u]) \leq (in[v], out[v])$
+
+$\{u,v\}$ je pøíèná hrana $\Leftrightarrow in[v] < out[v] < in[u] < out[u]$
+\bye
--- /dev/null
+P=8-grafy
+
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+\input ../lecnotes.tex
+
+\prednaska{9}{Prùchod grafem}{(zapsali T. Hubík, V. Kolomièenko, L. Slinták)}
+
+\h{1. do hloubky (DFS)}
+
+\>$\<in(v)>$, $\<out(v)>$
+
+\>Hrany rozdìlíme do nekolika tøíd
+\itemize\ibull
+\:{\I stromové} postupují do neoznaèených vrcholù
+\:{\I zpìtné} vedou do u¾ oznaèených vrcholù
+\:{\I dopøedné} vedou do vrcholù, odkud ji¾ jsme se vrátili
+\:{\I pøíèné} pouze zprava doleva
+\endlist
+
+\>{\I Pozn.: u neorientovaných grafù mohou být pouze stromové a zpìtné hrany}
+
+{\I Slo¾itost $\O(m+n)$, n ... poèet vrcholù, m ... poèet hran}
+
+\h{2. do ¹íøky (BFS)}
+{\I Slo¾itost $\O(m+n)$}
+
+\s{Definice:} {\I Lineární uspoøádání} $<$ vrcholù grafu G je topologické $\Longleftrightarrow \forall (u,v) \in E(G) : u < v$
+
+\>{\I Pozorování: Pro cyklické grafy topologické uspoøádání neexistuje.}
+
+\s{Vìta:} Libovolný acyklický orientovaný graf (DAG) má topologické uspoøádání a lze ho sestrojit v $\O(m+n)$.
+
+\proof
+Spustíme DFS, $u < v$ právì tehdy, kdy¾ $\<out(u)> > \<out(v)>$
+
+hrany:
+\itemize\ibull
+\:stromová - $\<out(u)> > \<out(v)>$
+\:zpìtná - existuje cyklus a to je spor
+\:dopøedná - $\<out(u)> > \<out(v)>$
+\:pøíèná - $\<out(u)> > \<in(u)> > \<out(v)>$
+\endlist
+\>{\I Dùsledek:} Orientovaný graf $G$ je cyklický $\Longleftrightarrow$ DFS najde zpìtnou hranu, existuje zpìtná hrana $\Rightarrow$ existuje cyklus, neexistuje zpìtná hrana $\Rightarrow$ existuje topologické uspoøádání $\Rightarrow$ neexistuje cyklus.
+\qed
+
+
+\h{Nejdel¹í cesta v DAGU:}
+
+(z $S$ do ka¾dého vrcholu)
+
+\algo
+\:Zvolíme topologické uspoøádání na $G$, BÚNO $S$ je minimum.
+\:$\forall v: D(v)=-\<inf> ; D(S)=0$
+\:Postupnì procházíme vrcholy $v \in V(G)$ v topologickém poøadí a pro $\forall v$ spoèítáme $D(v)$
+\:$D(v)=\<max>(D(n))+1$
+
+$u:(u,v) \in E(G)$
+\endalgo
+
+\>{\I Pozorování:} Ka¾dý DAG obsahuje {\I zdroj} (vycházejí z nìj cesty) a {\I stok} (cesty do nìj vstupují)
+
+\s{Rozkládání orientovaných grafù na komponenty souvislosti}
+
+\s{Definice:} $R$ bude relace na $V(G)$ tak, ¾e $uRv \Longleftrightarrow$ existuje orientovaná cesta v $G$ z $u$ do $v$ a opaène
+
+\s{Definice:} $G$ je silnì souvislý $\Longleftrightarrow \forall (u,v) \in V(G) : uRv$
+
+\>{\I Pozorování:} $R$ je ekvivalence, $u...v$ $v...w$ je $u$-$w$ sled $\Rightarrow$ existuje $u$-$v$ cesta
+
+{\narrower
+\s{Definice:} Komponenty silné souvislosti grafu $G \Longleftrightarrow$ ekvivalence tøídy relace $R$
+}
+
+\s{Lemma:} $\forall (G)$ a $\forall (u,v) \in V(G)$: existuje $u$-$v$ sled $\Longleftrightarrow$ existuje $u$-$v$ cesta
+
+\proof
+Buï $S$ $u$-$v$ sled, na kterém se opakuje vrchol $z$.
+Uva¾me sled $u...z$ $z...v$, co¾ je také $u$-$v$ sled, ale je krat¹í.
+Iterujeme $\Rightarrow$ zbude cesta.
+\qed
+
+\s{Definice:} Graf komponent $C(G)$
+
+$V(C(G))$: SSK grafu $G$
+
+$(c_1,c_2) \in E(C(G)) \Longleftrightarrow \exists v_1 \in c_1, v_2 \in c_2 : (v_1, v_2) \in E(G) $
+
+\s{Lemma:} $\forall (G) : C(G)$ je DAG.
+
+\proof
+obrázkem a sporem, existuje cyklus $c_1, c_2, ..., c_k$ v $C(G)$.
+\qed
+
+\h{Algoritmus hladání silnì souvislých komponent grafu v \O(n)}
+
+\>{\I Pozorování 1:} DFS z vrcholu $v$ projde $\{w, \exists v$-$w$ cesta$ \}$.
+
+\>{\I Pozorování 2:} Je-li $C$ stoková komponenta a $v \in V(C)$, pak DFS z $v$ projde právì $C$.
+
+\>{\I Pozorování 3:} Spustíme-li DFS na celý $G$, pak vchol $v$ s maximální $\<out(v)>$ le¾í ve zdrojové SSK.
+
+\>{\I Pozorování 3*:} Nech» $C_1, C_2$ jsou SSK a $C_1, C_2 \in E(C(G)$, pak $\<max>\{\<out(u)>, u \in V(C_1)\} > \<max>\{\<out(v)>, v \in V(C_2)\}$.
+
+\>{\I Pozorování 4:} $G$ a $G^R$ mají stejné SSK ( $G^R$ má proti $G$ otoèené hrany ).
+
+\s{Algoritmus:}
+
+\algo
+\:Sestrojíme $G^R$ \quad $\O(m+n)$
+\:DFS na $G^R \rightarrow \<out(v)>$ \quad $\O(m+n)$
+\:$T$ bude uspoøádání vrcholù podle klesajících $\<out(v)>$ \quad $\O(n)$
+\:$\forall v \in V(G)$ v poøadí podle $T$ \quad $\O(m+n)$
+\:pokud $v$ je¹tì nepatøí do ¾ádné komponenty, tak
+\::spustím DFS z $v$ a dosa¾ené vrcholy prohlásím za dal¹í SSK \quad $\O(m+n)$
+\endalgo
+
+\s{Vìta:} Ka¾dý orientovaný graf lze rozlo¾it na SSK v èase $\O(m+n)$
+
+\proof
+Pozorovani 3*:
+\itemize\ibull
+\:1. DFS vstoupí nedøíve do $C_1$ : Z $C_2$ vyleze urèitì døív ne¾ z $C_1 \Rightarrow \<out>(C_1) > \<out>(C_2)$.
+\:2. nejdøíve do $C_2$: Nejdøív se vrátím z celé $C_2$, a¾ pak nìkdy zase vlezu do $C_1 \Rightarrow \<out>(C_1) > \<out>(C_2)$.
+\endlist
+
+\bye
+
--- /dev/null
+P=9-dfs
+
+include ../Makerules
+++ /dev/null
-\input ../lecnotes.tex
-\input epsf.tex
-
-\prednaska{3}{Rozdìl a panuj II}{(zapsali Luká¹ Hermann, Vincent Krí¾, Oto Petøík)}
-
-\h{Násobení matic n$\times$n a Strassenùv algoritmus}
-
-Nejdøíve si pøipomeneme definici násobení dvou ètvercových matic typu $n \times n$. Platí, ¾e prvek v $i$-tém øádku a $j$-tém sloupci výsledné matice $Z$ se rovná standardnímu skalárnímu souèinu $i$-tého øádku první matice $X$ a $j$-tého sloupce druhé matice~$Y$. Formálnì zapsáno:
-
-$$ Z_{ij} = \sum_{k=1}^n X_{ik} \cdot Y_{kj}. $$
-
-\figure{nasobeni-matic.eps}{Násobení matic}{\epsfxsize}
-
-Algoritmus, který by násobil matice podle této definice, by mìl èasovou slo¾itost $ \Theta(n^3) $, proto¾e poèet prvkù ve výsledné matici je $n^2$ a jeden skalární souèin vektorù dimenze $n$ vy¾aduje lineární poèet operací.
-
-My se s touto èasovou slo¾itostí ov¹em nespokojíme a budeme postupovat podobnì jako pøi vylep¹ování algoritmu na násobení velkých èísel. Bez újmy na obecnosti pøedpokládejme, ¾e budeme násobit dvì matice typu $n \times n$, kde $n=2^k, k \in \bb N$. Obì tyto matice rozdìlíme na ètvrtiny a tyto èásti postupnì oznaèíme u matice $X$ písmeny $A$, $B$, $C$ a $D$, u matice $Y$ písmeny $P$, $Q$, $R$ a $S$. Z definice násobení matic zjistíme, ¾e ètvrtiny výsledné matice $Z$ mù¾eme zapsat pomocí souèinù èástí násobených matic. Levá horní ètvrtina bude odpovídat výsledku operací $AP+BR$, pravá horní ètvrtina bude $AQ+BS$, levá dolní $CP+DR$ a zbylá $CQ+DS$ (viz obrázek).
-
-\figure{nasobeni-matic-2.eps}{Násobení rozètvrcených matic}{\epsfxsize}
-
-\break
-
-Pøevedli jsme tedy problém násobení ètvercových matic øádu $n$ na násobení ètvercových matic øádu ${n/2}$. Tímto rozdìlováním bychom mohli pokraèovat, dokud bychom se nedostali na matice øádu 1, jejich¾ vynásobení je triviální. Dostali jsme tedy klasický algoritmus typu {\it rozdìl a panuj}. Pomohli jsme si ale nìjak ? V ka¾dém kroku provádíme 8 násobení matic polovièního øádu a navíc konstantní poèet operací na $n^2$ prvcích. Dostáváme tedy rekurentní zápis èasové slo¾itosti:
-
-$$ T(n) = 8T\left({n \over 2}\right) + \O(n^2). $$
-
-Pou¾itím Master Theoremu lehce dojdeme k závìru, ¾e slo¾itost je stále $\Theta(n^3)$, tedy stejná jako pøi násobení matic z definice. Zdánlivì jsme si tedy nepomohli, ale stejnì jako tomu bylo u násobení velkých èísel, i teï mù¾eme zredukovat poèet násobení matic polovièního øádu, které nejvíce ovlivòuje èasovou slo¾itost algoritmu. Není to bohu¾el nic triviálního, a proto si radìji rovnou øekneme správné øe¹ení. Jedná se o Strassenùv algoritmus, který redukuje potøebný poèet násobení na~7, a je¹tì pøed tím, ne¾ si uká¾eme, jak funguje, doká¾eme si, jak nám to s èasovou slo¾itostí vlastnì pomù¾e:
-
-$$ T(n) = 7T\left({n \over 2}\right) + \O(n^2) \Longrightarrow \O(n^{\log_2 7}) = \O(n^{2.808}). $$
-
-Výsledná slo¾itost Strassenova algoritmu je tedy $\O(n^{2.808})$, co¾ je sice malé, ale pro velké matice znatelné zlep¹ení oproti algoritmu vycházejícímu pøímo z~definice.
-
-\s{Lemma:} (vzorce pro násobení blokových matic ve~Strassenovì algoritmu)
-
-\def\\{\noalign{\vskip 7pt}}
-
-$$
-\pmatrix{A & B \cr\\ C & D \cr}
-\cdot
-\pmatrix{P & Q \cr\\ R & S \cr}
-=
-\pmatrix{
-T_1 + T_4 - T_5 + T_7 &
-T_3 + T_5 \cr\\
-T_2 + T_4 &
-T_1 - T_2 + T_3 + T_6 \cr
-},$$
-
-kde:
-
-$$\vbox{\halign{$#$\hfil\qquad &$#$\hfil\qquad \cr
-T_1 = (A+D)\cdot(P+S) & T_5 = (A+B)\cdot S \cr\\
-T_2 = (C+D)\cdot P & T_6 = (C-A)\cdot (P+Q) \cr\\
-T_3 = A\cdot(Q-S) & T_7 = (B-D)\cdot (R+S) \cr\\
-T_4 = D\cdot(R-P) \cr
-}}$$
-
-\medskip
-
-\proof Do ètvercù $4 \times 4$ si napí¹eme znaky $+$ nebo $-$ podle toho, jestli se pøi výpoètu dané matice pøièítá nebo odeèítá pøíslu¹ný souèin dvou matic. Øádky znamenají matice $A$, $B$, $C$ a $D$ a sloupce znaèí matice $P$, $Q$, $R$ a $S$. Pokud se tedy v prvním øádku a prvním sloupci vyskytuje znak $+$, znamená to ¾e pøièteme souèin matic $A$~a~$P$. Nejdøíve si spoèítáme pomocné matice $T_1$ a¾ $T_7$ a z nich pak dopoèítáme, co bude na pøíslu¹ných místech ve výsledné matici.
-
-\def\bbb#1{\vbox to 10pt{\vss\hbox to 10pt{\hss\tenrm #1\hss}\vss}}
-\def\bb#1{\ifx#1.\bbb{$\cdot$}\else\bbb#1\fi}
-\def\zz#1#2#3#4{\bb#1\bb#2\bb#3\bb#4}
-\def\qq#1#2#3#4{{\offinterlineskip\vcenter{\halign{\vrule ##\vrule \cr\noalign{\hrule}\zz#1\cr\zz#2\cr\zz#3\cr\zz#4\cr\noalign{\hrule}}}}}
-
-$$
-T_1 = \qq{+..+}{....}{....}{+..+} \qquad
-T_2 = \qq{....}{....}{+...}{+...} \qquad
-T_3 = \qq{.+.-}{....}{....}{....} \qquad
-T_4 = \qq{....}{....}{....}{-.+.}
-$$
-$$
-T_5 = \qq{...+}{...+}{....}{....} \qquad
-T_6 = \qq{--..}{....}{++..}{....} \qquad
-T_7 = \qq{....}{..++}{....}{..--}
-$$
-
-\medskip
-
-\def\\{\noalign{\vskip 7pt}}
-$$
-\eqalign{
-T_1 + T_4 - T_5 + T_7 &= \qq{+...}{..+.}{....}{....} = AP + BR \cr\\
-T_3 + T_5 &= \qq{.+..}{...+}{....}{....} = AQ + BS \cr\\
-T_2 + T_4 &= \qq{....}{....}{+...}{..+.} = CP + DR \cr\\
-T_1 - T_2 + T_3 + T_6 &= \qq{....}{....}{.+..}{...+} = CQ + DS \cr
-}
-$$
-
-% konec slidu z prednasky
-
-Jak je vidìt, výsledná matice je tvoøena stejnými èástmi jako pøi obyèejném násobení. Touto kapitolou jsme tedy dokázali následující vìtu:
-
-\s{Vìta:}
-Strassenùv algoritmus pro násobení matic $n \times n$ má èasovou slo¾itost
-v nejhor¹ím pøípadì $\O(n^{2.808})$. \qed
-
-\s{Poznámka:}
-Zatím nejlep¹í dokázaný algoritmus má èasovou slo¾itost $\O(n^{2.376})$, $ $ ov¹em s velkou multiplikativní konstantou.
-
-\h{Hledání $k$-tého nejmen¹ího prvku (mediánu)}
-
-V tomto oddílu se budeme zabývat tím, jak co nejrychleji najít v jakékoli posloupnosti $n$ èísel $k$-tý nejmen¹í prvek popøípadì medián. Pro ty, kdo medián neznají, tu máme definici:
-
-\s{Definice:}
-Medián posloupnosti $a_1, a_2,\ldots , a_n$ je takové $m=a_i$, kde nejvý¹e $n/2$ prvkù je men¹ích ne¾ $m$ a nejvý¹e $n/2$ prvkù je vìt¹ích ne¾ $m$.
-
-Nejjednodu¹¹ím øe¹ením by urèitì bylo celou posloupnost nejdøíve setøídit a pak u¾ jednodu¹e vyhmátnout po¾adovaný prvek. To bychom dokázali v celkem slu¹ném èase $\O(n\log n)$, ale u¾ teï mù¾eme prozradit, ¾e to jde v èase $\O(n)$. Jak?
-
-Pou¾ívat budeme metodu {\it rozdìl a panuj}. Nìjakým zpùsobem si zvolíme jeden prvek posloupnosti, který nazveme {\it pivot}. Poté rozdìlíme zadanou posloupnost na~tøi disjunktní mno¾iny. Do první dáme v¹echny prvky men¹í ne¾ pivot, do druhé stejné jako pivot a do tøetí vìt¹í ne¾ pivot. Tímto máme zaji¹tìno, ¾e prvky z první mno¾iny jsou urèitì men¹í ne¾ prvky z druhé a ty ne¾ prvky z tøetí.
-O tom, jak jsou prvky uspoøádány uvnitø tìchto mno¾in, ale nic nevíme.
-
-V posledním kroku na¹eho algoritmu se pak rozhodneme, na kterou mno¾inu svùj algoritmus rekurzivnì zavoláme. Pokud je $k$ men¹í ne¾ velikost první mno¾iny, pokraèujeme v první mno¾inì, pokud je $k$ men¹í ne¾ souèet velikostí první a druhé mno¾iny, pak hledaným prvkem je právì vybraný pivot a algoritmus skonèí, a nakonec pokud ani jedna podmínka splnìna nebyla, pustíme se do hledání ve tøetí mno¾inì, ov¹em u¾ nehledáme $k$-tý nejmen¹í prvek, ale $l$-tý, kde $l$ se rovná $k$ minus velikost prvních dvou mno¾in. Pro vìt¹í názornost zapí¹eme tento algoritmus formálnìji:
-
-{\bo Select($k,X$):} (Hledání $k$-tého nejmen¹ího prvku v mno¾ine $X$)
-\algo
-\:Jestli¾e $\vert X\vert \le 1$, vyøe¹íme triviálnì.
-\:Zvolíme pivota $p \in X$.
-\:Rozdìlíme mno¾inu X na tøi podmno¾iny: $M = \{x \in X; x < p\},$ $ P = \{x \in X; x = p\}, V = \{x \in X; x > p\}$.
-\:Jestli¾e $k \le \vert M\vert$, vrátím výsledek funkce \<Select>($k$, $M$).
-\:Jestli¾e $k \le \vert M\vert + \vert P\vert$, vrátíme pivota $p$.
-\:Jinak vrátíme výsledek funkce \<Select>($k - \vert M\vert - \vert P\vert, V$).
-\endalgo
-
-Na první pohled je vidìt, ¾e se algoritmus zastaví (vstup se v¾dy zmen¹í alespoò o 1) a ¾e vydá v¾dy správný výsledek. Jak je to ov¹em s èasovou slo¾itostí? Rozdìlení do mno¾in a podmínky v druhém a tøetím kroku mají lineární slo¾itost, èemu¾ se nevyhneme. Pøi ne¹»astné volbì pivota se nám mù¾e stát, ¾e poèet rekurencí mù¾e být a¾ $n$, tedy celková slo¾itost v nejhor¹ím pøípadì je $\Theta(n^2)$, èím¾ jsme si oproti prostému setøídìní je¹tì pohor¹ili. Co s tím? Jak je vidìt, velmi dùle¾itá je volba pivota. Tu mù¾eme provést nìkolika zpùsoby:
-
-a) Pivot by se v setøídìné posloupnosti vyskytoval uprostøed, vstup by se tedy stále pùlil. Èasovou slo¾itost vypoèteme z rekurentního zápisu:
-
-$$ T(n) = T\left({n \over 2}\right) + \O(n) = \O\left(n + {n \over 2} + {n \over 4} + \dots\right) = \O(n). $$
-
-To by bylo sice skvìlé, ale nalezení takového pivota je vlastnì vyøe¹ení úlohy hledání mediánu, o co¾ se sna¾íme. Tedy jsme si vùbec nepomohli.
-
-\break
-
-b) Pivot by se v setøídìné posloupnosti náchazel v prostøedních dvou ètvrtinách (nazvìme tento prvek "l¾imedián"). Tím bychom v ka¾dém kroku urèitì odstranili mno¾inu velikosti ètvrtiny vstupu. Èasová slo¾itost tohoto øe¹ení by byla:
-
-$$ T(n) = T\left({3 \over 4}n\right) + \O(n) = \O\left(n + {3 \over 4}n + {9 \over 16}n + \dots\right) = \O(n). $$
-
-Tímto bychom tedy také dosáhli lineární èasové slo¾itosti. Ale jak vybrat pivota tak, aby se nacházel v prostøedních dvou ètvrtinách, aby nám nám to nepokazilo lineární slo¾itost?
-
-c) Pivot bude v¾dy prostøední prvek zadané posloupnosti. V tomto pøípadì je zøejmé, ¾e existují urèité vstupy, pro které bude èasová slo¾itost $\Omega(n^2)$, ale v~prùmìrném pøípadì, bude èasová slo¾itost $\O(n)$, co¾ doká¾eme v pøí¹tí kapitole.
-
-d) Pivot bude náhodnì zvolený prvek zadané posloupnosti. Tímto dosáhneme nìèeho podobného jako v pøedchozím pøípadì, ale u¾ nebude existovat vstup, pro který by èasová slo¾itost byla $\Omega(n^2)$, jen nìkteré prùbìhy tohoto algoritmu na rùzných vstupech budou trvat takto dlouho. Prùmìrná èasová slo¾itost je tedy zase $O(n)$.
-
-My se ale nespokojíme pouze s lineární prùmìrnou èasovou slo¾itostí. Chceme dosáhnout lineární slo¾itosti i v nejhor¹ím pøípadì. Ne¾ si ale prozradíme takové øe¹ení, rozmyslíme si, ¾e u¾ teï známe 3 èasové slo¾itosti, i kdy¾ ne v¹echny je¹tì umíme analyzovat. Je to èasová slo¾itost v nejhor¹ím pøídadì, v prùmìrném pøípadì pøes v¹echny vstupy a v prùmìrném pøípadì pøes v¹echny bìhy programu. V¹imnìte si hlavnì rozdílu posledních dvou.
-
-A nyní u¾ slibované øe¹ení s lineární nejhor¹í èasovou slo¾itostí. Vyjdeme z~mo¾nosti b), tedy pokusíme se najít takového pivota, který by na pøí¹tí krok zaruèenì omezil velikost analyzované mno¾iny. Dosáhneme toho tímto algoritmem:
-
-\s{Volba pivota:}
-\algo
-\:Rozdìlíme vstup na pìtice \dots $\O(n)$
-\:Spoèteme medián ka¾dé pìtice \dots $\O(n)$
-\:Spoèteme medián mediánù pìtic tak, ¾e rekurzivnì zavoláme tentý¾ algoritmus \<Select>($n / 10$, {mediány pìtic}) a to je pivot.
-\endalgo
-
-Abychom dokázali, ¾e tento algoritmus bude mít opravdu lineární èasovou slo¾itost, musíme si nejdøíve dokázat následující lemma:
-
-\s{Lemma:} V ka¾dém kroku vypadne alespoò ${3/10}\cdot n$ prvkù.
-
-\proof Ten provedeme obrázkem. Pøedstavme si vybrané pìtice seøazené podle velikosti od nejvìt¹ího prvku a zakresleme je do sloupcù. Jejich mediány tedy vyplòují prostøední øadu. Tyto pìtice pak seøaïme podle velikosti jejich mediánù (nejmen¹í vlevo). Hledaný pivot se tedy nachází (pokud pøedpokládáme pro jednoduchost lichý poèet pìtic) pøesnì uprostøed. O prvcích nad pivotem a napravo od nìj mù¾eme urèitì øíct, ¾e jsou vìt¹í nebo rovny pivotu, prvky pod ním a nalevo od nìj jsou zase urèitì men¹í nebo rovny pivotu.
-
-Podle konstrukce algoritmu tedy zaruèenì vypadne jedna nebo druhá skupina prvkù. Obì tyto skupiny pøitom obsahují, jak je vidìt z obrázku, alespoò $3/10 \cdot n$ prvkù.
-\qed
-
-\figure{petice.eps}{Pìtice}{125mm}
-
-Nyní u¾ se tedy mù¾eme pustit do výpoètu èasové slo¾itosti. V ka¾dém kroku funkce zavolá sama sebe nejdøíve na vstup velikosti ${n \over 5}$ a poté na vstup velikosti nejvý¹e ${7 \over 10}n$. Ostatní operace zùstávají lineární. Èasovou slo¾itost v nejhor¹ím pøípadì tedy mù¾eme zapsat rekurentním vzorcem:
-
-$$ T(n) = \O(n) + T\left({n \over 5}\right) + T\left({7 \over 10}n\right). $$
-
-Tento rekurentní vzorec ale zatím neumíme obecnì øe¹it. Mohli bychom ho postupnì rozepsat, ale trvalo by dlouho, ne¾ bychom z toho nìco vykoukali. Lep¹í bude rovnou dokázat, ¾e tento rekurentní vzorec odpovídá lineární slo¾itosti, zkusíme tedy dosadit $T(n) = cn$:
-
-$$ cn = n + {1 \over 5}cn + {7 \over 10}cn \quad\Longleftrightarrow\quad c = 10. $$
-
-Tímto jsme tedy dokázali, ¾e èasová slo¾itost tohoto algoritmu v nejhor¹ím pøípadì je $\O(n)$.
-Jeliko¾ to rychleji urèitì není mo¾né, mù¾eme na závìr této kapitoly zformulovat tuto vìtu:
-
-\s{Vìta:} Hledání k-tého nejmen¹ího prvku v posloupnosti délky n má èasovou slo¾itost v nejhor¹ím pøípadì $\Theta(n)$. \qed
-
-\bye
+++ /dev/null
-P=3-strassen
-
-include ../Makerules
+++ /dev/null
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+++ /dev/null
-\input ../lecnotes.tex
-
-\def\E{{\bb E}}
-
-\prednaska{4}{Prùmìrná èasová slo¾itost}{(R. Cinkais, R. Barczi a R. Vansa)}
-
-\h{První pomoc z~teorie pravdìpodobnosti}
-
-\s{Definice:} {\I Diskrétní pravdìpodobnostní prostor} je uspoøádaná dvojice
-$(\Omega,P)$, kde $\Omega$ je nejvý¹e spoèetná mno¾ina {\I elementárních jevù}
-a $P:\Omega\rightarrow[0,1]$ je funkce, pro kterou platí:
-$$\sum_{x\in\Omega}P(x)=1.$$
-Hodnotám $P(x)$ budeme øíkat {\I pravdìpodobnosti} jednotlivých elementárních jevù.
-
-\s{Pøíklad:} Mno¾ina elementárních jevù pøi vrhu kostkou je $\Omega=\{1,2,3,4,5,6\}$ a pokud je kostka spravedlivá, platí $\forall x\in\Omega:P(x)=1/6$.
-
-\s{Definice:} {\I Jevem} $A$ rozumíme libovolnou mno¾inu $A\subseteq\Omega$. {\I Pravdìpodobnost}
-jevu $A$ poèítáme jako: $$\Pr[A]=\sum_{a\in A}P(a).$$
-
-\s{Pøíklad:}
-Jaká je pravdìpodobnost slo¾eného jevu \uv{na kostce padne sudé èíslo}? Oznaèíme
-$A=\{2,4,6\}\subseteq\Omega$. Hledanou pravdìpodobností je pak
-$P(A)=1/6+1/6+1/6=1/2$.
-
-\s{Definice:} {\I Náhodná promìnná} $X$ je funkce $X:\Omega\rightarrow{\bb
-R}$. {\I Støední hodnotou} náhodné promìnné $X$ rozumíme výraz
-$$\E{}X=\sum_{x\in\Omega}P(x)\cdot X(x).$$
-
-\s{Pøíklad:} Nech» $H$ je náhodná promìnná pøiøazující vrhu kostkou dosa¾enou
-hodnotu, tedy $\forall x\in\Omega:H(x)=x$. Støední (oèekávanou) hodnotou této
-promìnné je $\E{}H= 1/6\cdot1 +1/6\cdot2 +1/6\cdot3 +1/6\cdot4 +1/6\cdot5
-+1/6\cdot6=7/2$.
-
-\s{Vìta:} ({\I Linearita støední hodnoty})
-Nech» $A,B:\Omega\rightarrow\bb R$ jsou náhodné promìnné a $\alpha\in\bb R$ je libovolná konstanta. Potom:
-\numlist\ndotted
-\:$\E{}(A+B)=\E{}(A)+\E{}(B)$,
-\:$\E{}(\alpha\cdot A)=\alpha\cdot\E{}(A)$.
-\endlist
-
-\proof
-\numlist\ndotted
-\:Podle definice støední hodnoty je:
-$$\eqalign{
-\E{}(A+B)&=\sum_{x\in\Omega}(A(x)+B(x))\cdot P(x)=\sum_{x\in\Omega}(A(x)\cdot P(x)+B(x)\cdot P(x))=\cr
- &=\sum_{x\in\Omega}A(x)\cdot P(x)+\sum_{x\in\Omega}B(x)\cdot P(x)=\E{}A+ \E{}B.
-}$$
-\:Opìt pøímo z definice støední hodnoty:
-$$\E{}(\alpha\cdot A)=\sum_{x\in\Omega}\alpha\cdot A(x)\cdot P(x)=\alpha\cdot \sum_{x\in\Omega}A(x)\cdot P(x)=\alpha\cdot\E{}A.$$
-\endlist
-\qed
-
-\s{Pøíklad:}
-Házíme dvìmi odli¹nými hracími kostkami. Uva¾ovaný prostor elementárních jevù
-je proto $\Omega=\{(1,1),(1,2),\dots, (6,5),(6,6)\}$ a $\forall
-(a,b)\in\Omega:P((a,b))=1/36$. Mìjme náhodnou promìnnou $S$ pøiøazující vrhu
-kostkami souèet dosa¾ených hodnot, tedy $\forall (a,b)\in\Omega:S((a,b))=a+b$.
-Chceme urèit $\E{}S$.
-
-Oznaème $A$ náhodnou promìnnou, která pøi vrhu kostkami vrací hodnotu na první
-kostce, tedy $\forall (a,b)\in\Omega:A((a,b))=a$. Podobnì oznaème $B$ náhodnou
-promìnnou, která vrací hodnotu na druhé kostce, tedy $\forall
-(a,b)\in\Omega:B((a,b))=b$. V~pøedcházejícím pøíkladì jsme si ukázali, ¾e
-$\E{}A=\E{}B=7/2$. Proto¾e $S((a,b))=a+b=A((a,b))+B((a,b))$, pou¾itím vìty
-o linearitì støední hodnoty dostáváme $\E{}S=\E{}(A+B)=\E{}A+\E{}B=7/2+7/2=7$.
-
-\s{Poznámka:}
-Pro ka¾dou náhodnou promìnnou $Y$ platí: $$\E{}Y=\sum_{x\in{\bb R}}x\cdot\Pr[Y=x].$$
-\proof
-Uvedený vztah plyne okam¾itì z definice støední hodnoty, slouèíme-li dohromady v¹echny èleny $x\in\Omega$ se stejnou hodnotou $Y(x)$.
-\qed
-
-\s{Prùmìrná slo¾itost algoritmu}
-
-Prùmìrnou slo¾itost algoritmu mù¾eme poèítat pøes v¹echny vstupy nebo pøes náhodné
-volby vstupù. V prvním pøípadì je algoritmus pravdìpodobnostní (\uv{hází mincí}) a slo¾itost poèítáme jako prùmìr pøes v¹echny vstupy.
-Ve druhém je zase algoritmus deterministický a zajímá nás pouze slo¾itost v nejhor¹ím pøípadì.
-Pokud jde o koneèný výsledek, jsou oba uvedené zpùsoby ekvivalentní.
-
-Na¹im cílem bude odvodit prùmìrnou slo¾itost algoritmu $\<Select>(k,X)$ z~minulé pøedná¹ky.
-K~tomu si uká¾eme algoritmus slou¾ící k výbìru pivota pro $\<Select>(k,X)$ ($\<Select>(k,X)$ jej pøi svém bìhu
-bude spou¹tìt jako podprogram). Bez újmy na~obecnosti pøedpokládejme,
-¾e v¹echny prvky mno¾iny $X$ jsou navzájem rùzné.
-
-\s{Algoritmus:} ({\I Hledání l¾imediánu v mno¾inì $X$})
-\algo
-\:Zvol náhodnì $x\in X$ (v¹echna~$x$ se stejnou pravdìpodobností).
-\:Spoèítej $A=\vert \{y\in X:y<x\} \vert$ a $B=\vert\{y\in X: y>x\}\vert$.
-\:Pokud $A\geq \vert X\vert/4$ a~zároveò $B\geq \vert X\vert/4$, vra» $x$, jinak skoè na~1.
-\endalgo
-
-\s{Pozorování:}
-Uva¾ujme, ¾e házíme spravedlivou mincí. Aby nedo¹lo k ¾ádnému mezinárodnímu sporu, oznaème strany mince jako tuèòák ($T$) a ryba ($R$). Tedy pravdìpodobnost pádu na nìjakou stranu je $1/2$.
-Budeme zkoumat, jaká je støední hodnota èekání na pád tuèòáka $\E{}T$. Mno¾inu v¹ech jevù (posloupností vrhù), které mohou nastat, oznaème jako $\Omega=\{T,RT,RRT,\dots,RRR\dots\}$. Platí:
-$$\Pr[T]={1\over 2}, \Pr[RT]={1\over 4}, \Pr[R^{k}T]={1\over 2^{k+1}}, \Pr[RRR\dots]=\lim_{k\rightarrow\infty}{1\over 2^k}=0.$$
-Podle pozorování máme: $$\E{}T=\sum_{k=1}^{\infty}k\cdot\underbrace{\Pr[T=k]}_{1\over 2^k}=\sum_{k=1}^{\infty}{k\over 2^k}.$$
-Tuto sumu je v¹ak trochu obtí¾né spoèítat (kdo chce si to mù¾e zkusit), proto provedeme následující trik:
-$$\E{}T=1+{1\over 2}\cdot0+{1\over 2}\cdot\E{}T,$$
-to znamená, ¾e po jednom tahu jsme s polovièní pravdìpodobností vyhráli
-a s polovièní pravdìpodobností se situace nezmìnila. Jednoduchou úpravou tedy
-zjistíme, ¾e: $$\E{}T=2.$$
-
-\s{Lemma:}
-Èekání na jev, který nastane s pravdìpodobností $p$, trvá prùmìrnì $1/p$.
-
-\proof
-$\E{}T=1+(1-p)\E{}T$, tedy $\E{}T=1/p$.
-\qed
-
-Pøi hledání l¾imediánu máme polovièní ¹anci, ¾e se trefíme pøi náhodném výbìru
-do prostøedních dvou ètvrtin -- prùmìrnì tedy vykonáme dva prùchody cyklem, pøièem¾
-jeden prùchod trvá $\Theta(n)$. Celkem bude tedy hledání l¾imediánu trvat prùmìrnì
-$\Theta(n)$.
-
-\s{Vìta:}
-\<Select> s~náhodnou volbou pivota má v prùmìru èasovou slo¾itost $\Theta(n)$.
-
-\proof
-Rozdìlíme algoritmus na {\I fáze} $\equiv$ posloupnosti iterací konèící dobrou
-volbou pivota (jako l¾imedián). V¹imneme si, ¾e jedna fáze zmen¹í vstup na nejvý¹e ${3\over 4}\cdot n$
-a~trvá v~prùmìru $\Theta(n)$ (prùmìrnì 2~iterace po $\Theta(n)$).
-Tedy $\E{}T=\E{}T_1+\E{}T_2+{\dots}=\Theta(n)$, kde $\E{}T_i$ oznaèuje $i$-tou fázi.
-\qed
-
-\s{Vìta:}
-\<Select> s~pivotem rovným prostøednímu prvku má v~prùmìru pøes permutace na vstupu èasovou slo¾itost $\Theta(n)$.
-
-\proof
-Uká¾eme, ¾e algoritmus s~pevnou volbou pivota se na~náhodném vstupu chová
-stejnì, jako algoritmus s~náhodnou volbou pivota na~pevném vstupu, jeho¾
-prùmìrnou slo¾itost odhaduje pøedchozí vìta.
-
-Pokud je na~vstupu náhodná permutace~$\pi$, jsou v¹echny hodnoty pivota~$p$ stejnì pravdìpodobné.
-Zbývá ukázat, ¾e dal¹í iterace algoritmu dostane na vstupu prvky vìt¹í nebo men¹í ne¾~$p$
-opìt v~náhodném poøadí (s~rovnomìrným rozdìlením pravdìpodobností).
-K~tomu sestrojíme bijekci mezi mno¾inou v¹ech permutací na~$\{1,\ldots,n\}$ a mno¾inou ètveøic $(p, L, \pi_M, \pi_V)$, kde:
-
-\itemize\ibull
-\:$p\in\{1,\ldots,n\}$ je prostøední prvek permutace~$\pi$ (pivot), tedy $\pi[l]$ le¾ící na~pozici $l=\lfloor {(1+n)/2} \rfloor$,
-\:$L\in{{\{1,{\dots},l-1,l+1,{\dots},n\}} \choose {p-1}}$ je mno¾ina pozic, na~nich¾ jsou
- umístìny prvky men¹í ne¾~$p$ (ty, které ná¹ algoritmus umístí do~mno¾iny~$M$),
-\:$\pi_M$ permutace na~$\{1,\ldots, p-1\}$ popisující poøadí men¹ích prvkù a
-\:$\pi_V$ permutace na~$\{p+1,\ldots, n\}$, tedy poøadí vìt¹ích prvkù.
-\endlist
-
-Jakmile zvolíme~$p$, jsou volby~$L$, $\pi_M$ a $\pi_V$ navzájem nezávislé. Ka¾dé volbì~$\pi_M$
-tedy odpovídá stejný poèet permutací~$\pi$, a~proto jsou v¹echny~$\pi_M$ stejnì pravdìpodobné.
-Analogicky pro~$\pi_V$.
-
-\qed
-
-\bye
+++ /dev/null
-P=4-avg
-
-include ../Makerules
+++ /dev/null
-\input ../lecnotes.tex
-
-\prednaska{5}{QuickSort a slo¾itost tøídìní}{(Michal Sta¹a, Jan Návrat)}
-
-Dostaneme posloupnost, její¾ prvky dovedeme porovnávat, a sna¾íme se co
-nejefektivnìji posloupnost setøídit. Uká¾eme si tøídící algoritmus QuickSort
-(pro pøátele QSort nebo QS) zalo¾ený na metodì Rozdìl a panuj:
-
-\s{Algoritmus:} (QuickSort)
-\def\concat{\mathop{\hbox{.}}}
-
-\algo
-\:Pokud $\vert X\vert \leq 1$, vrátíme~$X$.
-\:Vybereme pivota $p \in X$.\foot{pozdìji upøesníme, jak}
-\:$M \leftarrow \{x \in X ; x \le p\}$.
-\:$P \leftarrow \{x \in X ; x = p\}$.
-\:$V \leftarrow \{x \in X ; x \ge p\}$.
-\:Vrátíme $\<Quicksort>(M) \concat P \concat \<Quicksort>(V)$. \foot{Operátor \uv{$\concat$} znaèí zøetìzení seznamù}
-\endalgo
-
-\s{Rozbor:}
-V¹imnìme si, ¾e QS se urèitì zastaví a také ¾e vydá
-správný výsledek. To mù¾eme ovìøit napøíklad indukcí podle $\vert X\vert$.
-
-Podobnì jako u vybírání $k$-tého nejmen¹ího prvku v minulých pøedná¹kách
-i zde èasová slo¾itost závisí hlavnì na volbì pivota. Kdybychom za pivota
-zvolili medián, vy¹la by èasová slo¾itost stejnì jako u MergeSortu:
- $$ T(n) = 2T(n/2) + O(n) = O(n\log n). $$
-Pokud naopak budeme volit pivota ne¹ikovnì, dostaneme:
- $$ T(n) = T(n - 1) + \Theta(n) = \Theta(n^2). $$
-V ideálním pøípadì bychom tedy chtìli za pivota zvolit medián, av¹ak jeho pøímým výpoètem
-bychom algoritmus pøíli¹ zpomalili.
-Pou¾ívá se proto mnoho zpùsobù, jak vybrat rychle pivota blízkého mediánu.
-Èasto pou¾ívanou metodou je náhodný výbìr, v praxi realizovaný nìjakým pseudonáhodným
-generátorem. Uká¾eme, ¾e v tomto pøípadì je QS v prùmìrném pøípadì rychlý:
-
-\s{Vìta:} QS s náhodnou volbou pivota má èasovou slo¾itost v prùmìru $\O(n\log n)$.
-
-\s{Poznámka:} Stejnì jako u výbìru $k$-tého nejmen¹ího prvku bychom také
-mohli ukázat, ¾e QS s pevnou volbou pivota spu¹tìný na náhodnou permutaci
-má tuté¾ èasovou slo¾itost. Detaily nicménì vynecháme.
-
-\noindent {\sl Dùkaz vìty:}
-
-Dùkaz provádíme rozdìlením algoritmu na fáze.
-Rekurzivní volání QSortu zle zobrazit jako¾to strom, pøièem¾ fází rozumíme cestu ve stromu volání,
-která sleduje vìt¹í díl a konèí, kdy¾ se povede vybrat za pivota l¾imedián.
-
-\figure{strom-dukaz.eps}{Dùkaz rozdìlením na fáze}{0.3\hsize}
-
-\figure{Faze.eps}{Fáze}{0.3\hsize}
-
-Ka¾dá fáze pøitom rozdìlí vstup na disjunktní èásti $X_1, \ldots, X_k$ ($k\ge 2$)
-a pivoty, kteøí je oddìlují.
-Oznaème si $n$ za velikost vstupu (poèet prvkù vstupní posloupnosti) a $n_i$ za velikost $i$-té èásti.
-Nahlédneme, ¾e platí $\sum_i n_i \leq n$.
-Velikost ka¾dé èásti je navíc nejvý¹e $3/4 \cdot n$ (na konci fáze to platí proto, ¾e jsme
-zvolili l¾imedián, pøedtím jsme v¾dy oddìlili men¹í z èástí, èili nejvý¹e $n/2$).
-
-Jedna iterace algoritmu trvá $\O(n)$ a jeliko¾ l¾imedián vybereme s pravdìpodobností alespoò $1/2$,
-je v jedné fázi v prùmìru $\O(1)$ iterací a celá fáze proto v prùmìru trvá èas $\O(n)$.
-Z toho dostaneme následující rekurenci pro prùmìrnou èasovou slo¾itost celého algoritmu:
-$${\bb E}T(n) = \sum_i {\bb E}T (n_i) + \O(n).$$
-
-Tento typ rekurence jsme je¹tì nepotkali a Kuchaøková vìta\foot{MasterTheorem} na ni nezabere,
-ov¹em mù¾eme si pomoci jednoduchou úvahou. Pøedstavme si, ¾e v na¹em stromu
-rekurzivních volání zkomprimujeme ka¾dou fázi do jednoho vrcholu. Tím vznikne
-strom, který odpovídá algoritmu, jen¾ v jedné iteraci v prùmìrnì lineárním
-èase rozdìlí vstup na nìkolik èástí a rekurzivnì se na nì zavolá.
-
-\figure{KomprimovanyStrom.eps}{Komprimovaný Strom}{0.3\hsize}
-
-\figure{Komprimace.eps}{Zpùsob vytvoøení komprimovaného stromu}{0.3\hsize}
-
-Nový strom má logaritmickou hloubku, proto¾e na ka¾dé dal¹í hladinì jsou
-délky posloupností nejvý¹e $3/4$ délek z pøedchozí hladiny. Navíc souèet
-délek pøes ka¾dou hladinu je maximálnì $n$. Proto na ka¾dé hladinì
-trávíme èas v prùmìru $\O(n)$ a v celém stromu tedy $\O(n\log n)$.
-\qed
-
-\s{Pozorování:} Na¹e první verze QS spotøebuje lineární mno¾ství pamìti
-na pomocné pole a na zásobník. Na pøedná¹ce jsme ukazovali rùzná jeho
-praktická vylep¹ení, které staèí pomocná pamì» o velikosti $\O(\log n)$.
-Detaily viz webové stránky pøedná¹ky.
-
-Známe u¾ nìkolik tøídících algoritmù s èasovou slo¾itostí $\O(n\log n)$.
-Následující vìta ukazuje, ¾e efektivnìj¹í algoritmus v obecném pøípadì
-nese¾eneme.
-
-\s{Vìta:}
-Ka¾dý tøídící algoritmus zalo¾ený na porovnávání a prohazování prvkù
-potøebuje na vstup délky $n$ v nejhor¹ím pøípadì $\Omega (n \log n)$ porovnání.
-
-\proof
-Bez újmy na obecnosti budeme nejdøíve pøedpokládat o algoritmu dvì vìci:
-Jednak to, ¾e algoritmus nejprve porovnává, a teprve potom prohazuje.\foot{Algoritmus
-mù¾eme upravit tak, aby si pamatoval aktuální permutaci prvkù a podle ní prohazoval a¾ na konci.}
-Také pøedpokládáme, ¾e vstup algoritmu je permutace na mno¾inì $\{1, \ldots, n\}$.
-
-Chování tohoto algoritmu popí¹eme rozhodovacím stromem. V rozhodovacím stromu vnitøní vrcholy
-urèují jednotlivá porovnání prvkù a listy odpovídají okam¾ikùm, kdy algoritmus pøestal porovnávat a zaèal prohazovat.
-
-\figure{RozhodovaciStrom.eps}{Rozhodovací Strom}{0.3\hsize}
-
-Vstup je tedy permutace $n$ prvkù, a víme ¾e poèet rùzných permutací je $n!$. Existuje tedy právì $n!$ rùzných vstupù.
-Dále si v¹imneme, ¾e nemohou existovat dvì vstupní permutace, pro které by algoritmus skonèil v tém¾e listu rozhodovacího stromu.
-Listù stromu je tedy alespoò tolik, kolik je rùzných vstupù, tedy $n!$.
-
-{\narrower
- \s{Lemmátko:} Binární strom hloubky $k$ má nejvý¹e $2^k$ listù.
- \par\noindent {\sl Dùkazík:} Uva¾me binární strom hloubky $k$ s maximálním poètem
- listù. V takovém stromu budou v¹echny listy urèitì le¾et na poslední hladinì
- (kdyby nele¾ely, mù¾eme pod nìkterý list na vy¹¹í hladinì pøidat dal¹í dva vrcholy a získat
- tak \uv{listnatìj¹í} strom stejné hloubky). Jeliko¾ na $i$-té hladinì je nejvý¹e $2^i$
- vrcholù, v¹ech listù je nejvý¹e $2^k$.
- \qed
-}
-
-\>Z~tohoto lemmátka plyne, ¾e rozhodovací strom musí být hluboký alespoò $\log n!$.
-
-\>Zbytek je u¾ snadné cvièení z diskrétní matematiky:
-
-{\narrower
- \s{Lemmátko:} $ n! \ge n^{n / 2}$.
- \par\noindent {\sl Dùkazík:} $n! = \sqrt{(n!)^2} = \sqrt{1(n-1)\cdot 2(n-2) \cdot \ldots \cdot n\cdot 1}$,
- co¾ mù¾eme také zapsat jako $\sqrt{1(n-1)}\cdot \sqrt{2(n-2)} \cdot \ldots \cdot \sqrt{n\cdot 1}$.
- Pøitom pro ka¾dé $1\le k\le n$ je $k(n+1-k) = kn + k - k^2 = n + (k-1)n + k(1-k) = n + (k-1)(n-k) \ge n$.
- Proto je ka¾dá z~odmocnin vìt¹í nebo rovna $n^{1/2}$ a $n!\ge (n^{1/2})^n = n^{n/2}$.
- \qed
-}
-
-\>Hloubka stromu tedy èiní minimálnì $\log n! \ge \log(n^{n/2}) = n/2 \cdot \log n = \Omega(n\log n)$,
-co¾ také zdola odhaduje poèet porovnání, který algoritmus provede v nejhor¹ím pøípadì.
-\qed
-
-\bye
+++ /dev/null
-P=5-qs
-
-include ../Makerules
+++ /dev/null
-\input ../lecnotes.tex
-
-\prednaska{6}{Tøídìní v lineárním èase}{(zapsal M.~Kupec, J.~Volec, J.~Ivánek)}
-
-Na minulých pøedna¹kách jsme si ukázali tøídìní v èase $ \O(N\log{N}) $ a také dokázali, ¾e líp to v obecném pøípadì nejde. Jak je tedy mo¾né tøídit v lineárním èase?
-
-V¹echny dosud pøedvedené alrogitmy tøídily pouze pomocí porovnávní hodnot na vstupu. Ale co kdy¾ o vstupu víme víc, tøeba rozsah hodnot na vstup, nedá se toho vyu¾ít?
-
-Následující algoritmy vyu¾ívají znalosti rozsahu hodnot na vstupu a toho, ¾e tento rozsah je malý.
-
-\h{Counting sort}
-
-Counting sort je algoritmus pro tøídìní $N$ celých èísel s maximálním rozsahem hodnot $R$. Tøídí v èase $ \O(N + R) $ s~pamì»ovou nároèností $ \O(R) $.
-
-Algoritmus postupnì prochází vstup a poèítá si ke ka¾dému prvku z~rozsahu kolikrát jej vidìl. Poté a¾ projde celý vstup, projde poèítadla a indexy u~tech, kde nìco napoèítal vydá jako výstup.
-
-\s{Algoritmus:} (tøídìní posloupnosti $ \left ( x_i \right )_0^N $ pomocí {\sc Counting sort}u)
-
-\algo
-\:Pro $ i \leftarrow 1 $ do $R$ opakuj:
-\::$ p_i \leftarrow 0 $
-\:Pro $ i \leftarrow 1 $ do $N$ opakuj:
-\::$ p_{x_i} \leftarrow p_{x_i} + 1 $
-\:$ j \leftarrow 1 $
-\:Pro $ i \leftarrow 1 $ do $R$ opakuj:
-\::Pokud $ p_i \neq 0 $
-\:::$ v_j \leftarrow i $
-\:::$ j \leftarrow j + 1 $
-\:Vra» $v$
-\endalgo
-
-\h{Pøihrádkové tøídìní}
-
-Counting sort nám moc nepomù¾e, pokud chceme tøídit neco jiného ne¾ celá èísla, ale i to se dá øe¹it. Pøihrádkové, popøípadì kbelíkové tøídìní, neboli bucket-sort umí tøídit mno¾inu prvkù oèíslovaných èísly $ 1 \ldots R $.
-
-\>Potøebuje k tomu èas $ \O(N + R) $ a pamì» $ \O(N + R) $.
-
-Bucket sort ma je¹tì jednu pøíjemnou vlastnost: jedná se o stabilní tøídìní.
-
-\s{Definice:} {\it Stabilní tøídìní} je takové, ¾e ka¾dé 2 prvky vstupu se stejnou hodnotou klíèe jsou na výstupu ve stejnìm poøadí jako na vstupu.
-
-\s{Algoritmus:} (tøídìní mno¾iny $ x_1,x_2,\ldots,x_n $ oèíslované $ c_1,c_2,\ldots,c_n $ pomocí {\sc bucket-sort}u)
-\algo
-\:Pro $ k \leftarrow 1 $ do $R$ opakuj:
-\::$ p_k \leftarrow 0 $
-\:Pro $ i \leftarrow 1 $ do $N$ opakuj:
-\::$ p_{c_i} \leftarrow p_{c_i} + 1 $
-\:$ b_1 \leftarrow 1 $
-\:Pro $ k \leftarrow 2 $ do $R$ opakuj:
-\::$ b_k \leftarrow b_{k - 1} + p_{b - 1} $
-\:Pro $ i \leftarrow 1 $ do $N$ opakuj:
-\::$ t \leftarrow c_i $
-\::$ v_{b_t} \leftarrow x_i $
-\::$ b_t \leftarrow b_t + 1 $
-\:Vra» $v$
-\endalgo
-
-\h{Lexikografické tøídìní k-tic}
-Mìjme $n$ uspoøádaných $k$-tic prvkù z mno¾iny $ \{1 \ldots R \}^k $. Úkol zní seøadit \hbox{$k$-tice} slovníkovì (lexikograficky).
-Mù¾eme pou¾ít metodu rozdìl a panuj, tak¾e prvky setøídíme nejprve podle první souøadnice $k$-tic a pak se rekurzivnì zavoláme na ka¾dou pøihrádku a tøídíme podle následující souøadnice.
-% a dal??
-Nebo mù¾eme vyu¾ít toho, ¾e bucket-sort je stabilní a tøídit takto:
-
-\s{Algoritmus: } (tøídìní $k$-tic $ x_1, x_2, \ldots ,x_n $ lexikograficky pomocí {\sc Bucket-sort}u)
-\algo
-\:$ S \leftarrow \{x_1, x_2, \dots, x_n\} $
-\:Pro $ i \leftarrow k $ do $ 1 $ opakuj:
-\::$ S \leftarrow $ bucket-sort $ S $ podle $i$-té souøadnice
-\:vysledek $ S $
-\endalgo
-
-Pro pøehlednost v následujícím pozorování oznaème $ l = k - i + 1 $, co¾ pøesnì odpovídá tomu, v kolikátém prùchodu cyklu jsme.
-
-\s{Pozorování:}
-Po $l$-tém prùchodu cyklem jsou prvky uspoøádány lexikograficky podle $i$-té a¾ $k$-té souøadnice.
-
-\proof
-Indukcí podle $l$
-% FIXME nevim jak presne by mel byt list
-\itemize
-\next Pro $ l = 1 $ jsou prvky uspoøádány podle poslední souøadnice.
-\next Po $l$ prùchodech ji¾ máme prvky setøídìny lexikograficky podle $i$-té a¾ $k$-té souøadnice a spou¹tíme $(l + 1)$-ní prùchod, tj. budeme tøídít podle $(i - 1)$-ní souøadnice.
-Proto¾e bucket-sort tøídí stabilnì, zùstanou prvky se stejnou $(i - 1)$-ní souøadnicí vùèi sobì seøazeny tak, jak byly seøazeny na vstupu.
-Z IP tam v¹ak byly seøazeny lexikograficky podle $i$-té a¾ $k$-té souøadnice. Tudí¾ po $(l + 1)$-ním prùchodu jsou prvky seøazeny podle $(i - 1)$-ní a¾ $k$-té souøadnice.
-\endlist
-\qed
-
-Èasová slo¾itost je $\O( k \cdot (n + R))$, co¾ je lineární s délkou vstupu $(k \cdot n)$ pro pevné $k$ a pamì»ová slo¾itost je $\O(n + R)$.
-
-\h{Tøídìní èísel $ 1 \ldots R $ podruhé (Radix sort)}
-Zvolíme základ $Z$ a èísla zapí¹eme v soustavì o základu $Z$, èím¾ získáme $ ( \lfloor \log_z{R} \rfloor +1) $-tice, na které spustíme pøedcházející algoritmus.
-Díky tomu budeme tøídít v èase $\O({\log{R} \over \log{Z}} \cdot (n + Z))$. A jak zvolit vhodnì $Z$?
-
-Pokud bychom zvolili $ Z = konstanta $, èasová slo¾itost bude $\O(\log{R} \cdot n) $, co¾ mù¾e být $ n * \log{n} $ nebo i víc.
-Zvolíme-li $ Z = n $, dostáváme $ \O({\log{R} \over \log{n}} \cdot n) $, co¾ pro $ R \leq n^\alpha $ znamená $ \O(\alpha \cdot n) $.
-
-\h{Tøídìní øetìzcù}
-Mìjme $n$ øetìzcù $ r_1, r_2 \dots r_n $ dlouhých $ l_1, l_2 \dots l_n $
-Oznaème si $ L = \max_{1 \leq i \leq n} l_i $ délku nejdel¹ího øetìzce a $R$ poèet znakù abecedy. Problém je, ¾e øetìzce nemusí být stejnì dlouhé (pokud by byly, lze se na øetìzce dívat jako na $k$-tice, které u¾ tøídít umíme). S tím se mù¾eme pokusit vypoøádat doplnìním øetìzcù mezerami tak, aby mìly v¹echny stejnou délku, a spustit na nìj algoritmus pro $k$-tice. Tím dostaneme algoritmus, který bude mít èasovou slo¾itost $ \O(L * n) $, co¾ bohu¾el mù¾e být a¾ kvadratické vzhledem k velikosti vstupu.
-Pøíklad: na vstupu mìjme k øetìzcù, kde prvních $k-1$ z nich bude mít délku $1$ a poslední øetìzec bude dlouhý pøesnì $k$. Vstup má tedy celkovou délku $2k-1$ a my teï doplníme prvních $k-1$ øetìzcù mezerami. Vidíme, ¾e algoritmus teï bude pracovat v èase $ \O(k^2) $.
-To, co nám nejvíce zpùsobovalo problémy u pøedchozího pøíkladu, bylo velké mno¾ství èasu zabraného porovnáváním doplnìných mezer. Zkusíme proto øe¹it ná¹ problem trochu chytøeji a koncové mezery do øetìzù vùbec pøidávat nebudeme.
-Nejprve roztøídíme bucket-sortem øetìzce do pøihrádek (mno¾in) $ P_i $ podle jejich délek, kde $i$ znaèí délku øetìzcù v dané pøihrádce, neboli definujme $ P_i = \left\{ r_j \vert l_j = i \right\} $. Dále si zavedeme seznam setøídìných øetìzcù $S$ takový, ¾e v nìm po $k$-tém prùchodu tøídícím cyklem budou øetìzce s délkou alespoò $ L - k + 1 $ (oznaème $l$) a zároveò v nìm tyto øetìzce budou setøídìny lexikograficky od $l$-tého znaku po $L$-tý. Z definice tohoto seznamu je patrné, ¾e po $L$ krocích tøídícího cyklu bude tento seznam obsahovat v¹echny øetìzce a tyto øetìzce v nìm budou lexikograficky seøazeny. Zbývá u¾ jen popsat, jak tento cyklus pracuje. Nejprve vezme $l$-tou mno¾inu $ P_l $ a její øetìzce roztøídí do pøihrádek $ Q_j $ (kde index $j$ znaèí j-tý znak abecedy) podle jejich $l$-tého (neboli posledního) znaku. Dále vezme seznam $S$ a jeho øetìzce pøidá opìt podle jejich $l$-tého znaku do stejných pøihrádek $ Q_j $ za ji¾ døíve pøidané øetìzce z $ P_l $. Na závìr postupnì projde v¹echny pøihrádky $ Q_j $ a øetìzce v nich pøesune do seznamu $S$. Proto¾e øetìzce z pøihrádek $ Q_j $ bude brát ve stejném poøadí, v jakém do nich byly umístìny, a proto¾e ze seznamu $S$, který je setøízený podle $ (l + 1) $-ního znaku po $L$-tý, bude také brát øetìzce postupnì, bude seznam $S$ po $k$-tém prùchodu pøesnì takový, jaký jsme chtìli (indukcí bychom dokázali, ¾e cyklus pracuje skuteènì správnì). Zároveò z popisu algoritmu je jasné, ¾e bìhem tøídìní ka¾dý znak ka¾dého øetìzce pou¾ijeme právì jednou, tudí¾ algoritmus bude lineární s délkou vstupu (pro úplnost dodejme, ¾e popsaný algoritmus funguje v pøípadech, kdy abeceda má pevnou velikost).
-
-Algoritmus: (tøídìní øetìzcù)
-\algo
-\:$L \leftarrow \max(l_1, l_2,\ldots,l_n)$
-\:Pro $ i \leftarrow 1 $ do $L$ opakuj:
-\::$ P_i \leftarrow \emptyset $
-\:Pro $ i \leftarrow 1 $ do $ n $ opakuj:
-\::$ \<pridej>(P_{l_i}, r_i) $
-
-\:$ S \leftarrow \emptyset $
-\:Pro $ i \leftarrow L $ do $ 1 $ opakuj:
-\::Pro $ j \leftarrow 1 $ do $ R $ opakuj:
-\:::$ Q_j \leftarrow \emptyset $
-
-\::Pro $ j \leftarrow 1 $ do velikost($P_i$) opakuj:
-\:::$ \<vezmi>(P_i, r) $
-\:::$ \<pridej>(Q_{r[i]}, r) $
-\::Pro $ j \leftarrow 1 $ do velikost($S$) opakuj:
-\:::$ \<vezmi>(S, r) $
-\:::$ \<pridej>(Q_{r[i]}, r) $
-
-\::$ S \leftarrow \emptyset $
-\::Pro $ j \leftarrow 1 $ do $ R $ opakuj:
-\:::Pro $ k \leftarrow 1 $ do velikost($Q_j$) opakuj:
-\::::$ \<vezmi>(Q_j, r) $
-\::::$ \<pridej>(S, r) $
-
-\:výsledek $S$
-\endalgo
-
-Èasová slo¾itost tohoto algoritmu tedy bude $ \O(RN) $, kde $N$ je délka vstupu a $R$ poèet znakù abecedy.
-
-\h{Sorting Zoo}
-
-% FIXME pridat svisle cary..mozna i vodorovne
-
-$$
-\vbox {
-\halign{$#$ \hfil & \quad \hfil $#$ & \quad \hfil $#$ & \quad \hfil $#$ \cr
-& \O(1) & \O(\log{N}) & \O(N) \cr
-\noalign{\medskip\hrule\bigskip}
-\O(N^2) & Buble & & \cr
-\O(N\log{N}) & Heap & Quick & Merge \cr
-\O(N) & & & Bucket \cr
-}
-}
-$$
-
-\h{K èemu je vlastnì tøídìní dobré?}
-Díky nìmu mù¾eme rychle vyhledávat prvky v mno¾inì, konkrétnì v èase $ \O(\log(n)) $ napø. pomocí pùlení intervalù.
-Dal¹ím problémem, na který se hodí pou¾ít tøídìní, je zji¹tìní, zda se v posloupnosti nìjaké její hodnoty opakují. Dá se dokázat, ¾e tuto úlohu nelze vyøe¹it lépe (rychleji), ne¾ tak, ¾e hodnoty nejprve setøídíme a poté setøidìnou posloupnost projdeme.
-
-Ukázali jsme si tedy, ¾e v se setøízené mno¾inì jsme schopni vyhledávat prvky v èase $ \O(\log(n)) $. Bohu¾el do ní v èase $ \O(\log(n)) $ neumíme prvky pøidávat.
-Jedním ze zpùsobu, jak zrychlit pøidávání nových prvkù a zároveò nepøijít o logaritmické vyhledávání, je pou¾ít dal¹í datouvou strukturu, vyhledávací stromy.
-
-\h{Vyhledávací stromy}
-Problém: chceme udr¾ovat nìjaká data setøízená. To znamená, ¾e chceme udr¾ovat mno¾inu $X$ prvkù z lineárnì uspoøádaného universa $U$ s operacemi Find (najdi prvek), Insert (vlo¾ prvek), Delete (sma¾ prvek). K tomu se nám bude hodit reprezentace dat pomocí stromu.
-
-\s{Definice:} {\sc Binární strom} je zakoøenìný strom; ka¾dý vrchol má max. 2 syny, pravého a levého.
-
-Dále budeme pou¾ívat toto znaèení:
-$v$ ... vrchol
-$l(v)$ ... levý syn, $p(v)$ ... pravý syn
-$L(v)$ ... levý podstrom, $p(v)$ ... pravý podstrom
-$T(v)$ ... podstrom s koøenem $v$.
-
-\s{Definice:} {\sc Binární vyhledávací strom} je takový binární strom, ¾e ka¾dý vrchol obsahuje prvek z $U$
-a pro $ \forall v: \forall x \in L(v): x < v, \forall x \in P(v): x > v $.
-
-\s{Find($T(v)$,$x$)}
-Procházíme strom od koøene, na ka¾dém vrcholu se podle hodnoty $x$ rozhodneme jak budeme pokraèovat. Buï se hodnota vrcholu rovná $x$, pak konèíme (nalezli jsme hledaný prvek). V opaèném pøípadì pokraèujeme v hledání v levém podstromu ($x$ je men¹í ne¾ hodnota vrcholu), nebo v pravém podstromu ($x$ je vìt¹í ne¾ hodnota vrcholu). Pokud dojdeme do listu a prvek jsme nena¹li, znamená to ¾e tento prvek ve stromì není.
-Toto nám zabere $\O($Hloubka stromu$)$.
-
-\s{Insert($T(v)$,$x$)}
-Nejprve najdeme místo kam je potøeba prvek vlo¾it, k tomu pou¾ijeme operaci Find. Pokud Find najde $x$ neprovedeme nic (máme strom s jedineènými hodnotami). Jinak vlo¾íme prvek na místo, kde by mìl být (jako syna vrcholu ve kterém skonèil Find).
-Nejslo¾itìj¹í je operace Find, která trvá $\O($Hloubka stromu$)$.
-
-\s{Min($T(v)$)}
-Minimum se nachází v nejlevìj¹ím listu stromu. Jdeme tedy z koøene poøád doleva, dokud nenarazíme na list.
-Èasová slo¾itost je $\O($hloubka stromu$)$.
-
-\s{Delete($T(v)$,$x$)}
-Opìt pou¾ijeme Find, zjistíme zda prvek vùbec existuje a kde se nachází. Pokud existuje mù¾ou nastat tyto pøípady:
-% FIXME nevim jak presne by mel byt list
-\itemize\ibull
-\next $x$ je list: Prostì ho sma¾eme.
-\next $x$ má jednoho syna: Zru¹íme $x$ a syna pøipojíme na uzel, který ukazoval na $x$.
-\next $x$ má dva syny: Najdeme minimum $P(v)$, toto minimum pøehodíme s $x$ a $x$ sma¾eme.
-\endlist
-Èas je tentokrát malièko slo¾itìj¹í, Find nám trvá $\O($Hloubka stromu$)$ a v posledním pøípadì nalezení Min($P(v)$) trvá taky a¾ $\O($Hloubka stromu$)$, co¾ nám dohromady dává pìkných $\O($Hloubka stromu$)$.
-
-Strom se mù¾e pou¾íváním operací "kazit", napø. opakováním Insert na max. hodnotu co¾ vede a¾ k hloubce $\sim n$. Mù¾eme
-% FIXME nevim jak presne by mel byt list
-\itemize\ibull
-\next vyka¹lat se na to $\rightarrow$ potøeba dokázat prùmìrný pøípad
-\next vylep¹it Insert a Delete $\rightarrow$ vyva¾ování stromù
-\endlist
-
-\bye
+++ /dev/null
-P=6-trideni
-
-include ../Makerules
+++ /dev/null
-\input ../lecnotes.tex
-
-% Vkladani obrazku
-\input ../mjipe.tex
-\def\treepic#1{
-\medskip
-\IpeInput{treepic/t#1.ipe}
-\medskip
-}
-\def\abpic#1{
-\medskip
-\centerline{\epsfxsize=0.7\hsize\epsfbox{abpic/#1.eps}}
-\medskip
-}
-\def\abpics#1#2{
-\medskip
-\centerline{\epsfxsize=0.4\hsize\epsfbox{abpic/#1.eps}\qquad\epsfxsize=0.4\hsize\epsfbox{abpic/#2.eps}}
-\medskip
-}
-
-\prednaska{7}{Vyhledávací stromy}{zapsali M. Øezáè, ©. Masojídek, B. Urbancová}
-
-\h{Pár obrázkù, které by stály za pøesun do~pøedchozí kapitoly:}
-
-\treepic{1}
-\treepic{2}
-\treepic{3}
-
-\h{Vyvá¾ené binární vyhledávací stromy}
-
-V minulé kapitole jsme se zabývali problematikou pøidávání a ubírání prvkù
-binárního vyhledávacího stromu a jeho slo¾itostí a zjistili, ¾e v¹e zále¾í
-na~hloubce stromu. Víme, ¾e chceme hloubku logaritmickou, ale jak ji mù¾eme
-udr¾et pøi~operacích? Øe¹ením je následující definice:
-
-\s{Definice:} {\I Dokonale vyvá¾ení} je takové vyvá¾ení, ve~kterém pro v¹echny vrcholy~$v$ platí $\left\vert \vert L(v)\vert - \vert P(v)\vert \right \vert \leq 1 $.
-
-Toto nám jistì zaji¹»uje logaritmickou hloubku, ale je velmi pracné na udr¾ování.
-Zkusíme proto slab¹í podmínku:
-
-\s{Definice:} {\I Hloubkové vyvá¾ení} je takové vyvá¾ení, ve~kterém pro v¹echny vrcholy~$v$ platí $\left \vert h(L(v)) - h(P(v)) \right \vert \leq 1 $.
-Stromùm s hloubkovým vyvá¾ením se øíká {\I AVL stromy} a platí o nich následující lemma.
-
-\s{Lemma:} AVL strom o $n$ vrcholech má hloubku $ \O(\log{n}) $.
-
-\proof
-Uva¾me $a_k = $ minimální poèet vrcholù AVL stromu o~hloubce $k$.
-Lehce spoèteme:
-$$\eqalign{
-a_0 &= 0 \cr
-a_1 &= 1 \cr
-a_2 &= 2 \cr
-&\vdots \cr
-a_k &= 1 + a_{k - 1} + a_{k - 2}. \cr
-}$$
-
-Rekurentní vzorec jsme dostali rekurzivním stavìním stromu hloubky $k$: nový koøen a 2 podstromy o hloubce $k - 1$ a $k - 2$.
-
-Indukcí doká¾eme, ¾e $ a_k \geq 2^{k \over 2} $.
-První indukèní krok jsme si u¾ ukázali, teï pro $ k \geq 2 $ platí:
-$ a_k = 1 + a_{k - 1} + a_{k - 2} > 2^{{k - 1} \over 2} + 2^{{k - 2} \over 2} = 2^{k \over 2} \cdot (2^{-{1 \over 2}} + 2^{-1}) \cong 2^{k \over 2} \cdot 1.21 > 2^{k \over 2} $
-
-Tímto jsme dokázali, ¾e na ka¾dé hladinì je minimálnì exponenciálnì vrcholù, co¾ nám zaruèuje hloubku $ \O(\log{n})$
-\qed
-
-\s{Operace s AVL stromy}
-
-\<Find> se neli¹í od~operace find v~binárních stromech.
-
-Dùraz klademe na operace \<Insert> a \<Delete>, proto¾e pøi~nich musíme o¹etøit udr¾ení struktury AVL~stromù.
-
-První nutnou podmínkou je, ¾e si musíme {\I pamatovat stav} v~ka¾dém vrcholu tohoto stromu. A~to {\I vyvá¾ení} hloubky jeho podstromù.
-
-Umluvíme~se napø. na~tomto oznaèení:
-
-\>Dostaneme tøi typy vrcholù, které se mohou v~AVL~stromu vyskytnout:
-\itemize\ibull
-\:{\I Vrchol typu~$\oplus$}, pokud je pravý podstrom hlub¹í
-\:{\I Vrchol typu~$\ominus$}, pokud je levý podstrom hlub¹í a
-\:{\I Vrchol typu~$\odot$ (nulou)}, který má oba syny schodné hloubky.
-\endlist
-
-\s {Sestavení} \rm AVL stromu:
-
-Postupujeme po~struktuøe binárního stromu od~listù ke~koøeni a~kontrolujeme, zda jsou vrcholy v~jednom ze~tøí uvedených stavù. Pokud ne, opravíme ho operací jménem rotace.
-
-\s {Rotace}
-\treepic{4}
-\treepic{5}
-
-Jde o~pøevrácení hrany mezi pùvodním otcem (koøenem podstromu) a nevyvá¾eným vrcholem tak, aby byli i po pøeskupení synové vzhledem k~otcùm správnì uspoøádáni.
-
-
-\s {Insert} \rm - vlo¾ení vrcholu do~AVL~stromu.
-
-Vlo¾íme jej jako list. Nový list má v¾dy \uv{znaménko} nula $\odot$. Pøedpokládáme, ¾e patøí nalevo od posledního otce. Podíváme~se na~znaménko jeho otce:
-\itemize\ibull
-\:{\I mìl~$\odot$ (nemìl syna) $\rightarrow$ teï má~$\ominus$}, po struktuøe stromu nahoru posíláme informaci, ¾e se podstrom prohloubil o~1, co¾ mù¾e mít samozøejmì vliv na~znaménka vrcholù na~cestì ke~koøeni.
-\:{\I mìl~$\oplus$ (mìl pravého syna, který je listem) $\rightarrow$ teï má~$\odot$}, hloubka podstromu se~nemìní
-\:{\I mìl $\ominus$} -- nenastane, proto¾e v binární struktuøe nemohou být dva leví synové
-\endlist
-\>Pøipadne-li pøidaný list napravo, øe¹íme zrcadlovì.
-
-\treepic{6}
-\treepic{7}
-
-\>{\I Prohloubil-li se strom} vlo¾ením nového listu, musíme pracovat s vyvá¾ením:
-\itemize\ibull
-\:Informace o~prohloubení pøi¹la zleva {\I do~vrcholu typu~$\odot$} $\rightarrow$ mìní jej na~vrchol se~znaménkem~$\ominus$ a informace o~prohloubení je tøeba poslat o~úroveò vý¹.
-\:Informace o~prohloubení pøi¹la zleva {\I do~vrcholu typu~$\oplus$} $\rightarrow$ mìní jej na~vrchol se~znaménkem~$\odot$, hloubka je vyrovnána, dál nic neposíláme.
-\:Informace o~prohloubení pøi¹la zleva {\I do vrcholu s~$\ominus$} $\rightarrow$
-
-\>rozebereme na~tøi pøípady podle znaménka vrcholu, ze~kterého pøi¹la informace o~prohloubení:
-\itemize\ibull
-\:Informace pøi¹la {\I z~vrcholu typu~$\ominus$} $\rightarrow$ provedeme rotaci doprava tak, ¾e novým koøenem se~stane vrchol~$y$, ze~kterého pøi¹la informace o~prohloubení.
-
-\treepic{8}
-
-{\I Pozorování 1:} znaménko vrcholù~$y$ a~$x$ je~$\odot$\
-
-{\I Pozorování 2:} hloubka pøed vkládáním byla $h+1$ a~nyní je také $h+1$, tedy nemusíme dále posílat informaci o~prohloubení a mù¾eme skonèit
-\:Informace pøi¹la {\I z~vrcholu typu~$\oplus$}
-\itemize\ibull
-\:uva¾me je¹te vrchol~$z$ jako pravého syna vrcholu~$y$, ze~kterého pri¹la informace o~prohloubení, a~jeho podstromy~$B$ a~$C$
-\:vrcholy~$B$ a~$C$ mají hloubku~$h$ nebo $h-1$ $\rightarrow$ oznaème~ji tedy $h-$ (to zøejmì proto¾e vrchol~$y$ má znaménko~$\oplus$, tedy jeho pravý podstrom s~koøenem~$z$ má hloubku~$h+1$ )
-\:provedeme dvojrotaci tak, ¾e novým koøenem se stane vrchol~$z$
-\endlist
-\treepic{9}
-
-{\I Pozorování 1:} znaménko vrcholu~$z$ bude~$\odot$\
-
-{\I Pozorování 2:} znaménka vrcholu~$x$ a~$y$ se~dopoèítají v~závislosti na~hloubce~$B$ a~$C$\
-
-{\I Pozorování 3:} rozdíl hloubky pravého a~levého podstromu bude u~tìchto vrcholù $0$ nebo~$1$\
-
-{\I Pozorování 4:} hloubka pøed vkládáním byla $h+2$ a~nyní je také $h+2$, tedy nemusíme dále posílat informaci o~prohloubení a~mù¾eme skonèit
-\:informace pøi¹la {\I z~vrcholu typu~$\odot$} -- to nemù¾e nastat, proto¾e v~tom pøípadì by ne¹lo o~prohloubení
-\endlist
-\endlist
-
-\s {Delete} -- odebrání vrcholu z~AVL~stromu
-\> Buï ma¾eme list nebo ma¾eme vrchol, který mìl nìjaké syny.
-
-\itemize\ibull
-\:pokud ma¾eme list, podíváme~se na~typ otce. Pøedpokládáme mazání levého syna.
-\itemize\ibull
-\:byl typu $\ominus$ (nemìl pravého syna) $\rightarrow$ zmìní~se na~$\odot$ (vrchol teï nemá ¾ádné syny)
-\:byl typu $\odot$ (mìl oba syny) $\rightarrow$ zmìní~se na~$\oplus$
-\endlist
-(ma¾eme-li pravý list, øe¹íme zrcadlovì)
-\:ma¾eme vrchol s~jedním (levým nebo pravým) synem $\rightarrow$ syn nastupuje na~místo otce a~získává typ~$\odot$\
-
-\>V~obou pøípadech posílame informaci o~zmìnì hloubky stromu...
-\:mazaný vrchol mìl oba syny (listy) $\rightarrow$ vybereme jednoho ze~synù na~místo smazaného otce. Hloubka se nemìní.
-\:mazaný vrchol mìl syny podstromy $\rightarrow$ na~jeho místo vezmeme nejvìt¹í prvek levého podstromu (nebo nejmen¹í prvek pravého podstromu) a od~odebraného (nahrazujícího) listu kontrolujeme vyvá¾ení podstromu.
-\endlist
-
-
-
-
-\>{\I Úprava vyvá¾ení} stromu po~odebrání listu z~podstromu
-\itemize\ibull
-\:informace o~zmìnì hloubky pøi¹la z~levého podstromu do~vrcholu typu~$\odot$ $\rightarrow$ vrchol se~zmìní na~$\oplus$ a~dál se hloubka nemìní
-
-\:informace pøi¹la zleva do~vrcholu s~$\ominus$ $\rightarrow$ mìní~se na~$\odot$ a~posíláme informaci o~zmìnì hloubky.
-
-\treepic{10}
-\treepic{11}
-
-\:problémová situace nastává, kdy¾ informace o~zmìnì pøi¹la zleva do~vrcholu se~znaménkem~$\oplus$
-\endlist
-\>Rozebereme na~tøi~pøípady podle znaménka pravého syna nevyvá¾eného vrcholu
-\itemize\ibull
-\:{\I pravý syn je typu~$\oplus$} $\rightarrow$ provedeme rotaci vlevo, novým koøenem se~stává~$y$ (pravý syn), oba vrcholy zmìní typ na~$\odot$ a~posíláme informaci o~zmìnì hloubky.
-
-\treepic{12}
-
-\:{\I pravý syn je typu~$\odot$} $\rightarrow$ provedeme opìt rotaci vlevo, koøenem se~stává~$y$, následnì se u~$y$ zmìní typ na~$\ominus$ , u~vrcholu~$x$ se typ nemìní. Hloubka stromu se~nemìní, tudí¾ není tøeba posílat informaci.
-
-\treepic{12y}
-
-\:{\I pravý syn je typu~$\ominus$} $\rightarrow$ v~tomto pøípadì uva¾ujeme je¹tì vrchol~$z$ jako levého syna vrcholu~$y$, s~podstromy $B$ a~$C$, podstromy $B$ a~$C$ mají hloubku~$h$ nebo~$h-1$. Provedeme dvojrotaci, napøed vpravo rotujeme vrcholy $z$ a~$y$, potom vlevo vrcholy~$x$ a~$z$ tak, ¾e se $z$ stane novým koøenem, typ vecholu~$x$ bude potom~$\ominus$ nebo~$\odot$, typ~$y$~$\oplus$ nebo~$\odot$ (podle toho, jaké znaménko mìl pùvodnì vrchol~$z$), typ~$z$ bude~$\odot$ a~opìt posíláme informaci o~zmìnì hloubky stromu.
-
-\treepic{13}
-\endlist
-
-\h{Obecné vyhledávací stromy}
-
-Pøi ulo¾ení dat na~disku se~sna¾íme, aby~se ètení z~disku provádìlo pokud mo¾no
-co nejménìkrát a~nezále¾í nám tolik na~tom, kolik operací se~vykoná v~jednom
-uzlu. (Èasovì je operace porovnávání zanedbatelná oproti ètení z~disku.)
-
-\s{Definice:} {\I $(a,b)$-strom} pro parametry $a,b$, $a \geq 2$, $b\geq 2a - 1$ je zakoøenìný strom s~uspoøádanými syny a~vnìj¹ími vrcholy, pro který platí
-následující axiomy:
-\numlist\nparen
-\:Data jsou ulo¾ena ve~vnitøních vrcholech a~ka¾dý vrchol obsahuje o~1 ménì klíèù ne¾ má synù.
-\:Platí stromové uspoøádání, tedy ¾e $ A < x_1 < B < x_2 < C < x_3 < D $.
-\:Koøen má $2$ a¾~$b$ synù, ostatní vnitøní vrcholy $a$ a¾ $b$ synù.
-\:V¹echny vnìj¹í vrcholy jsou ve~stejné hloubce (vnìj¹í vrchol$=$list).
-\endlist
-\>{\I Poznámka:} kdekoli~by mohl být syn a~není, pøipojíme vrchol, kterému øíkáme vnìj¹í vrchol)
-
-\abpic{ab-strom11}
-
-\s{Lemma:} $(a,b)$-strom na~$n$~vrcholech má hloubku~$O(\log_a n)$.
-
-\proof
-Zjistíme jeho minimální poèet listù (oznaème jej $m$): ka¾dý vrchol a¾ na~koøen má alespoò $a$ synù, hloubku si oznaèíme~$d$ $\rightarrow$
-$$m\geq~a^{(d -1)}$$
-$$\log_a m \geq d -1$$
-$$d \leq 1+ \log_a m$$
-\centerline{co¾ je øádovì $O(\log_a n)$, kde $n$ je poèet vrcholù.}\
-
-\s{Operace s (a,b)-stromy:}
-
-\s{Find}
-\item{-}V¾dy zjistíme, mezi které 2 klíèe hledaný vrchol patøí a potom se zanoøíme hloubìji.\
-
-\>Èasová slo¾itost nalezení prvku v $(a,b)$-stromu je $O(\log b \cdot \log_a n)$, kde $\log b$ je èas strávený na~jednom vrcholu pro zji¹tení, mezi které 2 vrcholy hledaný patøí, $\log_a n$ je hloubka stromu.
-
-\s{Insert}
-
-\>Jako Find, pøièem¾ jestli¾e nena¹el, skonèí na~posledním patøe a~pøidáme klíè
-\itemize\ibull
-\:pokud pøidáním nepøesáhneme maximální poèet klíèù mù¾eme skonèit
-\abpic{insert1}
-\:pokud pøidáním pøesáhneme maximální poèet klíèù
-\endlist
-\algo
-\:rozdìlíme vrchol na~3 èásti: $L$,$x$,$P$
-\:$L$ a $P$ jsou nové vrcholy
-\:$x$ je hodnota mezi $L$ a $P$, kterou vlo¾íme o patro vý¹ jako klíè oddìlující novì vzniklé vrcholy $L$ a $P$
-\:tím jsme pøevedli problém o patro vý¹ a opakujeme algoritmus
-\endalgo
-
-\abpics{b-klicu1}{b-klicu2}
-
-\s{Poznámka:} Jestli¾e se dostaneme a¾ do koøene, rozdìlí se koøen na dvì èásti, vznikne nám nový koøen se dvìma syny (co¾ je povoleno) a celému stromu vzroste hloubka o jedna.
-
-\s{Korektnost:}
-Potøebujeme, aby
-$$\vert L\vert \geq a-1$$
-$$\vert P\vert \geq a-1$$
-po seètení obou nerovností a~priètení 1 na~obì strany rovnice:
-$$\vert L\vert +\vert P\vert +1\geq 2a-2+1=2a-1$$
-pravá strana je rovna $b$ a~to podle definice $\geq 2a-1$. \par
-\s{Èasová slo¾itost:} vkládání prvku do $(a,b)$-stromu je $O(b\cdot \log_a n)$.
-
-
-\s{Delete}
-\item{-} pøevedeme na~delete z~listu (stejný postup jako u~stromu: jestli¾e to není list, prohodíme tuto hodnotu s~nejmen¹í hodnotou podstromu jeho pravého syna) -- v tomto pøípadì na~klíè posledního vnitøního vrcholu, proto¾e listy jsou vnìj¹í vrcholy bez dat.
-\itemize\ibull
-\:pokud má vrchol, ze~kterého odebíráme stále $a-1$ klíèù, mù¾eme skonèit
-\:pokud má vrchol($V$), ze~kterého odebíráme $a-2$ klíèù a~jeho levý sousední vrchol($L$) alespoò $a$ klíèù (klíè otce oddìlující tyto vrcholy oznaème $x$):
-\endlist
-\algo
-\:sma¾eme nejvìt¹í klíè levého sousedního vrcholu ($L$) a~nahradíme tím klíè otce obou vrcholù (nahradíme $x$ za~tuto hodnotu)
-\:pùvodní klíè otce($x$) pøidáme jako nejmen¹í klíè odebíranému vrcholu ($V$)
-\:tím mají oba tyto vrcholy $a-1$ klíèù a mù¾eme skonèit
-\endalgo
-\abpics{delete21}{delete22}
-\itemize\ibull
-\:pokud má vrchol, z kterého odebíráme($V$) $a-2$ klíèù a jeho levý sousední vrchol($L$) $a-1$ klíèù (klíè otce oddìlující tyto vrcholy oznaème $x$):
-\endlist
-\algo
-\:slouèíme $V$,$x$,$L$ do jednoho vrcholu
-\:tím jsme problém pøevedli o patro vý¹ a opakujeme algoritmus \par
-\endalgo
-\abpics{delete31}{delete32}
-
-\>{\I Poznámka:} Dojdeme-li takto a¾ do koøene, na místo klíèe odebraného z koøene lze pou¾ít nejmen¹í nebo nejvìt¹í klíè novì slouèeného podstromu. Ten odebrat lze, proto¾e po slouèení (které bylo pøíèinou této situace), je v nejni¾¹ím vrcholu $2a-2$ klíèù.
-
-\>{\I Èasová slo¾itost:} $$O(b\cdot \log_a n)$$
-
-\bye
+++ /dev/null
-P=7-stromy
-
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+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%Creator: (ImageMagick)
-%%Title: (b-klicu1.eps)
-%%CreationDate: (Mon Nov 12 12:22:07 2007)
-%%BoundingBox: 0 0 517 375
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-%%DocumentData: Clean7Bit
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-
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-
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-%
-% Display a color image. The image is displayed in color on
-% Postscript viewers or printers that support color, otherwise
-% it is displayed as grayscale.
-%
-/DirectClassPacket
-{
- %
- % Get a DirectClass packet.
- %
- % Parameters:
- % red.
- % green.
- % blue.
- % length: number of pixels minus one of this color (optional).
- %
- currentfile color_packet readhexstring pop pop
- compression 0 eq
- {
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- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add 3 mul def
- } ifelse
- 0 3 number_pixels 1 sub
- {
- pixels exch color_packet putinterval
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/DirectClassImage
-{
- %
- % Display a DirectClass image.
- %
- systemdict /colorimage known
- {
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { DirectClassPacket } false 3 colorimage
- }
- {
- %
- % No colorimage operator; convert to grayscale.
- %
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { GrayDirectClassPacket } image
- } ifelse
-} bind def
-
-/GrayDirectClassPacket
-{
- %
- % Get a DirectClass packet; convert to grayscale.
- %
- % Parameters:
- % red
- % green
- % blue
- % length: number of pixels minus one of this color (optional).
- %
- currentfile color_packet readhexstring pop pop
- color_packet 0 get 0.299 mul
- color_packet 1 get 0.587 mul add
- color_packet 2 get 0.114 mul add
- cvi
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- compression 0 eq
- {
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- }
- {
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- {
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- pixels 0 number_pixels getinterval
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-
-/GrayPseudoClassPacket
-{
- %
- % Get a PseudoClass packet; convert to grayscale.
- %
- % Parameters:
- % index: index into the colormap.
- % length: number of pixels minus one of this color (optional).
- %
- currentfile byte readhexstring pop 0 get
- /offset exch 3 mul def
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- color_packet 1 get 0.587 mul add
- color_packet 2 get 0.114 mul add
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- {
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- {
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- {
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- pixels 0 number_pixels getinterval
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-
-/PseudoClassPacket
-{
- %
- % Get a PseudoClass packet.
- %
- % Parameters:
- % index: index into the colormap.
- % length: number of pixels minus one of this color (optional).
- %
- currentfile byte readhexstring pop 0 get
- /offset exch 3 mul def
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- {
- currentfile byte readhexstring pop 0 get
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-
-/PseudoClassImage
-{
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- % Display a PseudoClass image.
- %
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- class 0 gt
- {
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- columns rows depth
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- ]
- { currentfile grays readhexstring pop } image
- }
- {
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- %
- currentfile buffer readline pop
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- /colors colors 3 mul def
- /colormap colors string def
- currentfile colormap readhexstring pop pop
- systemdict /colorimage known
- {
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- [
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- ]
- { PseudoClassPacket } false 3 colorimage
- }
- {
- %
- % No colorimage operator; convert to grayscale.
- %
- columns rows 8
- [
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- rows neg 0 rows
- ]
- { GrayPseudoClassPacket } image
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-
-/DisplayImage
-{
- %
- % Display a DirectClass or PseudoClass image.
- %
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- % x & y scale.
- % label pointsize.
- % image label.
- % image columns & rows.
- % class: 0-DirectClass or 1-PseudoClass.
- % compression: 0-none or 1-RunlengthEncoded.
- % hex color packets.
- %
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- /pixels 768 string def
-
- currentfile buffer readline pop
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- x y translate
- currentfile buffer readline pop
- token pop /x exch def
- token pop /y exch def pop
- currentfile buffer readline pop
- token pop /pointsize exch def pop
- /Times-Roman findfont pointsize scalefont setfont
- x y scale
- currentfile buffer readline pop
- token pop /columns exch def
- token pop /rows exch def pop
- currentfile buffer readline pop
- token pop /class exch def pop
- currentfile buffer readline pop
- token pop /compression exch def pop
- class 0 gt { PseudoClassImage } { DirectClassImage } ifelse
- grestore
-} bind def
-%%EndProlog
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-%%PageTrailer
-%%Trailer
-%%EOF
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%Creator: (ImageMagick)
-%%Title: (b-klicu2.eps)
-%%CreationDate: (Mon Nov 12 12:22:07 2007)
-%%BoundingBox: 0 0 694 407
-%%HiResBoundingBox: 0 0 694 407
-%%DocumentData: Clean7Bit
-%%LanguageLevel: 1
-%%Pages: 1
-%%EndComments
-
-%%BeginDefaults
-%%EndDefaults
-
-%%BeginProlog
-%
-% Display a color image. The image is displayed in color on
-% Postscript viewers or printers that support color, otherwise
-% it is displayed as grayscale.
-%
-/DirectClassPacket
-{
- %
- % Get a DirectClass packet.
- %
- % Parameters:
- % red.
- % green.
- % blue.
- % length: number of pixels minus one of this color (optional).
- %
- currentfile color_packet readhexstring pop pop
- compression 0 eq
- {
- /number_pixels 3 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add 3 mul def
- } ifelse
- 0 3 number_pixels 1 sub
- {
- pixels exch color_packet putinterval
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/DirectClassImage
-{
- %
- % Display a DirectClass image.
- %
- systemdict /colorimage known
- {
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { DirectClassPacket } false 3 colorimage
- }
- {
- %
- % No colorimage operator; convert to grayscale.
- %
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { GrayDirectClassPacket } image
- } ifelse
-} bind def
-
-/GrayDirectClassPacket
-{
- %
- % Get a DirectClass packet; convert to grayscale.
- %
- % Parameters:
- % red
- % green
- % blue
- % length: number of pixels minus one of this color (optional).
- %
- currentfile color_packet readhexstring pop pop
- color_packet 0 get 0.299 mul
- color_packet 1 get 0.587 mul add
- color_packet 2 get 0.114 mul add
- cvi
- /gray_packet exch def
- compression 0 eq
- {
- /number_pixels 1 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add def
- } ifelse
- 0 1 number_pixels 1 sub
- {
- pixels exch gray_packet put
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/GrayPseudoClassPacket
-{
- %
- % Get a PseudoClass packet; convert to grayscale.
- %
- % Parameters:
- % index: index into the colormap.
- % length: number of pixels minus one of this color (optional).
- %
- currentfile byte readhexstring pop 0 get
- /offset exch 3 mul def
- /color_packet colormap offset 3 getinterval def
- color_packet 0 get 0.299 mul
- color_packet 1 get 0.587 mul add
- color_packet 2 get 0.114 mul add
- cvi
- /gray_packet exch def
- compression 0 eq
- {
- /number_pixels 1 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add def
- } ifelse
- 0 1 number_pixels 1 sub
- {
- pixels exch gray_packet put
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/PseudoClassPacket
-{
- %
- % Get a PseudoClass packet.
- %
- % Parameters:
- % index: index into the colormap.
- % length: number of pixels minus one of this color (optional).
- %
- currentfile byte readhexstring pop 0 get
- /offset exch 3 mul def
- /color_packet colormap offset 3 getinterval def
- compression 0 eq
- {
- /number_pixels 3 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add 3 mul def
- } ifelse
- 0 3 number_pixels 1 sub
- {
- pixels exch color_packet putinterval
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/PseudoClassImage
-{
- %
- % Display a PseudoClass image.
- %
- % Parameters:
- % class: 0-PseudoClass or 1-Grayscale.
- %
- currentfile buffer readline pop
- token pop /class exch def pop
- class 0 gt
- {
- currentfile buffer readline pop
- token pop /depth exch def pop
- /grays columns 8 add depth sub depth mul 8 idiv string def
- columns rows depth
- [
- columns 0 0
- rows neg 0 rows
- ]
- { currentfile grays readhexstring pop } image
- }
- {
- %
- % Parameters:
- % colors: number of colors in the colormap.
- % colormap: red, green, blue color packets.
- %
- currentfile buffer readline pop
- token pop /colors exch def pop
- /colors colors 3 mul def
- /colormap colors string def
- currentfile colormap readhexstring pop pop
- systemdict /colorimage known
- {
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { PseudoClassPacket } false 3 colorimage
- }
- {
- %
- % No colorimage operator; convert to grayscale.
- %
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { GrayPseudoClassPacket } image
- } ifelse
- } ifelse
-} bind def
-
-/DisplayImage
-{
- %
- % Display a DirectClass or PseudoClass image.
- %
- % Parameters:
- % x & y translation.
- % x & y scale.
- % label pointsize.
- % image label.
- % image columns & rows.
- % class: 0-DirectClass or 1-PseudoClass.
- % compression: 0-none or 1-RunlengthEncoded.
- % hex color packets.
- %
- gsave
- /buffer 512 string def
- /byte 1 string def
- /color_packet 3 string def
- /pixels 768 string def
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+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%Creator: (ImageMagick)
-%%Title: (delete21.eps)
-%%CreationDate: (Mon Nov 12 12:22:07 2007)
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-%
-% Display a color image. The image is displayed in color on
-% Postscript viewers or printers that support color, otherwise
-% it is displayed as grayscale.
-%
-/DirectClassPacket
-{
- %
- % Get a DirectClass packet.
- %
- % Parameters:
- % red.
- % green.
- % blue.
- % length: number of pixels minus one of this color (optional).
- %
- currentfile color_packet readhexstring pop pop
- compression 0 eq
- {
- /number_pixels 3 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add 3 mul def
- } ifelse
- 0 3 number_pixels 1 sub
- {
- pixels exch color_packet putinterval
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/DirectClassImage
-{
- %
- % Display a DirectClass image.
- %
- systemdict /colorimage known
- {
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { DirectClassPacket } false 3 colorimage
- }
- {
- %
- % No colorimage operator; convert to grayscale.
- %
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { GrayDirectClassPacket } image
- } ifelse
-} bind def
-
-/GrayDirectClassPacket
-{
- %
- % Get a DirectClass packet; convert to grayscale.
- %
- % Parameters:
- % red
- % green
- % blue
- % length: number of pixels minus one of this color (optional).
- %
- currentfile color_packet readhexstring pop pop
- color_packet 0 get 0.299 mul
- color_packet 1 get 0.587 mul add
- color_packet 2 get 0.114 mul add
- cvi
- /gray_packet exch def
- compression 0 eq
- {
- /number_pixels 1 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add def
- } ifelse
- 0 1 number_pixels 1 sub
- {
- pixels exch gray_packet put
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/GrayPseudoClassPacket
-{
- %
- % Get a PseudoClass packet; convert to grayscale.
- %
- % Parameters:
- % index: index into the colormap.
- % length: number of pixels minus one of this color (optional).
- %
- currentfile byte readhexstring pop 0 get
- /offset exch 3 mul def
- /color_packet colormap offset 3 getinterval def
- color_packet 0 get 0.299 mul
- color_packet 1 get 0.587 mul add
- color_packet 2 get 0.114 mul add
- cvi
- /gray_packet exch def
- compression 0 eq
- {
- /number_pixels 1 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add def
- } ifelse
- 0 1 number_pixels 1 sub
- {
- pixels exch gray_packet put
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/PseudoClassPacket
-{
- %
- % Get a PseudoClass packet.
- %
- % Parameters:
- % index: index into the colormap.
- % length: number of pixels minus one of this color (optional).
- %
- currentfile byte readhexstring pop 0 get
- /offset exch 3 mul def
- /color_packet colormap offset 3 getinterval def
- compression 0 eq
- {
- /number_pixels 3 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add 3 mul def
- } ifelse
- 0 3 number_pixels 1 sub
- {
- pixels exch color_packet putinterval
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/PseudoClassImage
-{
- %
- % Display a PseudoClass image.
- %
- % Parameters:
- % class: 0-PseudoClass or 1-Grayscale.
- %
- currentfile buffer readline pop
- token pop /class exch def pop
- class 0 gt
- {
- currentfile buffer readline pop
- token pop /depth exch def pop
- /grays columns 8 add depth sub depth mul 8 idiv string def
- columns rows depth
- [
- columns 0 0
- rows neg 0 rows
- ]
- { currentfile grays readhexstring pop } image
- }
- {
- %
- % Parameters:
- % colors: number of colors in the colormap.
- % colormap: red, green, blue color packets.
- %
- currentfile buffer readline pop
- token pop /colors exch def pop
- /colors colors 3 mul def
- /colormap colors string def
- currentfile colormap readhexstring pop pop
- systemdict /colorimage known
- {
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { PseudoClassPacket } false 3 colorimage
- }
- {
- %
- % No colorimage operator; convert to grayscale.
- %
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { GrayPseudoClassPacket } image
- } ifelse
- } ifelse
-} bind def
-
-/DisplayImage
-{
- %
- % Display a DirectClass or PseudoClass image.
- %
- % Parameters:
- % x & y translation.
- % x & y scale.
- % label pointsize.
- % image label.
- % image columns & rows.
- % class: 0-DirectClass or 1-PseudoClass.
- % compression: 0-none or 1-RunlengthEncoded.
- % hex color packets.
- %
- gsave
- /buffer 512 string def
- /byte 1 string def
- /color_packet 3 string def
- /pixels 768 string def
-
- currentfile buffer readline pop
- token pop /x exch def
- token pop /y exch def pop
- x y translate
- currentfile buffer readline pop
- token pop /x exch def
- token pop /y exch def pop
- currentfile buffer readline pop
- token pop /pointsize exch def pop
- /Times-Roman findfont pointsize scalefont setfont
- x y scale
- currentfile buffer readline pop
- token pop /columns exch def
- token pop /rows exch def pop
- currentfile buffer readline pop
- token pop /class exch def pop
- currentfile buffer readline pop
- token pop /compression exch def pop
- class 0 gt { PseudoClassImage } { DirectClassImage } ifelse
- grestore
-} bind def
-%%EndProlog
-%%Page: 1 1
-%%PageBoundingBox: 0 0 694 404
-userdict begin
-DisplayImage
-0 0
-694 404
-12.000000
-694 404
-1
-1
-1
-8
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-%%PageTrailer
-%%Trailer
-%%EOF
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%Creator: (ImageMagick)
-%%Title: (delete22.eps)
-%%CreationDate: (Mon Nov 12 12:22:08 2007)
-%%BoundingBox: 0 0 694 375
-%%HiResBoundingBox: 0 0 694 375
-%%DocumentData: Clean7Bit
-%%LanguageLevel: 1
-%%Pages: 1
-%%EndComments
-
-%%BeginDefaults
-%%EndDefaults
-
-%%BeginProlog
-%
-% Display a color image. The image is displayed in color on
-% Postscript viewers or printers that support color, otherwise
-% it is displayed as grayscale.
-%
-/DirectClassPacket
-{
- %
- % Get a DirectClass packet.
- %
- % Parameters:
- % red.
- % green.
- % blue.
- % length: number of pixels minus one of this color (optional).
- %
- currentfile color_packet readhexstring pop pop
- compression 0 eq
- {
- /number_pixels 3 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add 3 mul def
- } ifelse
- 0 3 number_pixels 1 sub
- {
- pixels exch color_packet putinterval
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/DirectClassImage
-{
- %
- % Display a DirectClass image.
- %
- systemdict /colorimage known
- {
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { DirectClassPacket } false 3 colorimage
- }
- {
- %
- % No colorimage operator; convert to grayscale.
- %
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { GrayDirectClassPacket } image
- } ifelse
-} bind def
-
-/GrayDirectClassPacket
-{
- %
- % Get a DirectClass packet; convert to grayscale.
- %
- % Parameters:
- % red
- % green
- % blue
- % length: number of pixels minus one of this color (optional).
- %
- currentfile color_packet readhexstring pop pop
- color_packet 0 get 0.299 mul
- color_packet 1 get 0.587 mul add
- color_packet 2 get 0.114 mul add
- cvi
- /gray_packet exch def
- compression 0 eq
- {
- /number_pixels 1 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add def
- } ifelse
- 0 1 number_pixels 1 sub
- {
- pixels exch gray_packet put
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/GrayPseudoClassPacket
-{
- %
- % Get a PseudoClass packet; convert to grayscale.
- %
- % Parameters:
- % index: index into the colormap.
- % length: number of pixels minus one of this color (optional).
- %
- currentfile byte readhexstring pop 0 get
- /offset exch 3 mul def
- /color_packet colormap offset 3 getinterval def
- color_packet 0 get 0.299 mul
- color_packet 1 get 0.587 mul add
- color_packet 2 get 0.114 mul add
- cvi
- /gray_packet exch def
- compression 0 eq
- {
- /number_pixels 1 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add def
- } ifelse
- 0 1 number_pixels 1 sub
- {
- pixels exch gray_packet put
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/PseudoClassPacket
-{
- %
- % Get a PseudoClass packet.
- %
- % Parameters:
- % index: index into the colormap.
- % length: number of pixels minus one of this color (optional).
- %
- currentfile byte readhexstring pop 0 get
- /offset exch 3 mul def
- /color_packet colormap offset 3 getinterval def
- compression 0 eq
- {
- /number_pixels 3 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add 3 mul def
- } ifelse
- 0 3 number_pixels 1 sub
- {
- pixels exch color_packet putinterval
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/PseudoClassImage
-{
- %
- % Display a PseudoClass image.
- %
- % Parameters:
- % class: 0-PseudoClass or 1-Grayscale.
- %
- currentfile buffer readline pop
- token pop /class exch def pop
- class 0 gt
- {
- currentfile buffer readline pop
- token pop /depth exch def pop
- /grays columns 8 add depth sub depth mul 8 idiv string def
- columns rows depth
- [
- columns 0 0
- rows neg 0 rows
- ]
- { currentfile grays readhexstring pop } image
- }
- {
- %
- % Parameters:
- % colors: number of colors in the colormap.
- % colormap: red, green, blue color packets.
- %
- currentfile buffer readline pop
- token pop /colors exch def pop
- /colors colors 3 mul def
- /colormap colors string def
- currentfile colormap readhexstring pop pop
- systemdict /colorimage known
- {
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { PseudoClassPacket } false 3 colorimage
- }
- {
- %
- % No colorimage operator; convert to grayscale.
- %
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { GrayPseudoClassPacket } image
- } ifelse
- } ifelse
-} bind def
-
-/DisplayImage
-{
- %
- % Display a DirectClass or PseudoClass image.
- %
- % Parameters:
- % x & y translation.
- % x & y scale.
- % label pointsize.
- % image label.
- % image columns & rows.
- % class: 0-DirectClass or 1-PseudoClass.
- % compression: 0-none or 1-RunlengthEncoded.
- % hex color packets.
- %
- gsave
- /buffer 512 string def
- /byte 1 string def
- /color_packet 3 string def
- /pixels 768 string def
-
- currentfile buffer readline pop
- token pop /x exch def
- token pop /y exch def pop
- x y translate
- currentfile buffer readline pop
- token pop /x exch def
- token pop /y exch def pop
- currentfile buffer readline pop
- token pop /pointsize exch def pop
- /Times-Roman findfont pointsize scalefont setfont
- x y scale
- currentfile buffer readline pop
- token pop /columns exch def
- token pop /rows exch def pop
- currentfile buffer readline pop
- token pop /class exch def pop
- currentfile buffer readline pop
- token pop /compression exch def pop
- class 0 gt { PseudoClassImage } { DirectClassImage } ifelse
- grestore
-} bind def
-%%EndProlog
-%%Page: 1 1
-%%PageBoundingBox: 0 0 694 375
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-DisplayImage
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-FBFBFBFBFBFBFBend
-%%PageTrailer
-%%Trailer
-%%EOF
+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%Creator: (ImageMagick)
-%%Title: (delete31.eps)
-%%CreationDate: (Mon Nov 12 12:22:08 2007)
-%%BoundingBox: 0 0 731 404
-%%HiResBoundingBox: 0 0 731 404
-%%DocumentData: Clean7Bit
-%%LanguageLevel: 1
-%%Pages: 1
-%%EndComments
-
-%%BeginDefaults
-%%EndDefaults
-
-%%BeginProlog
-%
-% Display a color image. The image is displayed in color on
-% Postscript viewers or printers that support color, otherwise
-% it is displayed as grayscale.
-%
-/DirectClassPacket
-{
- %
- % Get a DirectClass packet.
- %
- % Parameters:
- % red.
- % green.
- % blue.
- % length: number of pixels minus one of this color (optional).
- %
- currentfile color_packet readhexstring pop pop
- compression 0 eq
- {
- /number_pixels 3 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add 3 mul def
- } ifelse
- 0 3 number_pixels 1 sub
- {
- pixels exch color_packet putinterval
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/DirectClassImage
-{
- %
- % Display a DirectClass image.
- %
- systemdict /colorimage known
- {
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { DirectClassPacket } false 3 colorimage
- }
- {
- %
- % No colorimage operator; convert to grayscale.
- %
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { GrayDirectClassPacket } image
- } ifelse
-} bind def
-
-/GrayDirectClassPacket
-{
- %
- % Get a DirectClass packet; convert to grayscale.
- %
- % Parameters:
- % red
- % green
- % blue
- % length: number of pixels minus one of this color (optional).
- %
- currentfile color_packet readhexstring pop pop
- color_packet 0 get 0.299 mul
- color_packet 1 get 0.587 mul add
- color_packet 2 get 0.114 mul add
- cvi
- /gray_packet exch def
- compression 0 eq
- {
- /number_pixels 1 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add def
- } ifelse
- 0 1 number_pixels 1 sub
- {
- pixels exch gray_packet put
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/GrayPseudoClassPacket
-{
- %
- % Get a PseudoClass packet; convert to grayscale.
- %
- % Parameters:
- % index: index into the colormap.
- % length: number of pixels minus one of this color (optional).
- %
- currentfile byte readhexstring pop 0 get
- /offset exch 3 mul def
- /color_packet colormap offset 3 getinterval def
- color_packet 0 get 0.299 mul
- color_packet 1 get 0.587 mul add
- color_packet 2 get 0.114 mul add
- cvi
- /gray_packet exch def
- compression 0 eq
- {
- /number_pixels 1 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add def
- } ifelse
- 0 1 number_pixels 1 sub
- {
- pixels exch gray_packet put
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/PseudoClassPacket
-{
- %
- % Get a PseudoClass packet.
- %
- % Parameters:
- % index: index into the colormap.
- % length: number of pixels minus one of this color (optional).
- %
- currentfile byte readhexstring pop 0 get
- /offset exch 3 mul def
- /color_packet colormap offset 3 getinterval def
- compression 0 eq
- {
- /number_pixels 3 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add 3 mul def
- } ifelse
- 0 3 number_pixels 1 sub
- {
- pixels exch color_packet putinterval
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/PseudoClassImage
-{
- %
- % Display a PseudoClass image.
- %
- % Parameters:
- % class: 0-PseudoClass or 1-Grayscale.
- %
- currentfile buffer readline pop
- token pop /class exch def pop
- class 0 gt
- {
- currentfile buffer readline pop
- token pop /depth exch def pop
- /grays columns 8 add depth sub depth mul 8 idiv string def
- columns rows depth
- [
- columns 0 0
- rows neg 0 rows
- ]
- { currentfile grays readhexstring pop } image
- }
- {
- %
- % Parameters:
- % colors: number of colors in the colormap.
- % colormap: red, green, blue color packets.
- %
- currentfile buffer readline pop
- token pop /colors exch def pop
- /colors colors 3 mul def
- /colormap colors string def
- currentfile colormap readhexstring pop pop
- systemdict /colorimage known
- {
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { PseudoClassPacket } false 3 colorimage
- }
- {
- %
- % No colorimage operator; convert to grayscale.
- %
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { GrayPseudoClassPacket } image
- } ifelse
- } ifelse
-} bind def
-
-/DisplayImage
-{
- %
- % Display a DirectClass or PseudoClass image.
- %
- % Parameters:
- % x & y translation.
- % x & y scale.
- % label pointsize.
- % image label.
- % image columns & rows.
- % class: 0-DirectClass or 1-PseudoClass.
- % compression: 0-none or 1-RunlengthEncoded.
- % hex color packets.
- %
- gsave
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-
- currentfile buffer readline pop
- token pop /x exch def
- token pop /y exch def pop
- x y translate
- currentfile buffer readline pop
- token pop /x exch def
- token pop /y exch def pop
- currentfile buffer readline pop
- token pop /pointsize exch def pop
- /Times-Roman findfont pointsize scalefont setfont
- x y scale
- currentfile buffer readline pop
- token pop /columns exch def
- token pop /rows exch def pop
- currentfile buffer readline pop
- token pop /class exch def pop
- currentfile buffer readline pop
- token pop /compression exch def pop
- class 0 gt { PseudoClassImage } { DirectClassImage } ifelse
- grestore
-} bind def
-%%EndProlog
-%%Page: 1 1
-%%PageBoundingBox: 0 0 731 404
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+++ /dev/null
-%!PS-Adobe-3.0 EPSF-3.0
-%%Creator: (ImageMagick)
-%%Title: (Insert1.eps)
-%%CreationDate: (Mon Nov 12 12:22:07 2007)
-%%BoundingBox: 0 0 764 79
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-%%DocumentData: Clean7Bit
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-
-%%BeginDefaults
-%%EndDefaults
-
-%%BeginProlog
-%
-% Display a color image. The image is displayed in color on
-% Postscript viewers or printers that support color, otherwise
-% it is displayed as grayscale.
-%
-/DirectClassPacket
-{
- %
- % Get a DirectClass packet.
- %
- % Parameters:
- % red.
- % green.
- % blue.
- % length: number of pixels minus one of this color (optional).
- %
- currentfile color_packet readhexstring pop pop
- compression 0 eq
- {
- /number_pixels 3 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add 3 mul def
- } ifelse
- 0 3 number_pixels 1 sub
- {
- pixels exch color_packet putinterval
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/DirectClassImage
-{
- %
- % Display a DirectClass image.
- %
- systemdict /colorimage known
- {
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { DirectClassPacket } false 3 colorimage
- }
- {
- %
- % No colorimage operator; convert to grayscale.
- %
- columns rows 8
- [
- columns 0 0
- rows neg 0 rows
- ]
- { GrayDirectClassPacket } image
- } ifelse
-} bind def
-
-/GrayDirectClassPacket
-{
- %
- % Get a DirectClass packet; convert to grayscale.
- %
- % Parameters:
- % red
- % green
- % blue
- % length: number of pixels minus one of this color (optional).
- %
- currentfile color_packet readhexstring pop pop
- color_packet 0 get 0.299 mul
- color_packet 1 get 0.587 mul add
- color_packet 2 get 0.114 mul add
- cvi
- /gray_packet exch def
- compression 0 eq
- {
- /number_pixels 1 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add def
- } ifelse
- 0 1 number_pixels 1 sub
- {
- pixels exch gray_packet put
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/GrayPseudoClassPacket
-{
- %
- % Get a PseudoClass packet; convert to grayscale.
- %
- % Parameters:
- % index: index into the colormap.
- % length: number of pixels minus one of this color (optional).
- %
- currentfile byte readhexstring pop 0 get
- /offset exch 3 mul def
- /color_packet colormap offset 3 getinterval def
- color_packet 0 get 0.299 mul
- color_packet 1 get 0.587 mul add
- color_packet 2 get 0.114 mul add
- cvi
- /gray_packet exch def
- compression 0 eq
- {
- /number_pixels 1 def
- }
- {
- currentfile byte readhexstring pop 0 get
- /number_pixels exch 1 add def
- } ifelse
- 0 1 number_pixels 1 sub
- {
- pixels exch gray_packet put
- } for
- pixels 0 number_pixels getinterval
-} bind def
-
-/PseudoClassPacket
-{
- %
- % Get a PseudoClass packet.
- %
- % Parameters:
- % index: index into the colormap.
- % length: number of pixels minus one of this color (optional).
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+++ /dev/null
-\input ../lecnotes.tex
-
-\prednaska{8}{Základní grafové algoritmy}{(Zapsali M. Petr, O.Tichý, ¥. Magic)}
-
-\h{Reprezentace grafu}
-Oznaème vrcholy grafu na následujícím obrázku písmeny A, B, C, D.
-Pokud bychom chtìli tento graf uchovat v pamìti poèítaèe, máme na výbìr
-hned nìkolik zpùsobù, jak to udìlat.
-\figure{img1_stvorec.eps}{}{\epsfxsize}
-
-\s{1. seznam hran}
-
-Jednodu¹e vypí¹eme seznam v¹ech dvojic vrcholù grafu spojených hranou.
-
-AB, BC, DC, CA, BC
-\medskip
-
-\s{2. matice sousednosti}
-
-Matice sousednosti je pole $A$ o velikosti $n \times n$, jeho¾ prvky na
-souøadnicích $i, j$ jsou dány následujícím pøedpisem:
-
-$$ A_{i,j} = \left\{ \matrix {0 \Leftrightarrow \{i,j\} \notin E \cr
- 1 \Leftrightarrow \{i,j\} \in E \cr
- }
-\right.$$
-
-Ná¹ graf z obrázku vý¹e by tedy v maticové reprezentaci vypadal takto:
-
-$$\bordermatrix{
- & A & B & C & D\cr
-A & 0 & 1 & 1 & 0\cr
-B & 1 & 0 & 1 & 1\cr
-C & 1 & 1 & 0 & 1\cr
-D & 0 & 1 & 1 & 0\cr
-}$$
-
-S touto matici se pracuje velmi snadno (napø. v¹echny sousedy i-tého vrcholu
-zjistíme jednodu¹e tak, ¾e projdeme i-tý øádek matice, co¾ pøedstavuje èasovou
-slo¾itost $O(n)$), ale má i jednu zøejmou nevýhodu: její velikost je v¾dy
-kvadratická bez ohledu na to, jak "øídký" je graf. U grafu s mnoha vrcholy, ale
-s malým poètem hran, tedy budeme zbyteènì plýtvat místem v pamìti.
-
-Nabízí se proto tøetí, úspornìj¹í, zpùsob, a to:
-\medskip
-
-\s{3. seznam sousedù}
-
-%$$\matrix{
-%A&\rightarrow&BC\quad \cr
-%B&\rightarrow&ACD\cr
-%C&\rightarrow&ABD\cr
-%D&\rightarrow&BC\quad \cr
-%}$$
-
-V pamìti poèítaèe by se dal seznam sousedù uchovávat dvìma poli: polem vrcholù
-$V$ grafu, jeho¾ prvky postupnì pro ka¾dý vrchol udávají index na zaèátek
-odpovídajícího úseku v poli $E$, ve kterém by byli ulo¾eni jeho sousedé.
-\figure{img4_susedia.eps}{Znázornìní polí seznamu sousedù.}{\epsfxsize}
-
-Na tuto reprezentaci u¾ staèí prostor $O(n + m)$, co¾ u¾ je, na rozdíl od
-pøedchozího kvadratického prostoru, docela pøíjemné.
-
-Je¹tì dodejme, ¾e aèkoliv jsme jednotlivé reprezentace ukazovali na pøíkladì
-neorientovaného grafu, orientované grafy se dají reprezentovat prakticky stejnì. U
-orientovaných grafù je toti¾ jediný rozdíl: existuje-li hrana z vrcholu $u$ do
-vrcholu $v$, neznamená to, ¾e existuje i hrana z $v$ do $u$ (tedy $(u, v)$ a $(v, u)$
-tvoøí uspoøádané dvojice).
-
-\h{Prùchod grafem}
-
-\s{Definice:} Vrchol je dosa¾itelný, pokud do nìj vede cesta z poèáteèního vrcholu.
-
-\s{Efektivní prùchod by mìl splòovat tyto podmínky:}
-\itemize\ibull
-\:ka¾dou hranu projdu právì jednou (resp. maximálnì dvakrát - 1x tam, 1x zpìt)
-\:hranou se vracím zpìt, a¾ kdy¾ u¾ z vrcholu, na kterém momentálnì jsem, nevede
- dosud neprozkoumaná hrana
-\:pokud jsem do¹el po hranì do ji¾ døíve nav¹tíveného vrcholu, hned se vrátím
-\endlist
-
-\s{Koneènost}
-
-\proof
-Plyne z prvního bodu.
-\qed
-
-\s{Korektnost} (algoritmus projde celý graf)
-
-\proof
-Sporem: Nech» existuje vrchol, který je dosa¾itelný, ale algoritmus ho nepro¹el.
-Pak jsme ale pøi prùchodu narazili na $\{u, v\}\in E$ takovou, ¾e vrchol $u$ je
-dosa¾itelný a $w$ dosa¾itelný není. Tzn., ¾e $\{u, w\}$ tvoøí neprozkoumanou
-hranu $\Rightarrow$ spor s druhým bodem.
-\qed
-
-\s{Èasová slo¾itost}
-
-Z toho, ¾e ¾ádná hrana ani vrchol se neprojde vícekrát (první podmínka
-efektivního prùchodu), plyne èasová slo¾itost $O(n + m)$.
-
-\>Nyní si nìco povíme o dvou zpùsobech, kterak mù¾eme procházet grafy.
-
-\s{Prùchod do hloubky} (Depth First Search, DFS)
-
-Prùchod do hloubky je prvním základním postupem pøi procházení grafem. Pokud bychom
-chtìli DFS trochu pøiblí¾it, mohli bychom ho pøirovnat k Théseovì postupu
-pøi prùchodu Minotaurovho bludi¹te za pomoci Ariadniny nitì.
-
-\s{Prùchod do ¹íøky} (Breadth First Search, BFS)
-
-Na rozdíl od prohledávání do hloubky vyu¾ívá prohledávání do ¹íøky namísto zásobníku
-frontu. Motivaèní úlohou by mohlo být napø. hledání nejkrat¹í cesty mezi dvìma vrcholy
-v grafu, nebo známý pøevozníkùv problém.
-\figure{img6_BFS.eps}{}{\epsfxsize}
-
-\s{Pø.} Koza, vlk a zelí
-\figure{img8_kvz.eps}{}{\epsfxsize}
-
-Øe¹ení ulohy by se dalo popsat jako hledání cesty grafem stavù od poèáteèního stavu
-(koza, vlk i zelí jsou na jednom bøehu) do koncového (v¹ichni byli bez újmy pøevezeni
-na druhý bøeh øeky).
-
-\h{DFS na neorientovaném grafu}
-
-Pøi prùchodu grafu do hloubky pou¾íváme zásobník. Na zaèátku se v nìm nachází pouze
-vstupní vrchol v. Dokud není zásobník prázdný, tak z nìj odebíráme vrchol; ten oznaèíme
-jako zpracovaný (abychom ho v budoucnu, kdybychom na nìj je¹tì nìkdy narazili,
-nezpracovávali znovu) a do zásobníku pøidáme v¹echny jeho sousedy takové, kteøí
-je¹tì nebyli oznaèeni jako zpracovaní.
-
-\>Vrcholy znaèíme tìmito barvami
-\itemize\ibull
-\:bílá - dosud nenav¹tívìné vrcholy (na zaèátku jsou tedy bílé v¹echny)
-\:¹edá - oznaèuje právì zpracovávaný vrchol
-\:èerná - u¾ zpracované vrcholy
-\endlist
-
-\>U ka¾dého vrcholu si dále také zapamatujeme èas pøíchodu a èas odchodu
-($in[v]$, $out [v]$).
-
-\s{Algoritmus}
-
-\algo
-\:procedure Projdi(v)
-\::color[v] := ¹edá
-\::time := time + 1
-\::in[v] := time
-\::$\forall$ w $\in$ Sousedi(v):
-\:::if color[w] = bílá $\Rightarrow$
-\::::Projdi(w)
-\::color[v] := èerná
-\::time := time + 1
-\::out[v] := time
-\::
-\:procedure DFS()
-\::$\forall$ v: color[v] := bílá
-\::time := 0
-\::$\forall$ v $\in$ V :
-\:::if color[v] = bílá $\Rightarrow$
-\::::Projdi(v)
-\endalgo
-
-Implementaci zásobníku jsme v tomto meta-algoritmu nahradili pou¾itím rekurze
-(která ale funguje na podobném principu, kdy se na zásobník vkládají jednotlivá
-volání rekurzivní funkce).
-
-Na následujících obrázcích je znázornìno, jak probíhá prùchod grafu pomocí DFS:
-
-\figure{img2_dfs.eps}{}{\epsfxsize}
-\figure{img3_dfs.eps}{Znázornìní prùbìhu algoritmu.}{\epsfxsize}
-
-\s{Hrany v tomto znázornìní dìlíme na:}
-\itemize\ibull
-\:stromové - v momentì prùchodu vedou do dosud nenav¹tíveného, bílého, vrcholu
-\:zpìtné - ostatní hrany
-\endlist
-
-\s{Pozorování:} plné hrany tvoøí strom (DFS strom, strom prùchodu do hloubky).
-
-\s{Aplikace:}
-\item{(1)} hledání kostry
-\item{(2)} hledání komponent souvislosti
-
-Èasová slo¾itost tìchto aplikací DFS je stejná jako slo¾itost pùvodního
-meta-algoritmu, tedy $O(n + m)$.
-
-\item{(3)} hledání artikulací
-
-Tuto aplikaci si teï rozebereme tro¹ku podrobnìji. Nejprve v¹ak nìkolik definic.
-
-\s{Definice:} {\I Artikulace} je vrchol v grafu, jeho¾ odebráním se graf rozpadne na alespoò
-dvì komponenty souvislosti.
-
-\s{Definice:} {\I Most} je hrana v grafu, jejím¾ odebráním se graf rozpadne na dvì
-komponenty souvislosti.
-
-\s{Definice:} Graf je {\I 2-souvislý,} jestli¾e po odstranìní libovolného vrcholu
-zùstane souvislý.
-
-\s{Definice:} {\I 2-souvislá komponenta} je maximální 2-souvislý podgraf.
-
-Jak bychom tedy pøi hledání artikulací v grafu mohli postupovat? Jako první
-se nabízí tento algoritmus:
-
-\s{Hloupý algoritmus}
-
-Berme si postupnì v¹echny vrcholy grafu; pro ka¾dý zjistíme, jestli se jeho
-odebráním zvý¹il poèet komponent. K tomu bychom mohli pou¾ít prùchod do hloubky,
-který jsme si u¾ ukázali. Procedura Projdi(v) projde a oznaèí v¹echny vrcholy
-dosa¾itelné z v, tedy v¹echny vrcholy komponenty obsahující vrchol v. Mù¾eme
-tedy udìlat takovou zmìnu, ¾e si budeme nav¹tívené vrcholy oznaèovat èíslem
-komponenty.
-
-Takový algoritmus by mìl slo¾itost $n \times O(n + m))$ (pro ka¾dý vrchol grafu
-bychom n-krát spou¹tìli DFS), co¾ se nám vùbec nelíbí. Pokusme se tedy najít
-nìco lep¹ího. Nejprve uèiòmì toto pozorování:
-
-\s{Pozorování:} v je artikulace $\Leftrightarrow$ v je koøen DFS stromu a má
-alespoò dva potomky nebo v není koøen a má syna w takového, ¾e ze ¾ádného
-jeho potomka nevede hrana do pøedchùdce v.
-
-\s{Definice:} Pøípustná cesta je taková cesta, pokud vede pøes stromové hrany
-DFS stromu (mù¾e konèit i skokem zpìt).
-
-\figure{img7_hrany.eps}{Pøíklady pøípustných cest.}{\epsfxsize}
-
-\>Dále si definujme funkci low pro ka¾dý vrchol v, takto:
-
-\s{low(v)} = min(in[w], z v do w vede pøípustná cesta)
-
-\s{Pozorování:} v je artikulace $\Leftrightarrow$ v je koøen DFS stromu a má
-alespoò dva potomky nebo má syna w takového, ¾e platí $low(w) \geq in[w]$.
-
-\s{Jak ale hodnotu funkce low(w) spoèítat?}
-
-low(v) = min(x, y, z)
-
-x = in[v]
-
-y = min(in[w], $\{w, v\}$ je zpìtná hrana)
-
-z = min(low(w), $\{v, w\}$ je stromová hrana)
-
-\h{DFS na orientovaném grafu}
-
-\figure{img5_dfso.eps}{Graf a znázornení prùbehu:}{\epsfxsize}
-
-\s{Hrany v dìlíme na: }
-\itemize\ibull
-\:stromové $\{A,B\}$, $\{B,C\}$, $\{B,D\}$
-\:zpìtné: vedou do pøedchùdce v DFS stromì (do ¹edého vrcholu)
-\:pøíèné: vedou do vrcholu v jiném podstromì (do èerného vrcholu)
-\:dopøedné: vedou do svého potomka (do èerného vrcholu v tém¾e podstromì)
-\endlist
-\bigskip
-
-\>Na závìr nìkolik \s{pozorování:}
-
-Intervaly (in[v], out[v]) $\forall v \in V $ tvoøí dobré uzávorkování.
-
-v je následník u $\Leftrightarrow (in[v], out[v]) \subseteq (in[u], out[u]) $
-
-$\{u,v\}$ je zpìtná hrana $\Leftrightarrow (in[u], out[u]) \leq (in[v], out[v])$
-
-$\{u,v\}$ je pøíèná hrana $\Leftrightarrow in[v] < out[v] < in[u] < out[u]$
-\bye
+++ /dev/null
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-\input ../lecnotes.tex
-
-\prednaska{9}{Prùchod grafem}{(zapsali T. Hubík, V. Kolomièenko, L. Slinták)}
-
-\h{1. do hloubky (DFS)}
-
-\>$\<in(v)>$, $\<out(v)>$
-
-\>Hrany rozdìlíme do nekolika tøíd
-\itemize\ibull
-\:{\I stromové} postupují do neoznaèených vrcholù
-\:{\I zpìtné} vedou do u¾ oznaèených vrcholù
-\:{\I dopøedné} vedou do vrcholù, odkud ji¾ jsme se vrátili
-\:{\I pøíèné} pouze zprava doleva
-\endlist
-
-\>{\I Pozn.: u neorientovaných grafù mohou být pouze stromové a zpìtné hrany}
-
-{\I Slo¾itost $\O(m+n)$, n ... poèet vrcholù, m ... poèet hran}
-
-\h{2. do ¹íøky (BFS)}
-{\I Slo¾itost $\O(m+n)$}
-
-\s{Definice:} {\I Lineární uspoøádání} $<$ vrcholù grafu G je topologické $\Longleftrightarrow \forall (u,v) \in E(G) : u < v$
-
-\>{\I Pozorování: Pro cyklické grafy topologické uspoøádání neexistuje.}
-
-\s{Vìta:} Libovolný acyklický orientovaný graf (DAG) má topologické uspoøádání a lze ho sestrojit v $\O(m+n)$.
-
-\proof
-Spustíme DFS, $u < v$ právì tehdy, kdy¾ $\<out(u)> > \<out(v)>$
-
-hrany:
-\itemize\ibull
-\:stromová - $\<out(u)> > \<out(v)>$
-\:zpìtná - existuje cyklus a to je spor
-\:dopøedná - $\<out(u)> > \<out(v)>$
-\:pøíèná - $\<out(u)> > \<in(u)> > \<out(v)>$
-\endlist
-\>{\I Dùsledek:} Orientovaný graf $G$ je cyklický $\Longleftrightarrow$ DFS najde zpìtnou hranu, existuje zpìtná hrana $\Rightarrow$ existuje cyklus, neexistuje zpìtná hrana $\Rightarrow$ existuje topologické uspoøádání $\Rightarrow$ neexistuje cyklus.
-\qed
-
-
-\h{Nejdel¹í cesta v DAGU:}
-
-(z $S$ do ka¾dého vrcholu)
-
-\algo
-\:Zvolíme topologické uspoøádání na $G$, BÚNO $S$ je minimum.
-\:$\forall v: D(v)=-\<inf> ; D(S)=0$
-\:Postupnì procházíme vrcholy $v \in V(G)$ v topologickém poøadí a pro $\forall v$ spoèítáme $D(v)$
-\:$D(v)=\<max>(D(n))+1$
-
-$u:(u,v) \in E(G)$
-\endalgo
-
-\>{\I Pozorování:} Ka¾dý DAG obsahuje {\I zdroj} (vycházejí z nìj cesty) a {\I stok} (cesty do nìj vstupují)
-
-\s{Rozkládání orientovaných grafù na komponenty souvislosti}
-
-\s{Definice:} $R$ bude relace na $V(G)$ tak, ¾e $uRv \Longleftrightarrow$ existuje orientovaná cesta v $G$ z $u$ do $v$ a opaène
-
-\s{Definice:} $G$ je silnì souvislý $\Longleftrightarrow \forall (u,v) \in V(G) : uRv$
-
-\>{\I Pozorování:} $R$ je ekvivalence, $u...v$ $v...w$ je $u$-$w$ sled $\Rightarrow$ existuje $u$-$v$ cesta
-
-{\narrower
-\s{Definice:} Komponenty silné souvislosti grafu $G \Longleftrightarrow$ ekvivalence tøídy relace $R$
-}
-
-\s{Lemma:} $\forall (G)$ a $\forall (u,v) \in V(G)$: existuje $u$-$v$ sled $\Longleftrightarrow$ existuje $u$-$v$ cesta
-
-\proof
-Buï $S$ $u$-$v$ sled, na kterém se opakuje vrchol $z$.
-Uva¾me sled $u...z$ $z...v$, co¾ je také $u$-$v$ sled, ale je krat¹í.
-Iterujeme $\Rightarrow$ zbude cesta.
-\qed
-
-\s{Definice:} Graf komponent $C(G)$
-
-$V(C(G))$: SSK grafu $G$
-
-$(c_1,c_2) \in E(C(G)) \Longleftrightarrow \exists v_1 \in c_1, v_2 \in c_2 : (v_1, v_2) \in E(G) $
-
-\s{Lemma:} $\forall (G) : C(G)$ je DAG.
-
-\proof
-obrázkem a sporem, existuje cyklus $c_1, c_2, ..., c_k$ v $C(G)$.
-\qed
-
-\h{Algoritmus hladání silnì souvislých komponent grafu v \O(n)}
-
-\>{\I Pozorování 1:} DFS z vrcholu $v$ projde $\{w, \exists v$-$w$ cesta$ \}$.
-
-\>{\I Pozorování 2:} Je-li $C$ stoková komponenta a $v \in V(C)$, pak DFS z $v$ projde právì $C$.
-
-\>{\I Pozorování 3:} Spustíme-li DFS na celý $G$, pak vchol $v$ s maximální $\<out(v)>$ le¾í ve zdrojové SSK.
-
-\>{\I Pozorování 3*:} Nech» $C_1, C_2$ jsou SSK a $C_1, C_2 \in E(C(G)$, pak $\<max>\{\<out(u)>, u \in V(C_1)\} > \<max>\{\<out(v)>, v \in V(C_2)\}$.
-
-\>{\I Pozorování 4:} $G$ a $G^R$ mají stejné SSK ( $G^R$ má proti $G$ otoèené hrany ).
-
-\s{Algoritmus:}
-
-\algo
-\:Sestrojíme $G^R$ \quad $\O(m+n)$
-\:DFS na $G^R \rightarrow \<out(v)>$ \quad $\O(m+n)$
-\:$T$ bude uspoøádání vrcholù podle klesajících $\<out(v)>$ \quad $\O(n)$
-\:$\forall v \in V(G)$ v poøadí podle $T$ \quad $\O(m+n)$
-\:pokud $v$ je¹tì nepatøí do ¾ádné komponenty, tak
-\::spustím DFS z $v$ a dosa¾ené vrcholy prohlásím za dal¹í SSK \quad $\O(m+n)$
-\endalgo
-
-\s{Vìta:} Ka¾dý orientovaný graf lze rozlo¾it na SSK v èase $\O(m+n)$
-
-\proof
-Pozorovani 3*:
-\itemize\ibull
-\:1. DFS vstoupí nedøíve do $C_1$ : Z $C_2$ vyleze urèitì døív ne¾ z $C_1 \Rightarrow \<out>(C_1) > \<out>(C_2)$.
-\:2. nejdøíve do $C_2$: Nejdøív se vrátím z celé $C_2$, a¾ pak nìkdy zase vlezu do $C_1 \Rightarrow \<out>(C_1) > \<out>(C_2)$.
-\endlist
-
-\bye
-
+++ /dev/null
-P=9-dfs
-
-include ../Makerules
projects / ads1.git / commitdiff
