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-
-% End
-
-end %% of Ipe figure
--- /dev/null
+\input ../lecnotes.tex
+
+\prednaska{5}{Nejkrat¹í cesty}{}
+
+Na~této pøedná¹ce budeme studovat problém hledání nejkrat¹ích cest
+v~orientovaných grafech ohodnocených reálnými èísly.
+
+\s{Situace:} Máme orientovaný graf~$G$ a funkci $l : E(G) \rightarrow {\bb R}$
+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} tedy co nejvíce jako metrika. Orientovanost
+grafù nám kazí symetriènost -- nemusí nutnì platit $d(x, y)=d(y, x)$. Budeme aspoò 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
+
+\s{Jednoduché pøípady}
+
+
+\itemize\ibull
+\:Pokud $l$ je konstantní funkce, pou¾ijeme BFS. Èasová slo¾itost bude $\Theta(m+n)$.
+\:Délky hran jsou malá pøirozená èísla $l(x, y) \in\{1, \dots, L\}$ podrozdìlíme hrany
+a pou¾ijeme BFS. Èasová slo¾itost $\Theta(Lm+n)$.
+\:DAG (orientovaný acyklický graf) indukcí pøes topologické uspoøádání v èase
+$\Theta(m+n)$.
+\endlist
+
+\s{Definice:} $D_k(v):=$ minimální délka ze sledù z $v_0$ do $v$ o právì k hranách.
+$D_0(v)=0$ pokud $v=v_0$, jinak $D_0(v)=\infty$.
+$d(v_0, v)=\min D_k(v)$
+
+Kdy¾ známe $D_0 \dots D_{k-1}$ spoèteme $D_k=\min D_{k-1}(u)+l(u, v)$.
+
+Naivní implementace pobì¾í $n(\sum_v deg^+(v))+n$ (musíme pøièíst na
+konec $n$ za izolované vrcholy), úpravíme $\sum_v deg^+(v)=m$. Bohu¾el
+spotøebujeme $\Theta(n^2)$ pamìti.
+
+Nevýhodou této implementace je pøistupování k hranám pozpátku.
+
+\s{Bellman-Fordùv algoritmus}
+\algo
+\:$D(*) \leftarrow \infty, D(s) \leftarrow 0$
+\:Pro $k=1, \dots, n-1$:
+\::$D_k(*) \leftarrow \infty$
+\::Pro $\forall v\in V(G)$:
+\:::Pro $\forall w (v, w)\in E(G)$:
+\::::$D(w)\leftarrow \min(D(w), D_{k-1}(v)+l(v, w))$
+\endalgo
+
+Pokud by i v $n$-tém kroku nìco vykonal, graf obsahuje záporný cyklus.
+
+\s{Vìta:} Bellman--Fordùv algoritmus najde v èase $\Theta(nm)$ vzdálenosti $d(v_0, v)$
+pro $\forall v\in V(G)$.
+
+\proof
+Invarianty:
+\itemize\ibull
+\:Koneèné $D(v)$ v¾dy odpovídá délce nìjakého sledu z $v_0 \rightarrow w$
+\:Na konci k-tého prùchodu vnìj¹m cyklem platí $D(w)\leq \min$ délka sledu z $v_0
+\rightarrow v$ o ménì ne¾ $k$ hranách z toho plyne ¾e na konci $D(w)\leq d(v_0, v)$
+
+Nech» nejkrat¹í sled $v_0\rightarrow w$ o ménì ne¾ $k$ hranách konèí hranou $(v, w)$,
+zastavme algoritmus v okam¾iku kdy v $k$-tém prùchodu zpravovává hranu $(v, w)$ tehdy
+$D(w)\leq D(v)+l(v, w)$ a $D(v)$ je dle indukèního pøedpokladu men¹í nebo rovný
+minimální délce sledu $v_0\rightarrow v$ o $\leq k$ hránách.
+\endlist
+\qed
+
+\s{"Prùzkumnický algoritmus"}
+Stav: $D(v)$ je ohodnoceníí, $S(v)$ stav (vidìn, otevøen -- od posledního prozkoumání
+se $D(v)$ zmìnilo, zavøen není potøeba zkoumat znovu, nic by se nezmìnilo).
+
+\algo
+\:$D(*)\leftarrow \infty, D(v_0)=0, S(*)\leftarrow N, S(v_0)\leftarrow O,
+P(*)\leftarrow ?$
+\:Dokud $\exists u$: $S(u)=O$ opakuj:
+\::$S(u)\leftarrow Z$
+\::Pro $\forall v: (u, v)\in E(G)$:
+\:::Je-li $D(u)+l(u, v)<D(v)$
+\::::$D(v)\leftarrow D(u)+l(u, v)$
+\::::$S(v)\leftarrow O$
+\endalgo
+
+\s{Invariant} $D(v)$ neroste a odpovídá délce nìjakého sledu z $v_0$ do $v$.
+
+\s{Lemma} algoritmus se v¾dy zastaví, èas pøi nevhodném zavírání mù¾e být a¾
+exponenciální. Toto lemma bude bez zanecháno bez dùkazu.
+
+Pokud se zastaví, pak dosa¾itelné vrcholy jsou zavøené a $S(v)=Z$ pak platí
+$D(v)=d(v_0, v)$.
+
+\proof
+Nech» cesta z $v_0$ do $v$ je nejmen¹í protipøíklad, pak musí existovat vrchol $u$,
+pro který neplatí $S(u)=Z$. Co¾ je spor, proto¾e takový vrchol musel ná¹ algoritmus
+projít.
+
+Vezmìme minimání protipøíklad co do poètu hran. Nech» $v$ je nejbli¾¹í vrchol takový
+¾e $D(v)\neq d(v_0, v)$, tudí¾ musí být vìt¹í (odpovídá délce nìjakého sledu). $u:=$
+pøedchùdce $v$ na nejkrat¹í cestì tedy $D(u)$ je správnì. Algoritmus zkoumal u na
+nastavil finální $D(u)$, tedy zpracoval hranu $(u, v)$ a $D(v)\leq D(u)+l(u, v)$, co¾
+je opravdová vzdálenost. Dostali jsme spor s tím, ¾e $D(x)$ neroste.
+\qed
+
+\bye
--- /dev/null
+P=5-cesty
+
+include ../Makerules
--- /dev/null
+\input ../lecnotes.tex
+
+% Vkladani obrazku
+\input ../mjipe.tex
+\def\treepic#1{
+\medskip
+\IpeInput{treepic/t#1.ipe}
+\medskip
+}
+
+\prednaska{8}{Datové struktury}{}
+
+Co si mù¾eme pøedstavit pod pojmem datová struktura? Mù¾eme tøeba chtít udr¾ovat
+mno¾inu $X$ prvkù z nìjakého universa $X\subseteq U$ (kde universum mohou být
+napøíklad pøirozená èísla).
+
+Na na¹ich datech budeme chtít provádìt následující operace:
+\itemize\ibull
+\:{\it Insert} -- vlo¾it novou polo¾ku
+\:{\it Delete} -- smazat polo¾ku
+\:{\it Find} -- najít polo¾ku
+\endlist
+
+Èasovou slo¾itost jednotlivých operací poèítejme vzhledem k poètu prvkù obsa¾ených v
+datové struktuøe.
+
+Také mù¾eme chtít udr¾ovat slovník, tedy mno¾inu dvojic $(k, v)$ kde $k\in U$ se
+nazývá klíè a $v$ hodnota. Dále pøedpokládejme, ¾e $U$ je lineárnì uspoøádaná a prvky
+porovnáme v konstantním èase.
+
+V následující tabulce jsou nìkteré mo¾né zpùsoby reprezentace na¹í datové struktury.
+
+\settabs 4 \columns
+\+ & Insert & Delete & Find \cr
+\+ pole & $\Theta (1)$ & $\Theta (n)$ & $\Theta (n)$ \cr
+\+ setøídìné pole & $\Theta (n)$ & $\Theta (n)$ & $\Theta (\log n)$ \cr
+\+ spojový seznam & $\Theta (1)$ & $\Theta (n)$ & $\Theta (n)$ \cr
+\+ setøídìný seznam & $\Theta (n)$ & $\Theta (n)$ & $\Theta (n)$ \cr
+\+ vyhledávací stromy & $\Theta (\log n)$ & $\Theta (\log n)$ & $\Theta (\log n)$ \cr
+
+\s{Pozorování:} Proces binárního vyhledávání v~setøídìném poli se dá reprezentovat
+binárním vyhledávacím stromem.
+
+\s{Definice:} {\I Binární strom:} Strom je binární, pokud je zakoøenìný a ka¾dý vrchol
+má nejvý¹e dva syny, u nich¾ rozli¹ujeme, který je levý a který pravý.
+
+\s{Definice:} Pro vrchol $v$ znaèíme:
+\itemize\ibull
+\:$l(v)$ a $p(v)$ -- levý a pravý syn vrcholu $v$
+\:$L(v)$ a $P(v)$ -- levý a pravý podstrom vrcholu $v$
+\:$S(v)$ -- pøíslu¹ný podstrom s~koøenem $v$
+\:$h(v)$ -- hloubka stromu $S(v)$, neboli délka nejdel¹í cesty z koøene do listu
+\endlist
+
+\s{Definice:} {\I Binární vyhledávací strom} (BVS): Binární strom je vyhledávací,
+pokud v~ka¾dém vrcholu je ulo¾ena dvojice (klíè, hodnota) [ztoto¾níme vrchol s~klíèem]
+a pro v¹echny vrcholy platí: $\left ( \forall u \in L(v) : u < v \right ) $ $ \&$ $
+\left ( \forall u \in P(v) : u > v \right)$.
+
+\s{Pøíklady binárních vyhledávacích stromù:}
+
+\treepic{2}
+
+\break
+
+Jak budou tedy vypadat operace {\it Find}, {\it Insert} a {\it Delete} na~binárním
+vyhledávacím stromu?
+
+{\bo Find$(v,x)$:}
+\algo
+\:Pokud $v = \emptyset \Rightarrow$ vrátíme $\emptyset$.
+\:Pokud $v = x \Rightarrow$ vrátíme $v$.
+\:Pokud $v < x \Rightarrow$ vrátíme $\<Find>(p(v),x)$.
+\:Pokud $v > x \Rightarrow$ vrátíme $\<Find>(l(v),x)$.
+\endalgo
+
+{\bo Insert$(v,x)$:}
+\algo
+\:Jako \<Find> a na~konci pøidáme list.
+\endalgo
+
+{\bo Delete$(v)$:}
+\algo
+\:Pokud $v$ je list $\Rightarrow$ jednodu¹e list utrhneme.
+\:Pokud $v$ má jednoho syna $\Rightarrow$ vrchol \uv{vyøízneme}.
+\:Jinak má $v$ oba syny $\Rightarrow$ do vrcholu vlo¾íme minimum z $P(v)$, co¾ bude list, a ten utrhneme.
+\endalgo
+
+\s{Poznámka:} Pokud má vrchol $v$ pøi operaci {\it Delete} oba syny, je vlo¾ení minima
+z $P(v)$ ekvivalentní s~vlo¾ením maxima z $L(v)$.
+
+\s{Pøíklady operací {\it Insert} a {\it Delete} na~BVS:}
+\treepic{1}
+\treepic{3}
+
+\break Èasová slo¾itost v¹ech tøí operací je $\Theta(\<hloubka stromu>)$, co¾ mù¾e být
+$\Theta(n)$, kdy¾ budeme mít smùlu a strom bude (témìø) lineární spojový seznam, nebo
+$\Theta(\log{n})$ kdy¾ bude strom pìknì vyvá¾enì vystavìný. Vidíme tedy, ¾e slo¾itost
+operací stojí a padá s~hloubkou stromu. Proto by se nám líbilo, aby mìl ná¹ strom v¾dy
+hloubku $\Theta(\log{n})$. Podívejme se tedy, jak se dá navrhnout binární vyhledávací
+strom, aby tuto podmínku splòoval \dots
+
+\h{Vyvá¾ené binární vyhledávací stromy}
+
+\s{Definice:} {\I Dokonalá vyvá¾enost:} Strom je dokonale vyvá¾ený, pokud pro v¹echny
+jeho vrcholy platí: $\left \vert \vert L(v)\vert - \vert P(v)\vert \right \vert \leq 1
+$.
+
+Takto definovaný binární strom bude mít urèitì logaritmickou hloubku. Jak takový strom
+ale konstruovat? To se nám podaøí buï staticky, nebo na~nìm budou operace dra¾¹í ne¾
+$\Theta(\log{n})$.
+
+\s{Statická konstrukce dokonale vyvá¾eného BVS:} Vybereme prostøední prvek ze
+setøídìného pole (tedy medián posloupnosti) a dáme ho do~koøene stromu. Jeho syny pak
+vystavíme rekurzivnì z~levé a pravé pùlky pole. Celá konstrukce tedy trvá $\O(n)$.
+
+\s{Lemma} buï Insert nebo Delete v dokonale vyvá¾eném BVS trvají $\Omega(n)$ (ve
+skuteènosti oba, ale dùkaz je mnohem obtí¾nìj¹í).
+
+\proof Nech» $n:=2^k-1$ pak má dokonale vyvá¾ený BVS urèený tvar a je jednoznaèné, co
+je v koøeni. Bez újmy na obecnosti mù¾eme pøedpokládat, ¾e nejmen¹í èíslo ve stromì je
+$x$ a nejvìt¹í je $x+n-1$.
+
+Proveïme posloupnost operací $$Insert(x+n), Delete(x), Insert(x+n+1), Delete(x+1)
+\dots$$ v¾dy se aspoò polovina listù posune o patro vý¹. Víme ¾e listù v na¹em stromì
+je $(n+1)/2$. Tedy aspoò jedna z operací Insert nebo Delete trvá $\Theta (n)$.
+
+Vidíme tedy, ¾e to ná¹ problém pøíli¹ neøe¹í. Potøebovali bychom, aby se strom dal
+také efektivnì udr¾ovat. Zkusíme proto slab¹í podmínku:
+
+\s{Definice:} {\I Hloubková vyvá¾enost:} Strom je hloubkovì vyvá¾ený, pokud pro
+v¹echny jeho vrcholy platí: $\left \vert h(L(v)) - h(P(v)) \right \vert \leq 1 $.
+
+Stromùm s~hloubkovým vyvá¾ením se øíká {\I AVL stromy} (objeviteli je ru¹tí
+matematikové G. M. Adelson-Velsky a E. M. Landis, podle nich jsou také pojmenovány) a
+platí o~nich následující lemma:
+
+\s{Lemma:} AVL strom na~$n$ vrcholech má hloubku $ \Theta(\log{n}) $.
+
+\proof Uva¾me posloupnost $A_k = $ minimální poèet vrcholù AVL stromù hloubky $k$.
+Staèí ukázat, ¾e $A_k$ roste exponenciálnì. Podívejme se na minimální AVL stromy:
+$$\eqalign{
+A_0 &= 1 \cr
+A_1 &= 2 \cr
+A_2 &= 4 \cr
+A_3 &= 7 \cr
+&\vdots \cr
+A_k &= 1 + A_{k - 1} + A_{k - 2}. \cr
+}$$
+
+Rekurentní vzorec jsme dostali rekurzivním stavìním stromu hloubky $k$: nový koøen a 2
+podstromy o~hloubkách $k - 1$ a $k - 2$.
+
+Vidíme tedy, ¾e $A_n = F_{n + 2} - 1$. (Mù¾eme dokázat napø. indukcí.) Teï nám ji¾
+staèí dokázat, ¾e posloupnost $A_k$ roste exponenciálnì S výhodou mù¾eme vyu¾ít toho,
+¾e na první pøedná¹ce jsme si ji¾ dokázali, ¾e Fibonacciho èísla rostou exponenciálnì.
+Nicménì pro zapomnìtlivé mù¾eme dùkaz ve struènosti zopakovat:
+
+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 $ \Theta(\log{n})$.
+\qed
+
+\bye
--- /dev/null
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projects / ads1.git / commitdiff