From 1630c1c1d8f29da725ee4680e252652a1a143a00 Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Tue, 19 May 2009 17:04:10 +0200
Subject: [PATCH] Nulta verze stromu.

---
 11-stromy/11-stromy.tex    | 113 ++++++
 11-stromy/Makefile         |   3 +
 11-stromy/treepic/t1.ipe   | 299 +++++++++++++++
 11-stromy/treepic/t10.ipe  | 389 +++++++++++++++++++
 11-stromy/treepic/t11.ipe  | 398 ++++++++++++++++++++
 11-stromy/treepic/t12.ipe  | 566 ++++++++++++++++++++++++++++
 11-stromy/treepic/t12y.ipe | 565 ++++++++++++++++++++++++++++
 11-stromy/treepic/t13.ipe  | 706 ++++++++++++++++++++++++++++++++++
 11-stromy/treepic/t2.ipe   | 465 +++++++++++++++++++++++
 11-stromy/treepic/t3.ipe   | 749 +++++++++++++++++++++++++++++++++++++
 11-stromy/treepic/t35.ipe  | 357 ++++++++++++++++++
 11-stromy/treepic/t4.ipe   | 424 +++++++++++++++++++++
 11-stromy/treepic/t5.ipe   | 540 ++++++++++++++++++++++++++
 11-stromy/treepic/t6.ipe   | 460 +++++++++++++++++++++++
 11-stromy/treepic/t7.ipe   | 398 ++++++++++++++++++++
 11-stromy/treepic/t8.ipe   | 579 ++++++++++++++++++++++++++++
 11-stromy/treepic/t9.ipe   | 705 ++++++++++++++++++++++++++++++++++
 17 files changed, 7716 insertions(+)
 create mode 100644 11-stromy/11-stromy.tex
 create mode 100644 11-stromy/Makefile
 create mode 100644 11-stromy/treepic/t1.ipe
 create mode 100644 11-stromy/treepic/t10.ipe
 create mode 100644 11-stromy/treepic/t11.ipe
 create mode 100644 11-stromy/treepic/t12.ipe
 create mode 100644 11-stromy/treepic/t12y.ipe
 create mode 100644 11-stromy/treepic/t13.ipe
 create mode 100644 11-stromy/treepic/t2.ipe
 create mode 100644 11-stromy/treepic/t3.ipe
 create mode 100644 11-stromy/treepic/t35.ipe
 create mode 100644 11-stromy/treepic/t4.ipe
 create mode 100644 11-stromy/treepic/t5.ipe
 create mode 100644 11-stromy/treepic/t6.ipe
 create mode 100644 11-stromy/treepic/t7.ipe
 create mode 100644 11-stromy/treepic/t8.ipe
 create mode 100644 11-stromy/treepic/t9.ipe

diff --git a/11-stromy/11-stromy.tex b/11-stromy/11-stromy.tex
new file mode 100644
index 0000000..7e598ec
--- /dev/null
+++ b/11-stromy/11-stromy.tex
@@ -0,0 +1,113 @@
+\input ../lecnotes.tex
+
+% Vkladani obrazku
+\input ../mjipe.tex
+\def\treepic#1{
+\medskip
+\IpeInput{treepic/t#1.ipe}
+\medskip
+}
+
+\prednaska{9}{Vyhledávací stromy}{}
+
+Pøedstavme si následující problém: cheme si udr¾ovat urèitá setøídìná data, napøíklad slovník. Polo¾kami jsou uspoøádané dvojice (klíè, hodnota). Klíèe jsou unikátní a jsou to prvky nìjakého lineárnì uspoøádaného universa.
+
+Na na¹ich datech budeme chtít provádìt následující operace: 
+\itemize\ibull
+\:{\it Insert} vlo¾it novou polo¾ku
+\:{\it Delete} smazat polo¾ku
+\:{\it Find} najít polo¾ku
+\:{\it Max $\&$ Min} vybrat polo¾ku s~nejvìt¹ím, resp. nejmen¹ím klíèem
+\:{\it Pred $\&$ Succ} vybrat polo¾ku s~klíèem o~jedna men¹ím, resp. vìt¹ím
+\endlist
+
+Jak by vypadalo nejjednodu¹¹í øe¹ení? Staticky bychom data mohli udr¾ovat v~setøídìném poli. Takové pole se vyrobí v~èase $\Theta(n \cdot log{n})$. Operace {\it Find} by trvala $\Theta(log{n})$ (vyhladávali bychom samozøejmì binárnì). Ale problém by nastal s~operacemi {\it Insert} a {\it Delete}, které by se takto implementovat nedaly, nebo by trvaly hodnì dlouho (napø. v~pøípadì {\it Insert} bychom museli celé pole pøestavìt).
+
+\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: $\forall u \in L(v) : u < v $ $\&$ $ \forall u \in P(v) : u > v$.
+
+\s{Pøíklady binárních vyhledávacích stromù:}
+
+\treepic{2}
+
+\break
+ 
+Jak budou tedy vypadat operace {\it Find}, {\it Insert} a {\it Delete} na~binárním vyhledávacím stromu?
+
+{\bo Find$(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 a nebo na~nìm budou operace dra¾¹í ne¾ $\Theta(\log{n})$.
+
+\s{Statická konstrukce dokonale vyvá¾eného BVS:} Vybereme medián posloupnosti a dáme ho do~koøene stromu. Jeho syny pak vystavíme rekurzivnì z~levé a pravé pùlky pole.
+
+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 $.
+
+Stromùm s~hloubkovým vyvá¾ením se øíká {\I AVL stromy} a platí o~nich následující lemma:
+
+\s{Lemma:} AVL strom na~$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~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 $ \Theta(\log{n})$.
+\qed
+
+\bye
diff --git a/11-stromy/Makefile b/11-stromy/Makefile
new file mode 100644
index 0000000..6f4f489
--- /dev/null
+++ b/11-stromy/Makefile
@@ -0,0 +1,3 @@
+P=11-stromy
+
+include ../Makerules
diff --git a/11-stromy/treepic/t1.ipe b/11-stromy/treepic/t1.ipe
new file mode 100644
index 0000000..84e02ad
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+-85.3436 131.562 lt
+% sk
+0 sg sk
+% End
+
+% Line
+% ss 0
+0.4 [] ss
+np % # 2
+41.6 116.8 mt
+22.4 131.2 lt
+% sk
+0 sg sk
+% End
+
+% Line
+% ss 0
+0.4 [] ss
+np % # 2
+-9.6 116.8 mt
+9.6 131.2 lt
+% sk
+0 sg sk
+% End
+
+% Line
+% ss 0
+0.4 [] ss
+np % # 4
+-68 120 mt
+-80 100 lt
+-56 100 lt
+-68 120 lt
+% sk
+0 sg sk
+% End
+
+% Text
+% xy -71.5466 102.579
+% sk 0
+% f 0 10
+% s D
+% End
+
+% Line
+% ss 0
+0.4 [] ss
+np % # 2
+-119.562 105.344 mt
+-112.438 94.6564 lt
+% sk
+0 sg sk
+% End
+
+% Line
+% ss 0
+0.4 [] ss
+% ar 2 7
+0 sg np % # 2
+-60 136 mt
+-24 136 7 36 0 at
+% sk
+0 sg sk
+% End
+
+% Text
+% xy -146.87 65.3268
+% sk 0
+% f 0 10
+% s $h$
+% End
+
+% Text
+% xy -96.1573 81.6676
+% sk 0
+% f 0 10
+% s $h+1$
+% End
+
+% Text
+% xy -130.24 40.8156
+% sk 0
+% f 0 10
+% s $h^-$
+% End
+
+% Text
+% xy -98.24 40.8156
+% sk 0
+% f 0 10
+% s $h^-$
+% End
+
+% Text
+% xy -159.948 108.902
+% sk 0
+% f 0 10
+% s $h+2$
+% End
+
+% Text
+% xy -65.4178 89.2933
+% sk 0
+% f 0 10
+% s $h$
+% End
+
+% Text
+% xy -129.416 147.031
+% sk 0
+% f 0 10
+% s $[h+2]$
+% End
+
+% Text
+% xy -34.87 66.4162
+% sk 0
+% f 0 10
+% s $h$
+% End
+
+% Text
+% xy -6.24 66.4162
+% sk 0
+% f 0 10
+% s $h^-$
+% End
+
+% Text
+% xy 25.76 66.4162
+% sk 0
+% f 0 10
+% s $h^-$
+% End
+
+% Text
+% xy 61.13 66.4162
+% sk 0
+% f 0 10
+% s $h$
+% End
+
+% Text
+% xy -162.718 122.52
+% sk 0
+% f 0 10
+% s $[h+1]$
+% End
+
+% Text
+% xy -5.65136 107.612
+% sk 0
+% f 0 10
+% s $h+1$
+% End
+
+% Text
+% xy 48.3251 123.609
+% sk 0
+% f 0 10
+% s $h+1$
+% End
+
+% Text
+% xy -18.1913 148.141
+% sk 0
+% f 0 10
+% s $h+2$
+% End
+
+% End
+
+end %% of Ipe figure
-- 
2.39.2