--- /dev/null
+\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
--- /dev/null
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projects / ads1.git / commitdiff