+++ /dev/null
-\input ../sgr.tex
-
-\hyphenation{mikro-strom mikro-stro-mo-vé}
-
-\prednaska{10}{Dekompozice stromù}{}
-
-V~této kapitole uká¾eme nìkolik datových struktur zalo¾ených
-na~my¹lence dekompozice problému na~dostateènì malé podproblémy,
-které u¾ umíme (obvykle vhodným kódováním èísly) øe¹it v~konstantním
-èase.
-
-\h{Union-Find problem}
-
-\s{Problém:} Udr¾ování tøíd ekvivalence: na~poèátku máme $N$ jednoprvkových ekvivalenèních
-tøíd, provádíme operace \<Find> (zji¹tìní, zda dva prvky jsou ekvivalentní) a \<Union>
-(slouèení dvou tøíd do~jedné). Také na~to lze pohlí¾et jako na~inkrementální udr¾ování
-komponent souvislosti neorientovaného grafu: \<Union> je pøidání hrany, \<Find> test,
-zda dva vrcholy le¾í v~té¾e komponentì. To se hodí v~mnoha algoritmech, kupøíkladu
-v~Kruskalovì algoritmu pro hledání minimální kostry.
-
-\s{Triviální øe¹ení:} Ka¾dé tøídì pøiøadíme unikátní barvu, kterou obarvíme prvky tøídy. Operace \<Find>
-porovnává barvy, operace \<Union> prvky jedné tøídy pøebarvuje.
-
-Operace \<Find> tak pracuje v~konstantním èase, \<Union> mù¾e zabrat a¾ lineární èas. Mù¾eme si
-pomoci tím, ¾e v¾dy pøebarvíme {\I men¹í} ze~sluèovaných ekvivalenèních tøíd (budeme
-si pro ka¾dou tøídu pamatovat seznam jejích prvkù a velikost). Tehdy mù¾e být ka¾dý
-prvek pøebarven jen $\O(\log n)$-krát, jeliko¾ ka¾dým pøebarvením se alespoò zdvojnásobí
-velikost tøídy, ve~které prvek le¾í. Posloupnost operací \<Union>, kterou vznikla tøída
-velikosti~$k$, tak trvá $\O(k\log k)$, tak¾e mù¾eme bezpeènì prohlásit, ¾e amortizovaná
-slo¾itost operace \<Union> je $\O(\log n)$.
-
-\s{Chytøej¹í øe¹ení:} Ka¾dou tøídu budeme reprezentovat zakoøenìným stromem s~hranami
-orientovanými smìrem ke~koøeni (jinými slovy pro ka¾dý prvek si pamatujeme jeho otce
-nebo ¾e je to koøen). \<Find> nalezne koøeny stromù a porovná je, \<Union> pøipojí koøen
-jedné tøídy pod koøen druhé. Aby stromy nedegenerovaly, pøidáme dvì podmínky:
-
-\itemize\ibull
-\:{\I Union by rank:} ka¾dý koøen $v$ si pamatuje svùj rank $r(v)$. Pokud spojujeme
-dva stromy s~koøeny $v$, $w$ a $r(v)<r(w)$, pøipojíme $v$ pod~$w$ a rank zachováme.
-Pokud $r(v)=r(w)$, pøipojíme libovolnì a nový koøen bude mít rank $r(v)+1$.%
-\foot{Stejnì by fungovalo pravidlo {\I Union by size,} které pøipojuje men¹í
-strom pod vìt¹í, ale ranky máme radìji, neb jsou skladnìj¹í a snáze se analyzují.}
-
-\:{\I Path compression:} pokud z~vrcholu vystoupíme do~koøene (napøíklad
-bìhem operace \<Find>), pøepojíme v¹echny vrcholy na~cestì, po~které jsme pro¹li,
-rovnou pod koøen.
-\endlist
-
-\s{Pozorování:} Samotné pravidlo Union by rank zajistí, ¾e strom ranku $r$ bude
-mít hloubku nejvý¹e $r$ a minimálnì $2^r$ vrcholù, tak¾e èasová slo¾itost operací
-bude omezena $\O(\log n)$.%
-\foot{Mimochodem, Path compression samotná by také na~slo¾itost $\O(\log n)$ amortizovanì staèila.}
-
-Ve~skuteènosti se popsaná struktura chová daleko lépe:
-
-\s{Vìta:} (Tarjan, van Leeuwen \cite{tarjan84setunion}) Kombinace Union by rank a Path compression vede k~amortizované
-slo¾itosti obou operací $\O(\alpha(n))$, kde $\alpha$ je inverzní Ackermannova funkce.%
-\foot{Existuje varianta tohoto algoritmu, která dosahuje stejné slo¾itosti i v~nejhor¹ím
-pøípadì; té¾ je známo, ¾e asymptoticky lep¹í slo¾itosti nelze dosáhnout.}
-
-\h{Union-Find s~pøedem známými Uniony}
-
-Dále nás bude zajímat speciální varianta Union-Find problemu, v~ní¾ dopøedu známe
-posloupnost Unionù, èili strom, který spojováním komponent vznikne.\foot{Kdy se to hodí?
-Tøeba v~Thorupovì lineárním algoritmu \cite{thorup:usssp} na~nejkrat¹í cesty nebo
-v~Weiheho takté¾ lineárním algoritmu \cite{weihe:paths} na~hledání hranovì disjunktních
-cest v~rovinných grafech.}
-Jiná interpretace tého¾ (jen pozpátku) je dekrementální udr¾ování komponent
-souvislosti lesa: na~poèátku je dán les a umíme smazat hranu a otestovat, zda jsou
-dva vrcholy v~tém¾e stromu.
-
-Popí¹eme algoritmus,
-který po~poèáteèním pøedzpracování v~èase $\O(n)$ zvládne \<Union> i \<Find> v~amortizovanì
-konstantním èase. Tento algoritmus je kombinací dekompozic popsaných Alstrupem v~\cite{alstrup97optimal}
-a \cite{alstrup98marked}.
-
-\s{Definice:} {\I (Microtree/Macrotree dekompozice)} Pro zakoøenìný strom $T$ o~$n$ vrcholech
-definujeme:
-\itemize\ibull
-\:{\I Koøeny mikrostromù} $R$ budou nejvy¹¹í vrcholy, pod~nimi¾ je nejvý¹e $\log n$ listù
-a které nejsou koøenem celého~$T$.
-\:{\I Mikrostromy} le¾í v~$T$ od~tìchto koøenù ní¾e.
-\:{\I Spojovací hrany} vedou z~koøenù mikrostromù do~jejich otcù.
-\:{\I Makrostrom} je tvoøen zbývajícími vrcholy a hranami stromu~$T$.
-\endlist
-
-\s{Pozorování:} Ka¾dý mikrostrom má nejvý¹e $\log n$ listù. Pod ka¾dým listem makrostromu le¾í
-alespoò jeden mikrostrom (mù¾e jich být i více, viz dekompozice hvìzdy na~obrázku), tak¾e
-listù makrostromu je nejvý¹e $n/\log n$.
-
-Vnitøních vrcholù makro- i mikrostromù ale mù¾e být ne¹ikovnì mnoho, proto¾e se ve~stromech mohou
-vyskytovat dlouhé cesty. Pomù¾eme si snadno: ka¾dou cestu si budeme pamatovat zvlá¹» a ve~stromu
-ji nahradíme hranou, která bude vlo¾ena právì tehdy, kdy¾ budou pøítomny v¹echny hrany cesty.
-
-\s{Pøíklad:} Následující obrázek ukazuje dekompozici nìkolika stromù za~pøepokladu,
-¾e $\log n=4$. Vrcholy mikrostromù jsou èerné, makrostromu bílé. Spojovací hrany kreslíme teèkovanì,
-hrany komprimovaných cest tuènì.
-
-\fig{mima.eps}{\epsfxsize}
-
-\s{Algoritmus pro cesty:} Cestu délky~$l$ rozdìlíme na~úseky délky $\log n$, pro nì¾ si ulo¾íme
-mno¾iny ji¾ pøítomných hran (po~bitech jako èísla). Pak si je¹tì pamatujeme zkomprimovanou cestu (hrany
-odpovídají úsekùm a jsou pøítomny právì tehdy, jsou-li pøítomny v¹echny hrany pøíslu¹ného úseku)
-délky $l/\log n$ a pro ni \uv{pøebarvovací} strukturu pro Union-Find.
-
-\>$\<Union>(x,y)$ (pøidání hrany $e=xy$ do~cesty):
-\algo
-\:Pøidáme $e$ do mno¾iny hran pøítomných v~pøíslu¹ném úseku.
-\:Pokud se tím úsek naplnil, pøidáme odpovídající hranu do~zkomprimované cesty.
-\endalgo
-
-\>$\<Find>(x,y):$
-\algo
-\:Pokud $x$ a $y$ jsou v~tém¾e úseku, otestujeme bitovými operacemi, zda
- jsou v¹echny hrany mezi $x$ a $y$ pøítomny.
-\:Pokud jsou v~rùzných úsecích, rozdìlíme cestu z~$x$ do~$y$ na~posloupnost celých úsekù,
- na~které nám odpoví zkomprimovaná cesta, a~dva dotazy v~krajních èásteèných úsecích.
-\endalgo
-
-Operace uvnitø úsekù pracují v~èase $\O(1)$, operace na~zkomprimované cestì v~$\O(\log l)$
-amortizovanì, ale za~dobu ¾ivota struktury je jich $\O(l/\log n)=\O(l/\log l)$, tak¾e celkovì zaberou lineární èas.
-
-\s{Cestová komprese:} Operace na~mikro/makro-stromech budeme následujícím zpùsobem
-pøevádìt na~operace s~jejich cestovì komprimovanými podobami a na~operace s~cestovými strukturami:
-
-\>$\<Union>(x,y)$:
-\algo
-\:Pokud $e=xy$ le¾í uvnitø nìjaké cesty, pøidáme ji do~cesty, co¾ buïto zpùsobí
- pøidávání jiné hrany, a~nebo u¾ jsme hotovi.
-\:Provedeme \<Union> v~komprimovaném stromu.
-\endalgo
-
-\>$\<Find>(x,y)$:
-\algo
-\:Pokud $x$ a $y$ le¾í uvnitø jedné cesty, zeptáme se cestové struktury a konèíme.
-\:Pokud $x$ le¾í uvnitø nìjaké cesty, zjistíme dotazem na~cestovou strukturu,
- ke~kterému krajnímu vrcholu cesty je pøipojen, a~$x$ nahradíme tímto vrcholem.
- Není-li pøipojen k~¾ádnému, je~evidentnì odpovìï na~celý \<Find> negativní;
- pokud k~obìma, vybereme si libovolný, proto¾e jsou stejnì v~cestovì komprimovaném
- stromu spojeny hranou. Analogicky pro~$y$.
-\:Zeptáme se struktury pro komprimovaný strom.
-\endalgo
-
-\s{Algoritmus pro mikrostromy:} Po~kompresi cest má ka¾dý mikrostrom nejvý¹e $2\log n$
-vrcholù, èili také nejvý¹e tolik hran. Hrany si oèíslujeme pøirozenými èísly, ka¾dou
-mno¾inu hran pak mù¾eme reprezentovat $2\log n$-bitovým èíslem a mno¾inové operace
-provádìt pomocí bitových v~konstantním èase.
-
-Pro ka¾dý mikrostrom si pøedpoèítáme pro v¹echny jeho vrcholy~$v$ mno¾iny~$P_v$ hran le¾ících
-na~cestì z~koøene mikrostromu do~$v$. Navíc si budeme pro celý mikrostrom pamatovat mno¾inu
-pøítomných hran~$F$.
-
-\>$\<Union>(x,y):$
-
-\algo
-\:Najdeme poøadové èíslo $i$ hrany $xy$ (máme pøedpoèítané).
-\:$F \leftarrow F \cup \{i\}$.
-\endalgo
-
-\>$\<Find>(x,y):$
-
-\algo
-\:$P \leftarrow P_x \mathop{\Delta} P_v$ (mno¾ina hran le¾ících na~cestì z~$x$ do~$y$).
-\:Pokud $P\setminus F=\emptyset$, le¾í $x$ a $y$ ve~stejnì komponentì, jinak ne.
-\endalgo
-
-\s{Algoritmus pro celý problém:} Strom rozlo¾íme na~mikrostromy, makrostromy a spojovací
-hrany. V~mikrostromech i makrostromech zkomprimujeme cesty. Pro cesty a mikrostromy pou¾ijeme
-vý¹e popsané struktury, pro ka¾dou spojovací hranu si budeme pamatovat jen znaèku,
-zda je pøítomna, a pro makrostrom pøebarvovací strukturu.
-
-\>$\<Union>(x,y)$:
-
-\algo
-\:Pokud $e=xy$ je spojovací, poznamenáme si, ¾e je pøítomna, a~konèíme.
-\:Nyní víme, ¾e $e$ le¾í uvnitø mikrostromu nebo makrostromu, a~tak provedeme \<Union>
- na~pøíslu¹né struktuøe.
-\endlist
-
-\>$\<Find>(x,y)$:
-
-\algo
-\:Le¾í-li $x$ a $y$ v~jednom mikrostromu, zeptáme se struktury pro~mikrostrom.
-\:Je-li $x$ uvnitø mikrostromu, zeptáme se mikrostruktury na~spojení s~koøenem mikrostromu.
- Není-li, odpovíme {\sc ne}, stejnì jako kdy¾ není pøítomna pøíslu¹ná spojovací hrana.
- Jinak $x$ nahradíme listem makrostromu, do~kterého spojovací hrana vede. Podobnì pro~$y$.
-\:Odpovíme podle struktury pro makrostrom.
-\endalgo
-
-\s{Analýza:} Operace \<Find> trvá konstantní èas, proto¾e se rozlo¾í na~$\O(1)$ \<Find>ù
-v~dílèích strukturách a ka¾dý z~nich trvá konstantnì dlouho. V¹ech $n$ operací \<Union>
-trvá $\O(n)$, jeliko¾ zpùsobí $\O(n)$ amortizovanì konstantních operací s~mikrostromy, spojovacími
-hranami a cestami a $\O(n/\log n)$ operací s~makrostromy, které trvají $\O(\log n)$ amortizovanì
-ka¾dá.%
-\foot{To je v~prùmìru $\O(1)$ na~operaci a dokonce i amortizovanì, pokud necháme inicializaci
-struktury, která je lineární, naspoøit potenciál $\O(n)$, ze~kterého budeme prùbì¾nì platit
-sluèování v~makrostromu.}
-
-\s{Cvièení:} Zkuste pomocí dekompozice vyøe¹it následující problém: je dán strom,
-jeho¾ ka¾dý vrchol mù¾e být oznaèený. Navrhnìte datovou strukturu, která bude umìt
-v~èase $\O(\log\log n)$ oznaèit nebo odznaèit vrchol a v~èase $\O(\log n/\log\log n)$ najít
-nejbli¾¹ího oznaèeného pøedchùdce.
-
-\h{Fredericksonova clusterizace}
-
-Mikro/makro-stromová dekompozice není jediný zpùsob, jak stromy rozkládat. Nìkdy
-se hodí napøíklad následující my¹lenka: \cite{frederickson91ambivalent}
-
-\s{Definice:} (Fredericksonova clusterizace) Nech» $G$ je graf s~vrcholy stupòù nejvý¹e~3
-a $c\ge 1$. Pak $c$-clusterizací grafu $G$ nazveme libovolný rozklad
-$G$ na~souvislé podgrafy (clustery) $C_1, C_2, \ldots, C_k$ takový, ¾e platí:
-\itemize\ibull
-\:$\forall v \in V \exists ! i: v \in C_i$.
-\:$\forall i: \vert C_i\vert \le c$.
-\:$\forall i$ je vnìj¹í stupeò $C_i$ (tj. poèet hran, které vedou mezi $C_i$ a zbytkem grafu)
-nejvý¹e~3. Navíc pokud je právì~3, je cluster triviální, èili $\vert C_i \vert = 1$.
-\:®ádné dva sousední clustery nelze spojit.
-\endlist
-
-\s{Vìta:} (Frederickson) Ka¾dá $c$-clusterizace grafu $G$ má $\O(V(G)/c)$ clusterù. Existuje
-algoritmus, který jednu takovou najde v~lineárním èase.
-
-\proof První èást rozborem pøípadù, druhá hladovì pomocí DFS. \qed
-
-\s{Pou¾ití:} Pøedchozí variantu Union-Find problemu bychom také mohli vyøe¹it nahrazením
-vrcholù stupnì $>3$ \uv{kruhovými objezdy bez jedné hrany}\foot{tzv. francouzský trik},
-nalezením $(\log n)$-clusterizace, pou¾itím bitové reprezentace mno¾in uvnitø clusterù
-a pøebarvovací struktury na~hrany mezi clustery.
-
-\h{Stromoví pøedchùdci}
-
-\s{Problém:} {\I (Least Common Ancestor alias LCA)} Chceme si pøedzpracovat zakoøenìný strom~$T$
-tak, abychom dokázali pro libovolné dva vrcholy $x,y$ najít co~nejrychleji jejich nejbli¾¹ího
-spoleèného pøedchùdce.
-
-\s{Triviální øe¹ení LCA:}
-\itemize\ibull
-\:Vystoupáme z~$x$ i $y$ do~koøene, oznaèíme vrcholy na~cestách a kde se poprvé
- potkají, tam je hledaný pøedchùdce. To je lineární s~hloubkou a nepotøebuje
- pøedzpracování.
-\:Vylep¹ení: Budeme stoupat z~$x$ a $y$ støídavì. Tak potøebujeme jen lineárnì mnoho
- krokù vzhledem ke~vzdálenosti spoleèného pøedchùdce.
-\:Pøedpoèítáme v¹echny mo¾nosti: pøedzpracování $\O(n^2)$, dotaz $\O(1)$.
-\:\dots\ co dál?
-\endlist
-
-\>Vìrni vtipùm o~matfyzácích a èlánku \cite{bender00lca} pøevedeme radìji tento problém na~jiný.
-
-\s{Problém:} {\I (Range Minimum Query alias RMQ)} Chceme pøedzpracovat posloupnost èísel
-$a_1,\ldots a_n$ tak, abychom umìli rychle poèítat $\min_{x\le i\le y} a_i$.%
-\foot{V¹imnìte si, ¾e pro sumu místo minima je tento problém velmi snadný.}
-
-\s{Lemma:} LCA lze pøevést na~RMQ s~lineárním èasem na~pøedzpracování a konstantním
-èasem na~pøevod dotazu.
-
-\proof Strom projdeme do~hloubky a poka¾dé, kdy¾ nav¹tívíme vrchol (v~inorderu),
-zapí¹eme jeho hloubku. ${\rm LCA}(x,y)$ pak bude nejhlub¹í vrchol mezi libovolnou
-náv¹tìvou~$x$ a libovolnou náv¹tìvou~$y$.
-\qed
-
-\s{Triviální øe¹ení RMQ:}
-\itemize\ibull
-\:Pøedpoèítáme v¹echny mo¾né dotazy: pøedzpracování $\O(n^2)$, dotaz $\O(1)$.
-\:Pro ka¾dé $i$ a $j\le \log n$ pøedpoèítáme $m_{ij} = \min\{ a_i, a_{i+1}, \ldots, a_{i+2^j-1} \}$,
-èili minima v¹ech blokù velkých jako nìjaká mocnina dvojky. Kdy¾ se poté nìkdo zeptá
-na~minimum bloku $a_i,a_{i+1},\ldots,a_{j-1}$, najdeme nejvìt¹í~$k$ takové, ¾e $2^k < j-i$
-a vrátíme:
-$$\min( \min\{ a_i, \ldots, a_{i+2^k-1} \}, \min\{ a_{j-2^k}, \ldots, a_{j-1} \} ).$$
-Tak zvládneme dotazy v~èase $\O(1)$ po~pøedzpracování v~èase $\O(n\log n)$.
-\endlist
-
-My si ov¹em v¹imneme, ¾e ná¹ pøevod z~LCA vytváøí dosti speciální instance problému RMQ,
-toti¾ takové, v~nich¾ je $\vert a_i - a_{i+1} \vert = 1$. Takovým instancím budeme
-øíkat RMQ${\pm}1$ a budeme je umìt øe¹it ¹ikovnou dekompozicí.
-
-\s{Dekompozice} pro RMQ${\pm}1$: Vstupní posloupnost rozdìlíme na~bloky velikosti $b=1/2\cdot \log n$,
-ka¾dý dotaz umíme rozdìlit na~èást týkající se celých blokù a maximálnì dva dotazy na~èásteèné bloky.
-
-V¹imneme si, ¾e aèkoliv blokù je mnoho, jejich mo¾ných typù (tj. posloupností klesání
-a stoupání) je pouze $2^{b-1}\le\sqrt n$ a bloky tého¾ typu se li¹í pouze posunutím
-o~konstantu. Vybudujeme proto kvadratickou strukturu pro jednotlivé typy a pro ka¾dý
-blok si zapamatujeme, jakého je typu a jaké má posunutí. Celkem strávíme èas
-$\O(n + \sqrt n \cdot \log^2 n) = \O(n)$ pøedzpracováním a $\O(1)$ dotazem.
-
-Mimo to je¹tì vytvoøíme komprimovanou posloupnost, v~ní¾ ka¾dý blok nahradíme
-jeho minimem. Tuto posloupnost délky $n/b$ budeme pou¾ívat pro èásti dotazù
-týkající se celých blokù a pøipravíme si pro ni \uv{logaritmickou} variantu
-triviální struktury. To nás bude stát $\O(n/b\cdot\log (n/b))=\O(n/\log n\cdot\log n)=\O(n)$ na~pøedzpracování
-a $\O(1)$ na~dotaz.
-
-Tak jsme získali algoritmus pro RMQ${\pm}1$ s~konstantním èasem na~dotaz po~lineárním
-pøedzpracování a vý¹e zmínìným pøevodem i algoritmus na~LCA se stejnými parametry.
-Je¹tì uká¾eme, ¾e pøevod mù¾e fungovat i v~opaèném smìru, a~tak mù¾eme získat
-i konstantní/lineární algoritmus pro obecné RMQ.
-
-\s{Definice:} {\I Kartézský strom} pro posloupnost $a_1,\ldots,a_n$ je strom,
-jeho¾ koøenem je $a_j=\min_i a_i$, jeho levý podstrom je kartézský strom pro
-$a_1,\ldots,a_{j-1}$ a pravý podstrom kartézský strom pro $a_{j+1},\ldots,a_n$.
-
-\s{Lemma:} Kartézský strom je mo¾né zkonstruovat v~lineárním èase.
-
-\proof Pou¾ijeme inkrementální algoritmus. V¾dy si budeme pamatovat
-kartézský strom pro ji¾ zpracované prvky a pozici posledního zpracovaného
-prvku v~tomto stromu. Kdy¾ pøidáváme dal¹í prvek, hledáme místo, kam ho
-pøipojit, od~tohoto oznaèeného prvku nahoru. Pov¹imnìme si, ¾e vzhledem
-k~potenciálu rovnému hloubce oznaèeného prvku je èasová slo¾itost pøidání
-prvku amortizovanì konstantní.
-\qed
-
-\s{Lemma:} RMQ lze pøevést na~LCA s~lineárním èasem na~pøedzpracování a konstantním
-èasem na~pøevod dotazu.
-
-\proof Sestrojíme kartézský strom a RMQ pøevedeme na~LCA v~tomto stromu.
-\qed
-
-Výsledky této podkapitoly mù¾eme shrnout do~následující vìty:
-
-\s{Vìta:} Problémy LCA i RMQ je mo¾né øe¹it v~konstantním èase na~dotaz
-po~pøedzpracování v~lineárním èase.
-
-\references
-\bye
+++ /dev/null
-P=10-decomp
-
-include ../Makerules
+++ /dev/null
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+\input ../sgr.tex
+
+\prednaska{10}{Suffixové stromy}{}
+
+V~této kapitole popí¹eme jednu pozoruhodnou datovou strukturu, pomocí ní¾ doká¾eme problémy týkající
+se øetìzcù pøevádìt na grafové problémy a tak je øe¹it v~lineárním èase.
+
+\h{Øetìzce, trie a suffixové stromy}
+
+\ss{Definice:}
+
+\nointerlineskip
+\halign{\qquad#\dotfill&~#\hfil\cr
+\hbox to 0.35\hsize{}\cr
+$\Sigma$ & koneèná abeceda -- mno¾ina znakù \cr
+\omit & (znaky budeme znaèit latinskými písmeny)\cr
+$\Sigma^*$ & mno¾ina v¹ech slov nad $\Sigma$ \cr
+\omit & (slova budeme znaèit øeckými písmeny)\cr
+$\varepsilon$ & prázdné slovo\cr
+$\vert\alpha\vert$ & délka slova $\alpha$\cr
+$\alpha\beta$ & zøetìzení slov $\alpha$ a $\beta$ ($\alpha\varepsilon=\varepsilon\alpha=\alpha$)\cr
+$\alpha^R$ & slovo $\alpha$ napsané pozpátku\cr
+$\alpha$ je {\I prefixem} $\beta$ & $\exists\gamma: \beta=\alpha\gamma$ ($\beta$ zaèíná na~$\alpha$)\cr
+$\alpha$ je {\I suffixem} $\beta$ & $\exists\gamma: \beta=\gamma\alpha$ ($\beta$ konèí na~$\alpha$)\cr
+$\alpha$ je {\I podslovem} $\beta$ & $\exists\gamma,\delta: \beta=\gamma\alpha\delta$ (znaèíme $\alpha \subset \beta$)\cr
+$\alpha$ je {\I vlastním prefixem} $\beta$ & je prefixem a $\alpha\ne\beta$ \cr
+\omit & (analogicky vlastní suffix a podslovo)\cr
+}
+
+\s{Pozorování:} Prázdné slovo je prefixem, suffixem i podslovem ka¾dého slova vèetnì sebe sama.
+Podslova jsou právì prefixy suffixù a také suffixy prefixù.
+
+\s{Definice:} {\I Trie ($\Sigma$-strom)} pro koneènou mno¾inu slov $X\subset\Sigma^*$ je orientovaný graf $G=(V,E)$, kde:
+\itemize\relax
+\:$V = \{\alpha: \alpha\hbox{ je prefixem nìjakého $\beta\in X$} \},$
+\:$(\alpha,\beta)\in E \equiv \exists x\in\Sigma: \beta=\alpha x$.
+\endlist
+
+\s{Pozorování:} Trie je strom s koøenem $\varepsilon$. Jeho listy jsou slova z $X$, která nejsou vlastními prefixy jiných slov z~$X$.
+Hrany si mù¾eme pøedstavit popsané písmeny, o~nì¾ prefix roz¹iøují, popisky hran na~cestì z~koøene do~vrcholu~$\alpha$ dávají právì slovo~$\alpha$.
+
+\s{Definice:} {\I Komprimovaná trie ($\Sigma^+$-strom)} vznikne z trie nahrazením maximálních nevìtvících se cest hranami. Hrany
+jsou tentokrát popsané øetìzci místo jednotlivými písmeny, pøièem¾ popisky v¹ech hran vycházejících z~jednoho vrcholu se li¹í v~prvním
+znaku. Vrcholùm \uv{uvnitø hran} (které padly za obì» kompresi) budeme øíkat {\I skryté vrcholy.}
+
+\medskip
+\tabskip=0pt plus 1fil
+\halign to\hsize{
+\hfil#\hfil&\hfil#\hfil&\hfil#\hfil\cr
+\epsfbox{trie.eps}&\epsfbox{trie-c.eps}&\epsfbox{trie-cd.eps}\cr
+Trie pro $\{\hbox{AULA, AUTO, AUTOBUS, BUS}\}$ & \dots komprimovaná & \dots odolarovaná\cr
+}
+\medskip
+
+\s{Definice:} {\I Suffixový strom (ST)} pro slovo $\sigma\in\Sigma^*$ je komprimovaná trie pro $X=\{\alpha: \hbox{$\alpha$ je suffixem $\sigma$}\}$.
+
+\s{Pozorování:} Vrcholy suffixového stromu (vèetnì skrytých) odpovídají prefixùm suffixù slova~$\sigma$,
+tedy v¹em jeho podslovùm. Listy stromu jsou suffixy, které se v~$\sigma$ ji¾ nikde jinde nevyskytují
+(takovým suffixùm budeme øíkat {\I nevnoøené}). Vnitøní vrcholy odpovídají {\I vìtvícím podslovùm,}
+tedy podslovùm $\alpha\subset\sigma$ takovým, ¾e $\alpha a\subset\sigma$ i $\alpha b\subset\sigma$
+pro nìjaké dva rùzné znaky~$a$,~$b$.
+
+Nìkdy mù¾e být nepraktické, ¾e nìkteré suffixy neodpovídají listùm (proto¾e jsou vnoøené), ale
+s~tím se mù¾eme snadno vypoøádat: pøidáme na~konec slova~$\sigma$ nìjaký znak~$\$$, který se nikde
+jinde nevyskytuje. Neprázdné suffixy slova $\sigma\$$ odpovídají suffixùm slova~$\sigma$
+a ¾ádný z~nich nemù¾e být vnoøený. Takový suffixový strom budeme znaèit ST\$.
+
+\figure{st-barbara.eps}{Suffixový strom pro slovo BARBARA}{0pt}
+
+Nyní jak je to s~konstrukcí suffixových stromù:
+
+\s{Lemma:} Suffixový strom pro slovo $\sigma$ délky $n$ je reprezentovatelný v~prostoru $\O(n)$.
+
+\proof Strom má $\O(n)$ listù a ka¾dý vnitøní vrchol má alespoò $2$ syny, tak¾e vnitøních
+vrcholù je také $\O(n)$. Hran je rovnì¾ lineárnì. Nálepky na~hranách staèí popsat
+poèáteèní a koncovou pozicí v~$\sigma$.
+\qed
+
+\s{Vìta:} Suffixový strom pro slovo $\sigma$ délky $n$ lze sestrojit v~èase $\O(n)$.
+
+\proof Ve~zbytku této kapitoly pøedvedeme dvì rùzné konstrukce v~lineárním èase.
+\qed
+
+\s{Aplikace} -- co v¹e doká¾eme v~lineárním èase, kdy¾ umíme lineárnì konstruovat ST:
+
+\nobreak
+
+\numlist\ndotted
+\:{\I Inverzní vyhledávání} (tj. pøedzpracujeme si v~lineárním èase text a pak umíme pro libovolné
+slovo~$\alpha$ v~èase $\O(\vert\alpha\vert)$ rozhodnout, zda se v~textu vyskytuje)\foot{Èili pøesný
+opak toho, co~umí vyhledávací automat -- ten si pøedzpracovává dotaz.} -- staèí sestrojit~ST
+a pak jej procházet od~koøene. Také umíme najít v¹echny výskyty (odpovídají suffixùm, které mají
+jako prefix hledané slovo, tak¾e staèí vytvoøit ST\$ a vypsat v¹echny listy pod
+nalezeným vrcholem) nebo pøímo vrátit jejich poèet (pøedpoèítáme si pomocí DFS pro ka¾dý vrchol,
+kolik pod ním le¾í listù).
+
+\:{\I Nejdel¹í opakující se podslovo} -- takové podslovo je v~ST\$ nutnì vìtvící, tak¾e staèí
+najít vnitøní vrchol s~nejvìt¹í {\I písmenkovou hloubkou} (tj. hloubkou mìøenou ve~znacích
+místo ve~hranách).
+
+\:{\I Histogram èetností podslov délky~$k$} -- rozøízneme ST v~písmenkové hloubce~$k$ a spoèítáme,
+kolik pùvodních listù je pod ka¾dým novým.
+
+\:{\I Nejdel¹í spoleèné podslovo} slov~$\alpha$ a $\beta$ -- postavíme ST pro slovo $\alpha\$_1\beta\$_2$,
+jeho listy odpovídají suffixùm slov $\alpha$ a $\beta$. Tak¾e staèí pomocí DFS najít nejhlub¹í vnitøní
+vrchol, pod kterým se vyskytují listy pro~$\alpha$ i $\beta$. Podobnì mù¾eme sestrojit ST\$ pro libovolnou
+mno¾inu slov.\foot{Jen si musíme dát pozor, abychom si moc nezvìt¹ili abecedu, ale to bude jasné,
+a¾ pøedvedeme konkrétní konstrukce.}
+
+\:{\I Nejdel¹í palindromické podslovo} (tj. takové $\beta\subset\alpha$, pro nì¾ je $\beta_R=\beta$)
+-- postavíme spoleèný ST\$ pro slova $\alpha$ a $\alpha_R$. Postupnì procházíme pøes v¹echny mo¾né støedy
+palindromického podslova a v¹imneme si, ¾e takové slovo je pro ka¾dý støed nejdel¹ím spoleèným
+prefixem podslova od~tohoto bodu do~konce a podslova od~tohoto bodu pozpátku k~zaèátku,
+èili nìjakého suffixu $\alpha$ a nìjakého suffixu $\alpha_R$. Tyto suffixy ov¹em odpovídají
+listùm sestrojeného ST a jejich nejdel¹í spoleèný prefix je nejbli¾¹ím spoleèným pøedchùdcem
+ve~stromu, tak¾e staèí pro strom vybudovat datovou strukturu pro spoleèné pøedchùdce
+a s~její pomocí doká¾eme jeden støed prozkoumat v~konstantním èase.
+
+\:{\I Burrows-Wheelerova Transformace} \cite{burrows:bwt} -- jejím základem je lexikografické setøídìní v¹ech
+rotací slova~$\sigma$, co¾ zvládneme sestrojením ST pro slovo~$\sigma\sigma$, jeho
+uøíznutím v~písmenkové hloubce~$\vert\sigma\vert$ a vypsáním novì vzniklých listù v~poøadí
+\uv{zleva doprava}.
+\endlist
+
+\s{Cvièení:} Zkuste vymyslet co nejlep¹í algoritmy pro tyto problémy bez pou¾ití~ST.
+
+\h{Suffix Array}
+
+\>V~nìkterých pøípadech se hodí místo suffixového stromu pou¾ívat kompaktnìj¹í datové struktury.
+
+\s{Notace:} Pro slovo $\sigma$ bude $\sigma[i]$ znaèit jeho $i$-tý znak (èíslujeme od~jednièky),
+$\sigma[i:j]$ pak podslovo slo¾ené z~$i$-tého a¾ $j$-tého znaku. Libovolnou z~mezí mù¾eme vynechat, proto
+$\sigma[i:{}]$ bude suffix od~$i$ do~konce a $\sigma[{}:j]$ prefix od~zaèátku do~$j$.
+Pokud $j<i$, definujeme $\sigma[i:j]$ jako prázdné slovo, tak¾e prázdný suffix mù¾eme
+napøíklad zapsat jako $\sigma[\vert\sigma\vert+1:{}].$
+
+${\rm LCP}(\alpha,\beta)$ bude znaèit délku nejdel¹ího spoleèného prefixu slov $\alpha$ a $\beta$,
+èili nejvìt¹í $i\le \vert\alpha\vert,\vert\beta\vert$ takové, ¾e $\alpha[{}:i]=\beta[{}:i]$.
+
+\s{Definice:} {\I Suffix Array} $A_\sigma$ pro slovo $\sigma$ délky~$n$ je posloupnost v¹ech suffixù
+slova~$\sigma$ v~lexikografickém poøadí. Mù¾eme ho reprezentovat napøíklad jako permutaci $A$ èísel
+$1,\ldots,n+1$, pro ní¾ $\sigma[A[1]:{}] < \sigma[A[2]:{}] < \ldots < \sigma[A[n+1]:]$.
+
+\s{Definice:} {\I Longest Common Prefix Array} $L_\sigma$ pro slovo $\sigma$ je posloupnost,
+v~ní¾ $L_\sigma[i]:={\rm LCP}(A_\sigma[i],A_\sigma[i+1])$.
+
+\s{Vìta:} Suffixový strom pro slovo $\sigma\$$ je lineárnì ekvivalentní s~dvojicí $(A_\sigma,L_\sigma)$.
+[Jinými slovy, kdy¾ máme jedno, mù¾eme z~toho v~lineárním èase spoèítat druhé, a naopak.]
+
+\proof Kdy¾ projdeme ST($\sigma$) do hloubky, poøadí listù odpovídá $A_\sigma$ a písmenkové hloubky vnitøních
+vrcholù v~inorderu odpovídají $L_\sigma$. Naopak ST($\sigma$) získáme tak, ¾e sestrojíme kartézský strom
+pro~$L_\sigma$ (získáme vnitøní vrcholy ST), doplníme do~nìj listy, pøiøadíme jim suffixy podle~$A_\sigma$
+a nakonec podle listù rekonstruujeme nálepky hran.
+\qed
+
+\h{Rekurzivní konstrukce}
+
+\>Tento algoritmus konstruuje pro slovo $\sigma$ délky~$n$ jeho suffix array a LCP array v~èase $\O(n+{\rm Sort}(n,\Sigma))$,
+kde ${\rm Sort}(\ldots)$ je èas potøebný pro setøídìní $n$ symbolù z~abecedy~$\Sigma$. V~kombinaci s~pøedchozími
+výsledky nám tedy dává lineární konstrukci ST($\sigma$) pro libovolnou fixní abecedu.
+
+\s{Algoritmus:} (Konstrukce $A$ a $L$ podle Kärkkäinena a Sanderse \cite{karkkainen03simple})
+
+\algo
+\:Redukujeme abecedu na~$1\ldots n$: ve~vstupním slovu je nejvý¹e $n$ rùzných znakù,
+tak¾e je staèí setøídit a pøeèíslovat.
+
+\:Definujeme slova $\sigma_0$, $\sigma_1$, $\sigma_2$ následovnì:
+$$\eqalign{
+\sigma_0[i] &:= \left<\sigma[3i],\sigma[3i+1],\sigma[3i+2]\right>\cr
+\sigma_1[i] &:= \left<\sigma[3i+1],\sigma[3i+2],\sigma[3i+3]\right>\cr
+\sigma_2[i] &:= \left<\sigma[3i+2],\sigma[3i+3],\sigma[3i+4]\right>\cr
+}$$
+V¹echna $\sigma_k$ jsou slova délky $\approx n/3$ nad~abecedou velikosti $n^3$. Dovolíme
+si mírnì zneu¾ívat notaci a pou¾ívat symbol $\sigma_k$ i jejich pøepis do~abecedy pùvodní.
+
+\:Zavoláme algoritmus rekurzivnì na slovo $\sigma_0\sigma_1$, èím¾ získáme $A_{01}$ a $L_{01}$.
+
+\:Z~$A_{01}$ a $L_{01}$ vydìlíme $A_0=A_{\sigma_0}$, $A_1$, $L_0$ a $L_1$. Také si pro ka¾dý prvek
+$A_i$ zapamatujeme, kde se v~$A_{01}$ vyskytoval.
+
+\:Dopoèítáme $A_2$: Jeliko¾ $\sigma_2[i:{}] = \sigma[3i+2:{}] = \sigma[3i+2]\sigma[3i+3:\nobreak{}\nobreak] = \sigma[3i+2]\sigma_0[i+1:{}]$
+a v¹echna $\sigma_0[i:{}]$ u¾ máme setøídìná, mù¾eme v¹echna $\sigma_2[i:{}]$ setøídit dvìma prùchody pøihrádkového tøídìní.
+
+\:Dopoèítáme $L_2$: Stejným trikem jako $A_2$ -- pokud jsou první písmena rùzná, je spoleèný prefix prázdný, jinak
+má délku $1+{\rm LCP}(\sigma_0[i+1:{}],\sigma_0[j+1:{}]) = 1+\min_{i+1\le k< j+1} L_0[k]$. Minimum zvládneme pro ka¾dou
+dvojici $i,j$ spoèítat v~konstantním èase pomocí datové struktury pro intervalová minima.
+
+\:$A_0,A_1,A_2\buildrel merge\over\longrightarrow A$ -- sléváme tøi setøídìné posloupnosti,
+tak¾e staèí umìt prvky libovolných dvou posloupností v~konstantním èase porovnat:
+$$\eqalign{
+\sigma_0[i:{}] < \sigma_1[j:{}] &~\hbox{podle zapamatovaných pozic v~$A_{01}$} \cr
+\sigma_0[i:{}] < \sigma_2[k:{}] &\equiv \sigma[3i]\,\sigma_1[i:{}] < \sigma[3k+2]\,\sigma_0[k+1:{}]\cr
+&\Leftrightarrow (\sigma[3i] < \sigma[3k+2]) \vee {} \cr&\hphantom{{}\Leftrightarrow{}} (\sigma[3i] = \sigma[3k+2] \wedge \sigma_1[i:{}] < \sigma_0[k+1:{}])\cr
+\sigma_1[j:{}]<\sigma_2[k:{}] &\equiv \sigma[3j+1]\,\sigma[3j+2]\,\sigma_0[j+1:{}] < \cr&\hphantom{{}\equiv{}} \sigma[3k+2]\,\sigma[3k+3]\,\sigma_1[k+1:{}]
+}$$
+
+\:Dopoèítáme $L$ -- pokud sousedí suffix ze~$\sigma_{0,1}$ se suffixem ze~$\sigma_{0,1}$,
+vyèteme výsledek pøímo z~$L_{01}$. Pokud sousedí $\sigma_2$ se $\sigma_2$, staèí pou¾ít
+u¾ spoèítané $L_2$. Pokud sousedí $\sigma_{0,1}$ se $\sigma_2$, odebereme první jeden
+nebo dva znaky, ty porovnáme samostatnì a v~pøípadì shody zbude suffix ze~$\sigma_0$
+a suffix ze~$\sigma_1$ (stejnì jako pøi slévání) a pro ty doká¾eme $L$ dopoèítat
+pomocí struktury pro intervalová minima v~$L_{01}$.
+
+\endalgo
+
+\s{Analýza èasové slo¾itosti:} Tøídìní v~prvním volání trvá ${\rm Sort}(n,\Sigma)$, ve~v¹ech
+ostatních voláních je lineární (trojice èísel velikosti $\O(n)$ mù¾eme tøídit tøíprùchodovým
+pøihrádkovým tøídìním s~$\O(n)$ pøihrádkami). Z~toho dostáváme:
+$$T(n) = T(2/3\cdot n) + \O(n),~\hbox{a tedy}~T(n)=\O(n).$$
+\qed
+
+\h{Ukkonenova inkrementální konstrukce}
+
+\>Ukkonenùv algoritmus \cite{ukkonen95line} konstruuje suffixový strom bez dolarù inkrementálnì: zaène se stromem
+pro prázdné slovo (ten má jediný vrchol, a to koøen) a postupnì pøidává dal¹í znaky na~konec
+slova. To zvládne v~èase $\O(1)$ amortizovanì na~pøidání jednoho znaku.
+Pro slovo~$\sigma$ tedy doká¾e sestrojit ST v~èase $\O(\vert\sigma\vert)$.
+
+Budeme pøedpokládat, ¾e hrany vedoucí z~jednoho vrcholu je mo¾né indexovat jejich
+prvními písmeny -- to bezpeènì platí, pokud je abeceda pevná; není-li, mù¾eme
+si pomoci hashováním.
+
+\s{Pozorování:} Kdy¾ slovo~$\sigma$ roz¹íøíme na~$\sigma a$, ST se zmìní následovnì:
+
+\numlist\ndotted
+\:Pokud $\beta$ byl nevnoøený suffix slova~$\sigma$, je i $\beta a$ nevnoøený suffix~$\sigma a$. Z~toho víme, ¾e listy
+zùstanou listy, pouze jim potøebujeme prodlou¾it nálepky. Pomù¾eme si snadno: zavedeme
+{\I otevøené hrany,} jejich¾ nálepka je \uv{od~pozice~$i$ do konce}. Listy se tak
+o~sebe postarají samy.
+\:Pokud $\beta$ bylo vìtvící slovo, zùstane nadále vìtvící -- tedy vnitøní vrcholy ve~stromu zùstanou.
+\:Pokud $\beta$ byl vnoøený suffix (tj. vnitøní èi skrytý vrchol), pak se $\beta a$ buïto
+vyskytuje v~$\sigma$, a tím pádem je to vnoøený suffix nového slova a strom není nutné
+upravovat, nebo se v~$\sigma$ nevyskytuje a tehdy pro nìj musíme zalo¾it novou odboèku
+a nový list s~otevøenou hranou.
+\endlist
+
+Víme tedy, co v¹echno musí algoritmus ve~stromu pøí roz¹íøení slova upravit, zbývá
+vyøe¹it, jak to udìlat efektivnì. K~tomu se hodí pár definic a lemmat:
+
+\s{Definice:} {\I Aktivní suffix} $\alpha(\sigma)$ øíkáme nejdel¹ímu vnoøenému suffixu slova~$\sigma$.
+
+\s{Lemma:} Suffix $\beta$ slova $\sigma$ je vnoøený $\Leftrightarrow$ $\vert\beta\vert \le \vert\alpha(\sigma)\vert.$
+
+\proof Ka¾dý suffix vnoøeného suffixu je opìt vnoøený. \qed
+
+\s{Lemma:} Pro ka¾dé $\sigma$, $a$ platí: $\alpha(\sigma a)$ je suffixem $\alpha(\sigma)a.$
+
+\proof $\alpha(\sigma a)$ i $\alpha(\sigma)a$ jsou suffixy slova $\sigma a$, a~proto staèí porovnat jejich délky.
+Slovo $\beta := \hbox{\uv{$\alpha(\sigma a)$ bez koncového~$a$}}$ je vnoøeným suffixem v~$\sigma$, tak¾e
+$\vert\beta\vert \le \vert\alpha(\sigma)\vert$, a~tedy také $\vert\alpha(\sigma a)\vert = \vert\beta a\vert \le \vert\alpha(\sigma)a\vert$.
+\qed
+
+\s{Definice:} Suffix $\beta a$ je {\I zralý} $\equiv$ $\beta$ je vnoøený suffix~$\sigma$, ale $\beta a$ není podslovem~$\sigma$
+(tedy musíme pro~nìj pøi pøidávání znaku~$a$ k~aktuálnímu slovu~$\sigma$ zakládat nový vrchol).
+
+\s{Lemma:} Suffix $\beta$ je zralý $\Leftrightarrow$ $\vert\alpha(\sigma)a\vert \ge \vert\beta a\vert > \vert\alpha(\sigma a)\vert$.
+
+\proof Jeliko¾ $\beta$ je vnoøeným suffixem $\sigma$, musí platit první nerovnost. Aby byl zralý,
+musí také nebýt vnoøeným suffixem $\sigma a$, a~tomu odpovídá druhá nerovnost.
+\qed
+
+\s{Idea algoritmu:} Udr¾ujeme si $\alpha=\alpha(\sigma)$ a pøi pøidání znaku $a$ zkontrolujeme, zda $\alpha a$ je
+stále vnoøený suffix. Pokud ano, nic se nemìní, pokud ne, pøidáme vnitøní vrchol, $\alpha$ zkrátíme
+zleva o~znak a testujeme dál.
+
+\s{Analýza:} Úprav stromu provedeme $\O(1)$ amortizovanì (ka¾dá úprava slovo $\alpha$ zkrátí,
+ka¾dé pøidání znaku ho~prodlou¾í o~znak, tak¾e v¹ech zkrácení je $\O(\vert\sigma\vert)$). Staèí
+tedy ukázat, jak provést úpravu v~(amortizovanì) konstantním èase, k~èemu¾ potøebujeme
+$\alpha$ reprezentovat ¹ikovnì a také si udr¾ovat pomocné informace (zpìtné hrany),
+abychom umìli rychle zkracovat.
+
+\s{Definice:} {\I Referenèní pár} je dvojice $(\pi,\tau)$, v~ní¾ $\pi$ je vrchol
+stromu a $\tau$ libovolné slovo. Tento pár popisuje slovo $\pi\tau$. Referenèní
+pár je {\I kanonický,} pokud neexistuje hrana vedoucí z~vrcholu $\pi$ s~nálepkou,
+která by byla prefixem slova~$\tau$.
+
+\s{Pozorování:} Ke~ka¾dému slovu existuje právì jeden kanonický referenèní pár,
+který ho popisuje. V¹imnìte si, ¾e je to ze~v¹ech referenèních párù pro toto slovo
+ten s~nejdel¹ím~$\pi$ (nejhlub¹ím vrcholem).
+
+\s{Definice:} Zpìtná hrana $\<back>[\pi]$ vede z~vrcholu $\pi$ do~vrcholu,
+který je ze~v¹ech vrcholù nejdel¹ím vlastním suffixem slova~$\pi$.
+
+\s{Pozorování:} Zpìtné hrany jsme sice zavedli stejnì obecnì, jako se to dìlá
+pøi konstrukci vyhledávacích automatù podle Aha a McCorasickové, ale v~na¹em
+pøípadì se \<back> pro vnitøní vrcholy chová daleko jednodu¹eji (a~na ¾ádné
+jiné ho potøebovat nebudeme): pokud je $\pi$ vnitøní vrchol, musí to být
+vìtvící podslovo, a~tím pádem ka¾dé jeho zkrácení zleva musí být také vìtvící
+podslovo. Tedy $\<back>(\pi)$ dá~$\pi$ bez prvního znaku, a~to se nám
+bude hodit pøi zkracování suffixù.
+
+\s{Algoritmus podrobnìji:} (Doplnili jsme detaily do~pøedchozího algoritmu.)
+
+\algo
+\:Vstup: $\alpha=\alpha(\sigma)$ reprezentovaný jako kanonický referenèní pár $(\pi,\tau)$, $T$ suffixový strom pro~$\sigma$ a jeho funkce \<back>, nový znak~$a$.
+\:Zjistíme, jestli $\alpha a$ je pøítomen ve~stromu, a pøípadnì ho zalo¾íme:
+\::Pokud $\tau=\varepsilon$: ($\alpha=\pi$ je vnitøní vrchol)
+\:::Vede-li z~vrcholu $\pi$ hrana s~nálepkou zaèínající znakem $a$, pak je pøítomen.
+\:::Nevede-li, není pøítomen, a~tak pøidáme novou otevøenou hranu vedoucí z~$\pi$ do~nového listu.
+\::Pokud $\tau\ne\varepsilon$: ($\alpha$ je skrytý vrchol)
+\:::Najdeme hranu, po~ní¾ z~$\pi$ pokraèuje slovo $\tau$ (která to je, poznáme podle prvního znaku slova~$\tau$).
+\:::Pokud v~popisce této hrany po~$\tau$ následuje znak~$a$, pak je $\alpha a$ pøítomen.
+\:::Pokud nenásleduje, tak nebyl pøítomen, èili tuto hranu rozdìlíme: pøidáme na~ni nový vnitøní vrchol,
+ do~nìj¾ povede hrana s~popiskou~$\tau$ a z~nìj zbytek pùvodní hrany a otevøená hrana do~nového listu.
+\:Pokud $\alpha a$ byl pøítomen, tak $\alpha$ zkrátíme a test opakujeme:
+\::Je-li $\pi\ne\varepsilon$, nastavíme $\pi := \<back>(\pi)$. V~opaèném pøípadì (jsme v~koøeni) zkrátíme $\tau$ o~znak zleva.
+\::Pár $(\pi,\tau)$ u¾ popisuje zkrácené slovo, ale nemusí být kanonický, tak¾e to je¹tì napravíme:
+\:::Dokud existuje hrana vedoucí z~$\pi$, její¾ popiska je prefixem slova $\tau$, tak se
+ po~této hranì posuneme, èili prodlou¾íme $\pi$ o~tuto popisku a zkrátíme o~ni~$\tau$.
+\::Zpìt na~krok 2.
+\:Pokud $\alpha a$ u¾ je pøítomen, zbývá pøidat $a$ k~$\alpha$ a zastavit se:
+\::$\tau := \tau a$.
+\::Kanonikalizace stejnì jako v~bodech 12--13.\foot{Dokonce jednodu¹¹í, proto¾e projdeme nejvý¹e jednu hranu.}
+\:Dopoèítáme zpìtné hrany (viz ní¾e).
+\:Výstup: $\alpha=\alpha(\sigma a)$ coby kanonický referenèní pár $(\pi,\tau)$, $T$ suffixový strom pro~$\sigma a$
+ a jeho funkce \<back>.
+\endalgo
+
+\s{Èasová slo¾itost:}
+
+Kanonikalizace pracuje v~amortizovanì konstantním èase, proto¾e ka¾dá její iterace
+zkrátí~$\tau$ a za~ka¾dé spu¹tìní algoritmu se~$\tau$ prodlou¾í jen jednou, a~to o~jeden znak.
+
+Prùchodù hlavním cyklem je, jak u¾ víme, amortizovanì konstantní poèet a ka¾dý prùchod
+zvládneme v~konstantním èase.
+
+Zbývá dodat, jak nastavovat novým vrcholùm jejich \<back>. To potøebujeme
+jen pro vnitøní vrcholy (na~zpìtné hrany z~listù se algoritmus nikdy neodkazuje)
+a v¹imneme si, ¾e pokud jsme zalo¾ili vrchol, odpovídá tento vrchol v¾dy souèasnému~$\alpha$
+a zpìtná hrana z~nìj povede do~zkrácení slova~$\alpha$ o~znak zleva, co¾ je
+pøesnì vrchol, který zalo¾íme (nebo zjistíme, ¾e u¾ existuje) v~pøí¹tí iteraci
+hlavního cyklu. V~dal¹í iteraci urèitì je¹tì nebudeme tuto hranu potøebovat,
+proto¾e $\pi$ v¾dy jen zkracujeme, a~tak mù¾eme vznik zpìtné hrany o~iteraci
+zpozdit a zvládnout to tak také v~èase $\O(1)$.
+
+Celkovì je tedy èasová slo¾itost inkrementálního udr¾ování suffixového
+stromu amortizovanì konstantní.
+\qed
+
+\references
+\bye
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+%%EOF
--- /dev/null
+P=10-suffix
+
+include ../Makerules
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: GIMP PostScript file plugin V 1.17 by Peter Kirchgessner
+%%Title: ST.eps
+%%CreationDate: Fri Jun 9 21:22:13 2006
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%Pages: 1
+%%BoundingBox: 14 14 200 137
+%%EndComments
+%%BeginProlog
+% Use own dictionary to avoid conflicts
+10 dict begin
+%%EndProlog
+%%Page: 1 1
+% Translate for offset
+14.173228346456687 14.173228346456687 translate
+% Translate to begin of first scanline
+0 121.99273036816345 translate
+184.98897637795278 -121.99273036816345 scale
+% Image geometry
+185 122 8
+% Transformation matrix
+[ 185 0 0 122 0 0 ]
+% Strings to hold RGB-samples per scanline
+/rstr 185 string def
+/gstr 185 string def
+/bstr 185 string def
+{currentfile /ASCII85Decode filter /RunLengthDecode filter rstr readstring pop}
+{currentfile /ASCII85Decode filter /RunLengthDecode filter gstr readstring pop}
+{currentfile /ASCII85Decode filter /RunLengthDecode filter bstr readstring pop}
+true 3
+%%BeginData: 15749 ASCII Bytes
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+\input ../sgr.tex
+
+\hyphenation{mikro-strom mikro-stro-mo-vé}
+
+\prednaska{9}{Dekompozice stromù}{}
+
+V~této kapitole uká¾eme nìkolik datových struktur zalo¾ených
+na~my¹lence dekompozice problému na~dostateènì malé podproblémy,
+které u¾ umíme (obvykle vhodným kódováním èísly) øe¹it v~konstantním
+èase.
+
+\h{Union-Find problem}
+
+\s{Problém:} Udr¾ování tøíd ekvivalence: na~poèátku máme $N$ jednoprvkových ekvivalenèních
+tøíd, provádíme operace \<Find> (zji¹tìní, zda dva prvky jsou ekvivalentní) a \<Union>
+(slouèení dvou tøíd do~jedné). Také na~to lze pohlí¾et jako na~inkrementální udr¾ování
+komponent souvislosti neorientovaného grafu: \<Union> je pøidání hrany, \<Find> test,
+zda dva vrcholy le¾í v~té¾e komponentì. To se hodí v~mnoha algoritmech, kupøíkladu
+v~Kruskalovì algoritmu pro hledání minimální kostry.
+
+\s{Triviální øe¹ení:} Ka¾dé tøídì pøiøadíme unikátní barvu, kterou obarvíme prvky tøídy. Operace \<Find>
+porovnává barvy, operace \<Union> prvky jedné tøídy pøebarvuje.
+
+Operace \<Find> tak pracuje v~konstantním èase, \<Union> mù¾e zabrat a¾ lineární èas. Mù¾eme si
+pomoci tím, ¾e v¾dy pøebarvíme {\I men¹í} ze~sluèovaných ekvivalenèních tøíd (budeme
+si pro ka¾dou tøídu pamatovat seznam jejích prvkù a velikost). Tehdy mù¾e být ka¾dý
+prvek pøebarven jen $\O(\log n)$-krát, jeliko¾ ka¾dým pøebarvením se alespoò zdvojnásobí
+velikost tøídy, ve~které prvek le¾í. Posloupnost operací \<Union>, kterou vznikla tøída
+velikosti~$k$, tak trvá $\O(k\log k)$, tak¾e mù¾eme bezpeènì prohlásit, ¾e amortizovaná
+slo¾itost operace \<Union> je $\O(\log n)$.
+
+\s{Chytøej¹í øe¹ení:} Ka¾dou tøídu budeme reprezentovat zakoøenìným stromem s~hranami
+orientovanými smìrem ke~koøeni (jinými slovy pro ka¾dý prvek si pamatujeme jeho otce
+nebo ¾e je to koøen). \<Find> nalezne koøeny stromù a porovná je, \<Union> pøipojí koøen
+jedné tøídy pod koøen druhé. Aby stromy nedegenerovaly, pøidáme dvì podmínky:
+
+\itemize\ibull
+\:{\I Union by rank:} ka¾dý koøen $v$ si pamatuje svùj rank $r(v)$. Pokud spojujeme
+dva stromy s~koøeny $v$, $w$ a $r(v)<r(w)$, pøipojíme $v$ pod~$w$ a rank zachováme.
+Pokud $r(v)=r(w)$, pøipojíme libovolnì a nový koøen bude mít rank $r(v)+1$.%
+\foot{Stejnì by fungovalo pravidlo {\I Union by size,} které pøipojuje men¹í
+strom pod vìt¹í, ale ranky máme radìji, neb jsou skladnìj¹í a snáze se analyzují.}
+
+\:{\I Path compression:} pokud z~vrcholu vystoupíme do~koøene (napøíklad
+bìhem operace \<Find>), pøepojíme v¹echny vrcholy na~cestì, po~které jsme pro¹li,
+rovnou pod koøen.
+\endlist
+
+\s{Pozorování:} Samotné pravidlo Union by rank zajistí, ¾e strom ranku $r$ bude
+mít hloubku nejvý¹e $r$ a minimálnì $2^r$ vrcholù, tak¾e èasová slo¾itost operací
+bude omezena $\O(\log n)$.%
+\foot{Mimochodem, Path compression samotná by také na~slo¾itost $\O(\log n)$ amortizovanì staèila.}
+
+Ve~skuteènosti se popsaná struktura chová daleko lépe:
+
+\s{Vìta:} (Tarjan, van Leeuwen \cite{tarjan84setunion}) Kombinace Union by rank a Path compression vede k~amortizované
+slo¾itosti obou operací $\O(\alpha(n))$, kde $\alpha$ je inverzní Ackermannova funkce.%
+\foot{Existuje varianta tohoto algoritmu, která dosahuje stejné slo¾itosti i v~nejhor¹ím
+pøípadì; té¾ je známo, ¾e asymptoticky lep¹í slo¾itosti nelze dosáhnout.}
+
+\h{Union-Find s~pøedem známými Uniony}
+
+Dále nás bude zajímat speciální varianta Union-Find problemu, v~ní¾ dopøedu známe
+posloupnost Unionù, èili strom, který spojováním komponent vznikne.\foot{Kdy se to hodí?
+Tøeba v~Thorupovì lineárním algoritmu \cite{thorup:usssp} na~nejkrat¹í cesty nebo
+v~Weiheho takté¾ lineárním algoritmu \cite{weihe:paths} na~hledání hranovì disjunktních
+cest v~rovinných grafech.}
+Jiná interpretace tého¾ (jen pozpátku) je dekrementální udr¾ování komponent
+souvislosti lesa: na~poèátku je dán les a umíme smazat hranu a otestovat, zda jsou
+dva vrcholy v~tém¾e stromu.
+
+Popí¹eme algoritmus,
+který po~poèáteèním pøedzpracování v~èase $\O(n)$ zvládne \<Union> i \<Find> v~amortizovanì
+konstantním èase. Tento algoritmus je kombinací dekompozic popsaných Alstrupem v~\cite{alstrup97optimal}
+a \cite{alstrup98marked}.
+
+\s{Definice:} {\I (Microtree/Macrotree dekompozice)} Pro zakoøenìný strom $T$ o~$n$ vrcholech
+definujeme:
+\itemize\ibull
+\:{\I Koøeny mikrostromù} $R$ budou nejvy¹¹í vrcholy, pod~nimi¾ je nejvý¹e $\log n$ listù
+a které nejsou koøenem celého~$T$.
+\:{\I Mikrostromy} le¾í v~$T$ od~tìchto koøenù ní¾e.
+\:{\I Spojovací hrany} vedou z~koøenù mikrostromù do~jejich otcù.
+\:{\I Makrostrom} je tvoøen zbývajícími vrcholy a hranami stromu~$T$.
+\endlist
+
+\s{Pozorování:} Ka¾dý mikrostrom má nejvý¹e $\log n$ listù. Pod ka¾dým listem makrostromu le¾í
+alespoò jeden mikrostrom (mù¾e jich být i více, viz dekompozice hvìzdy na~obrázku), tak¾e
+listù makrostromu je nejvý¹e $n/\log n$.
+
+Vnitøních vrcholù makro- i mikrostromù ale mù¾e být ne¹ikovnì mnoho, proto¾e se ve~stromech mohou
+vyskytovat dlouhé cesty. Pomù¾eme si snadno: ka¾dou cestu si budeme pamatovat zvlá¹» a ve~stromu
+ji nahradíme hranou, která bude vlo¾ena právì tehdy, kdy¾ budou pøítomny v¹echny hrany cesty.
+
+\s{Pøíklad:} Následující obrázek ukazuje dekompozici nìkolika stromù za~pøepokladu,
+¾e $\log n=4$. Vrcholy mikrostromù jsou èerné, makrostromu bílé. Spojovací hrany kreslíme teèkovanì,
+hrany komprimovaných cest tuènì.
+
+\fig{mima.eps}{\epsfxsize}
+
+\s{Algoritmus pro cesty:} Cestu délky~$l$ rozdìlíme na~úseky délky $\log n$, pro nì¾ si ulo¾íme
+mno¾iny ji¾ pøítomných hran (po~bitech jako èísla). Pak si je¹tì pamatujeme zkomprimovanou cestu (hrany
+odpovídají úsekùm a jsou pøítomny právì tehdy, jsou-li pøítomny v¹echny hrany pøíslu¹ného úseku)
+délky $l/\log n$ a pro ni \uv{pøebarvovací} strukturu pro Union-Find.
+
+\>$\<Union>(x,y)$ (pøidání hrany $e=xy$ do~cesty):
+\algo
+\:Pøidáme $e$ do mno¾iny hran pøítomných v~pøíslu¹ném úseku.
+\:Pokud se tím úsek naplnil, pøidáme odpovídající hranu do~zkomprimované cesty.
+\endalgo
+
+\>$\<Find>(x,y):$
+\algo
+\:Pokud $x$ a $y$ jsou v~tém¾e úseku, otestujeme bitovými operacemi, zda
+ jsou v¹echny hrany mezi $x$ a $y$ pøítomny.
+\:Pokud jsou v~rùzných úsecích, rozdìlíme cestu z~$x$ do~$y$ na~posloupnost celých úsekù,
+ na~které nám odpoví zkomprimovaná cesta, a~dva dotazy v~krajních èásteèných úsecích.
+\endalgo
+
+Operace uvnitø úsekù pracují v~èase $\O(1)$, operace na~zkomprimované cestì v~$\O(\log l)$
+amortizovanì, ale za~dobu ¾ivota struktury je jich $\O(l/\log n)=\O(l/\log l)$, tak¾e celkovì zaberou lineární èas.
+
+\s{Cestová komprese:} Operace na~mikro/makro-stromech budeme následujícím zpùsobem
+pøevádìt na~operace s~jejich cestovì komprimovanými podobami a na~operace s~cestovými strukturami:
+
+\>$\<Union>(x,y)$:
+\algo
+\:Pokud $e=xy$ le¾í uvnitø nìjaké cesty, pøidáme ji do~cesty, co¾ buïto zpùsobí
+ pøidávání jiné hrany, a~nebo u¾ jsme hotovi.
+\:Provedeme \<Union> v~komprimovaném stromu.
+\endalgo
+
+\>$\<Find>(x,y)$:
+\algo
+\:Pokud $x$ a $y$ le¾í uvnitø jedné cesty, zeptáme se cestové struktury a konèíme.
+\:Pokud $x$ le¾í uvnitø nìjaké cesty, zjistíme dotazem na~cestovou strukturu,
+ ke~kterému krajnímu vrcholu cesty je pøipojen, a~$x$ nahradíme tímto vrcholem.
+ Není-li pøipojen k~¾ádnému, je~evidentnì odpovìï na~celý \<Find> negativní;
+ pokud k~obìma, vybereme si libovolný, proto¾e jsou stejnì v~cestovì komprimovaném
+ stromu spojeny hranou. Analogicky pro~$y$.
+\:Zeptáme se struktury pro komprimovaný strom.
+\endalgo
+
+\s{Algoritmus pro mikrostromy:} Po~kompresi cest má ka¾dý mikrostrom nejvý¹e $2\log n$
+vrcholù, èili také nejvý¹e tolik hran. Hrany si oèíslujeme pøirozenými èísly, ka¾dou
+mno¾inu hran pak mù¾eme reprezentovat $2\log n$-bitovým èíslem a mno¾inové operace
+provádìt pomocí bitových v~konstantním èase.
+
+Pro ka¾dý mikrostrom si pøedpoèítáme pro v¹echny jeho vrcholy~$v$ mno¾iny~$P_v$ hran le¾ících
+na~cestì z~koøene mikrostromu do~$v$. Navíc si budeme pro celý mikrostrom pamatovat mno¾inu
+pøítomných hran~$F$.
+
+\>$\<Union>(x,y):$
+
+\algo
+\:Najdeme poøadové èíslo $i$ hrany $xy$ (máme pøedpoèítané).
+\:$F \leftarrow F \cup \{i\}$.
+\endalgo
+
+\>$\<Find>(x,y):$
+
+\algo
+\:$P \leftarrow P_x \mathop{\Delta} P_v$ (mno¾ina hran le¾ících na~cestì z~$x$ do~$y$).
+\:Pokud $P\setminus F=\emptyset$, le¾í $x$ a $y$ ve~stejnì komponentì, jinak ne.
+\endalgo
+
+\s{Algoritmus pro celý problém:} Strom rozlo¾íme na~mikrostromy, makrostromy a spojovací
+hrany. V~mikrostromech i makrostromech zkomprimujeme cesty. Pro cesty a mikrostromy pou¾ijeme
+vý¹e popsané struktury, pro ka¾dou spojovací hranu si budeme pamatovat jen znaèku,
+zda je pøítomna, a pro makrostrom pøebarvovací strukturu.
+
+\>$\<Union>(x,y)$:
+
+\algo
+\:Pokud $e=xy$ je spojovací, poznamenáme si, ¾e je pøítomna, a~konèíme.
+\:Nyní víme, ¾e $e$ le¾í uvnitø mikrostromu nebo makrostromu, a~tak provedeme \<Union>
+ na~pøíslu¹né struktuøe.
+\endlist
+
+\>$\<Find>(x,y)$:
+
+\algo
+\:Le¾í-li $x$ a $y$ v~jednom mikrostromu, zeptáme se struktury pro~mikrostrom.
+\:Je-li $x$ uvnitø mikrostromu, zeptáme se mikrostruktury na~spojení s~koøenem mikrostromu.
+ Není-li, odpovíme {\sc ne}, stejnì jako kdy¾ není pøítomna pøíslu¹ná spojovací hrana.
+ Jinak $x$ nahradíme listem makrostromu, do~kterého spojovací hrana vede. Podobnì pro~$y$.
+\:Odpovíme podle struktury pro makrostrom.
+\endalgo
+
+\s{Analýza:} Operace \<Find> trvá konstantní èas, proto¾e se rozlo¾í na~$\O(1)$ \<Find>ù
+v~dílèích strukturách a ka¾dý z~nich trvá konstantnì dlouho. V¹ech $n$ operací \<Union>
+trvá $\O(n)$, jeliko¾ zpùsobí $\O(n)$ amortizovanì konstantních operací s~mikrostromy, spojovacími
+hranami a cestami a $\O(n/\log n)$ operací s~makrostromy, které trvají $\O(\log n)$ amortizovanì
+ka¾dá.%
+\foot{To je v~prùmìru $\O(1)$ na~operaci a dokonce i amortizovanì, pokud necháme inicializaci
+struktury, která je lineární, naspoøit potenciál $\O(n)$, ze~kterého budeme prùbì¾nì platit
+sluèování v~makrostromu.}
+
+\s{Cvièení:} Zkuste pomocí dekompozice vyøe¹it následující problém: je dán strom,
+jeho¾ ka¾dý vrchol mù¾e být oznaèený. Navrhnìte datovou strukturu, která bude umìt
+v~èase $\O(\log\log n)$ oznaèit nebo odznaèit vrchol a v~èase $\O(\log n/\log\log n)$ najít
+nejbli¾¹ího oznaèeného pøedchùdce.
+
+\h{Fredericksonova clusterizace}
+
+Mikro/makro-stromová dekompozice není jediný zpùsob, jak stromy rozkládat. Nìkdy
+se hodí napøíklad následující my¹lenka: \cite{frederickson91ambivalent}
+
+\s{Definice:} (Fredericksonova clusterizace) Nech» $G$ je graf s~vrcholy stupòù nejvý¹e~3
+a $c\ge 1$. Pak $c$-clusterizací grafu $G$ nazveme libovolný rozklad
+$G$ na~souvislé podgrafy (clustery) $C_1, C_2, \ldots, C_k$ takový, ¾e platí:
+\itemize\ibull
+\:$\forall v \in V \exists ! i: v \in C_i$.
+\:$\forall i: \vert C_i\vert \le c$.
+\:$\forall i$ je vnìj¹í stupeò $C_i$ (tj. poèet hran, které vedou mezi $C_i$ a zbytkem grafu)
+nejvý¹e~3. Navíc pokud je právì~3, je cluster triviální, èili $\vert C_i \vert = 1$.
+\:®ádné dva sousední clustery nelze spojit.
+\endlist
+
+\s{Vìta:} (Frederickson) Ka¾dá $c$-clusterizace grafu $G$ má $\O(V(G)/c)$ clusterù. Existuje
+algoritmus, který jednu takovou najde v~lineárním èase.
+
+\proof První èást rozborem pøípadù, druhá hladovì pomocí DFS. \qed
+
+\s{Pou¾ití:} Pøedchozí variantu Union-Find problemu bychom také mohli vyøe¹it nahrazením
+vrcholù stupnì $>3$ \uv{kruhovými objezdy bez jedné hrany}\foot{tzv. francouzský trik},
+nalezením $(\log n)$-clusterizace, pou¾itím bitové reprezentace mno¾in uvnitø clusterù
+a pøebarvovací struktury na~hrany mezi clustery.
+
+\h{Stromoví pøedchùdci}
+
+\s{Problém:} {\I (Least Common Ancestor alias LCA)} Chceme si pøedzpracovat zakoøenìný strom~$T$
+tak, abychom dokázali pro libovolné dva vrcholy $x,y$ najít co~nejrychleji jejich nejbli¾¹ího
+spoleèného pøedchùdce.
+
+\s{Triviální øe¹ení LCA:}
+\itemize\ibull
+\:Vystoupáme z~$x$ i $y$ do~koøene, oznaèíme vrcholy na~cestách a kde se poprvé
+ potkají, tam je hledaný pøedchùdce. To je lineární s~hloubkou a nepotøebuje
+ pøedzpracování.
+\:Vylep¹ení: Budeme stoupat z~$x$ a $y$ støídavì. Tak potøebujeme jen lineárnì mnoho
+ krokù vzhledem ke~vzdálenosti spoleèného pøedchùdce.
+\:Pøedpoèítáme v¹echny mo¾nosti: pøedzpracování $\O(n^2)$, dotaz $\O(1)$.
+\:\dots\ co dál?
+\endlist
+
+\>Vìrni vtipùm o~matfyzácích a èlánku \cite{bender00lca} pøevedeme radìji tento problém na~jiný.
+
+\s{Problém:} {\I (Range Minimum Query alias RMQ)} Chceme pøedzpracovat posloupnost èísel
+$a_1,\ldots a_n$ tak, abychom umìli rychle poèítat $\min_{x\le i\le y} a_i$.%
+\foot{V¹imnìte si, ¾e pro sumu místo minima je tento problém velmi snadný.}
+
+\s{Lemma:} LCA lze pøevést na~RMQ s~lineárním èasem na~pøedzpracování a konstantním
+èasem na~pøevod dotazu.
+
+\proof Strom projdeme do~hloubky a poka¾dé, kdy¾ nav¹tívíme vrchol (v~inorderu),
+zapí¹eme jeho hloubku. ${\rm LCA}(x,y)$ pak bude nejhlub¹í vrchol mezi libovolnou
+náv¹tìvou~$x$ a libovolnou náv¹tìvou~$y$.
+\qed
+
+\s{Triviální øe¹ení RMQ:}
+\itemize\ibull
+\:Pøedpoèítáme v¹echny mo¾né dotazy: pøedzpracování $\O(n^2)$, dotaz $\O(1)$.
+\:Pro ka¾dé $i$ a $j\le \log n$ pøedpoèítáme $m_{ij} = \min\{ a_i, a_{i+1}, \ldots, a_{i+2^j-1} \}$,
+èili minima v¹ech blokù velkých jako nìjaká mocnina dvojky. Kdy¾ se poté nìkdo zeptá
+na~minimum bloku $a_i,a_{i+1},\ldots,a_{j-1}$, najdeme nejvìt¹í~$k$ takové, ¾e $2^k < j-i$
+a vrátíme:
+$$\min( \min\{ a_i, \ldots, a_{i+2^k-1} \}, \min\{ a_{j-2^k}, \ldots, a_{j-1} \} ).$$
+Tak zvládneme dotazy v~èase $\O(1)$ po~pøedzpracování v~èase $\O(n\log n)$.
+\endlist
+
+My si ov¹em v¹imneme, ¾e ná¹ pøevod z~LCA vytváøí dosti speciální instance problému RMQ,
+toti¾ takové, v~nich¾ je $\vert a_i - a_{i+1} \vert = 1$. Takovým instancím budeme
+øíkat RMQ${\pm}1$ a budeme je umìt øe¹it ¹ikovnou dekompozicí.
+
+\s{Dekompozice} pro RMQ${\pm}1$: Vstupní posloupnost rozdìlíme na~bloky velikosti $b=1/2\cdot \log n$,
+ka¾dý dotaz umíme rozdìlit na~èást týkající se celých blokù a maximálnì dva dotazy na~èásteèné bloky.
+
+V¹imneme si, ¾e aèkoliv blokù je mnoho, jejich mo¾ných typù (tj. posloupností klesání
+a stoupání) je pouze $2^{b-1}\le\sqrt n$ a bloky tého¾ typu se li¹í pouze posunutím
+o~konstantu. Vybudujeme proto kvadratickou strukturu pro jednotlivé typy a pro ka¾dý
+blok si zapamatujeme, jakého je typu a jaké má posunutí. Celkem strávíme èas
+$\O(n + \sqrt n \cdot \log^2 n) = \O(n)$ pøedzpracováním a $\O(1)$ dotazem.
+
+Mimo to je¹tì vytvoøíme komprimovanou posloupnost, v~ní¾ ka¾dý blok nahradíme
+jeho minimem. Tuto posloupnost délky $n/b$ budeme pou¾ívat pro èásti dotazù
+týkající se celých blokù a pøipravíme si pro ni \uv{logaritmickou} variantu
+triviální struktury. To nás bude stát $\O(n/b\cdot\log (n/b))=\O(n/\log n\cdot\log n)=\O(n)$ na~pøedzpracování
+a $\O(1)$ na~dotaz.
+
+Tak jsme získali algoritmus pro RMQ${\pm}1$ s~konstantním èasem na~dotaz po~lineárním
+pøedzpracování a vý¹e zmínìným pøevodem i algoritmus na~LCA se stejnými parametry.
+Je¹tì uká¾eme, ¾e pøevod mù¾e fungovat i v~opaèném smìru, a~tak mù¾eme získat
+i konstantní/lineární algoritmus pro obecné RMQ.
+
+\s{Definice:} {\I Kartézský strom} pro posloupnost $a_1,\ldots,a_n$ je strom,
+jeho¾ koøenem je $a_j=\min_i a_i$, jeho levý podstrom je kartézský strom pro
+$a_1,\ldots,a_{j-1}$ a pravý podstrom kartézský strom pro $a_{j+1},\ldots,a_n$.
+
+\s{Lemma:} Kartézský strom je mo¾né zkonstruovat v~lineárním èase.
+
+\proof Pou¾ijeme inkrementální algoritmus. V¾dy si budeme pamatovat
+kartézský strom pro ji¾ zpracované prvky a pozici posledního zpracovaného
+prvku v~tomto stromu. Kdy¾ pøidáváme dal¹í prvek, hledáme místo, kam ho
+pøipojit, od~tohoto oznaèeného prvku nahoru. Pov¹imnìme si, ¾e vzhledem
+k~potenciálu rovnému hloubce oznaèeného prvku je èasová slo¾itost pøidání
+prvku amortizovanì konstantní.
+\qed
+
+\s{Lemma:} RMQ lze pøevést na~LCA s~lineárním èasem na~pøedzpracování a konstantním
+èasem na~pøevod dotazu.
+
+\proof Sestrojíme kartézský strom a RMQ pøevedeme na~LCA v~tomto stromu.
+\qed
+
+Výsledky této podkapitoly mù¾eme shrnout do~následující vìty:
+
+\s{Vìta:} Problémy LCA i RMQ je mo¾né øe¹it v~konstantním èase na~dotaz
+po~pøedzpracování v~lineárním èase.
+
+\references
+\bye
--- /dev/null
+P=9-decomp
+
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-\input ../sgr.tex
-
-\prednaska{9}{Suffixové stromy}{}
-
-V~této kapitole popí¹eme jednu pozoruhodnou datovou strukturu, pomocí ní¾ doká¾eme problémy týkající
-se øetìzcù pøevádìt na grafové problémy a tak je øe¹it v~lineárním èase.
-
-\h{Øetìzce, trie a suffixové stromy}
-
-\ss{Definice:}
-
-\nointerlineskip
-\halign{\qquad#\dotfill&~#\hfil\cr
-\hbox to 0.35\hsize{}\cr
-$\Sigma$ & koneèná abeceda -- mno¾ina znakù \cr
-\omit & (znaky budeme znaèit latinskými písmeny)\cr
-$\Sigma^*$ & mno¾ina v¹ech slov nad $\Sigma$ \cr
-\omit & (slova budeme znaèit øeckými písmeny)\cr
-$\varepsilon$ & prázdné slovo\cr
-$\vert\alpha\vert$ & délka slova $\alpha$\cr
-$\alpha\beta$ & zøetìzení slov $\alpha$ a $\beta$ ($\alpha\varepsilon=\varepsilon\alpha=\alpha$)\cr
-$\alpha^R$ & slovo $\alpha$ napsané pozpátku\cr
-$\alpha$ je {\I prefixem} $\beta$ & $\exists\gamma: \beta=\alpha\gamma$ ($\beta$ zaèíná na~$\alpha$)\cr
-$\alpha$ je {\I suffixem} $\beta$ & $\exists\gamma: \beta=\gamma\alpha$ ($\beta$ konèí na~$\alpha$)\cr
-$\alpha$ je {\I podslovem} $\beta$ & $\exists\gamma,\delta: \beta=\gamma\alpha\delta$ (znaèíme $\alpha \subset \beta$)\cr
-$\alpha$ je {\I vlastním prefixem} $\beta$ & je prefixem a $\alpha\ne\beta$ \cr
-\omit & (analogicky vlastní suffix a podslovo)\cr
-}
-
-\s{Pozorování:} Prázdné slovo je prefixem, suffixem i podslovem ka¾dého slova vèetnì sebe sama.
-Podslova jsou právì prefixy suffixù a také suffixy prefixù.
-
-\s{Definice:} {\I Trie ($\Sigma$-strom)} pro koneènou mno¾inu slov $X\subset\Sigma^*$ je orientovaný graf $G=(V,E)$, kde:
-\itemize\relax
-\:$V = \{\alpha: \alpha\hbox{ je prefixem nìjakého $\beta\in X$} \},$
-\:$(\alpha,\beta)\in E \equiv \exists x\in\Sigma: \beta=\alpha x$.
-\endlist
-
-\s{Pozorování:} Trie je strom s koøenem $\varepsilon$. Jeho listy jsou slova z $X$, která nejsou vlastními prefixy jiných slov z~$X$.
-Hrany si mù¾eme pøedstavit popsané písmeny, o~nì¾ prefix roz¹iøují, popisky hran na~cestì z~koøene do~vrcholu~$\alpha$ dávají právì slovo~$\alpha$.
-
-\s{Definice:} {\I Komprimovaná trie ($\Sigma^+$-strom)} vznikne z trie nahrazením maximálních nevìtvících se cest hranami. Hrany
-jsou tentokrát popsané øetìzci místo jednotlivými písmeny, pøièem¾ popisky v¹ech hran vycházejících z~jednoho vrcholu se li¹í v~prvním
-znaku. Vrcholùm \uv{uvnitø hran} (které padly za obì» kompresi) budeme øíkat {\I skryté vrcholy.}
-
-\medskip
-\tabskip=0pt plus 1fil
-\halign to\hsize{
-\hfil#\hfil&\hfil#\hfil&\hfil#\hfil\cr
-\epsfbox{trie.eps}&\epsfbox{trie-c.eps}&\epsfbox{trie-cd.eps}\cr
-Trie pro $\{\hbox{AULA, AUTO, AUTOBUS, BUS}\}$ & \dots komprimovaná & \dots odolarovaná\cr
-}
-\medskip
-
-\s{Definice:} {\I Suffixový strom (ST)} pro slovo $\sigma\in\Sigma^*$ je komprimovaná trie pro $X=\{\alpha: \hbox{$\alpha$ je suffixem $\sigma$}\}$.
-
-\s{Pozorování:} Vrcholy suffixového stromu (vèetnì skrytých) odpovídají prefixùm suffixù slova~$\sigma$,
-tedy v¹em jeho podslovùm. Listy stromu jsou suffixy, které se v~$\sigma$ ji¾ nikde jinde nevyskytují
-(takovým suffixùm budeme øíkat {\I nevnoøené}). Vnitøní vrcholy odpovídají {\I vìtvícím podslovùm,}
-tedy podslovùm $\alpha\subset\sigma$ takovým, ¾e $\alpha a\subset\sigma$ i $\alpha b\subset\sigma$
-pro nìjaké dva rùzné znaky~$a$,~$b$.
-
-Nìkdy mù¾e být nepraktické, ¾e nìkteré suffixy neodpovídají listùm (proto¾e jsou vnoøené), ale
-s~tím se mù¾eme snadno vypoøádat: pøidáme na~konec slova~$\sigma$ nìjaký znak~$\$$, který se nikde
-jinde nevyskytuje. Neprázdné suffixy slova $\sigma\$$ odpovídají suffixùm slova~$\sigma$
-a ¾ádný z~nich nemù¾e být vnoøený. Takový suffixový strom budeme znaèit ST\$.
-
-\figure{st-barbara.eps}{Suffixový strom pro slovo BARBARA}{0pt}
-
-Nyní jak je to s~konstrukcí suffixových stromù:
-
-\s{Lemma:} Suffixový strom pro slovo $\sigma$ délky $n$ je reprezentovatelný v~prostoru $\O(n)$.
-
-\proof Strom má $\O(n)$ listù a ka¾dý vnitøní vrchol má alespoò $2$ syny, tak¾e vnitøních
-vrcholù je také $\O(n)$. Hran je rovnì¾ lineárnì. Nálepky na~hranách staèí popsat
-poèáteèní a koncovou pozicí v~$\sigma$.
-\qed
-
-\s{Vìta:} Suffixový strom pro slovo $\sigma$ délky $n$ lze sestrojit v~èase $\O(n)$.
-
-\proof Ve~zbytku této kapitoly pøedvedeme dvì rùzné konstrukce v~lineárním èase.
-\qed
-
-\s{Aplikace} -- co v¹e doká¾eme v~lineárním èase, kdy¾ umíme lineárnì konstruovat ST:
-
-\nobreak
-
-\numlist\ndotted
-\:{\I Inverzní vyhledávání} (tj. pøedzpracujeme si v~lineárním èase text a pak umíme pro libovolné
-slovo~$\alpha$ v~èase $\O(\vert\alpha\vert)$ rozhodnout, zda se v~textu vyskytuje)\foot{Èili pøesný
-opak toho, co~umí vyhledávací automat -- ten si pøedzpracovává dotaz.} -- staèí sestrojit~ST
-a pak jej procházet od~koøene. Také umíme najít v¹echny výskyty (odpovídají suffixùm, které mají
-jako prefix hledané slovo, tak¾e staèí vytvoøit ST\$ a vypsat v¹echny listy pod
-nalezeným vrcholem) nebo pøímo vrátit jejich poèet (pøedpoèítáme si pomocí DFS pro ka¾dý vrchol,
-kolik pod ním le¾í listù).
-
-\:{\I Nejdel¹í opakující se podslovo} -- takové podslovo je v~ST\$ nutnì vìtvící, tak¾e staèí
-najít vnitøní vrchol s~nejvìt¹í {\I písmenkovou hloubkou} (tj. hloubkou mìøenou ve~znacích
-místo ve~hranách).
-
-\:{\I Histogram èetností podslov délky~$k$} -- rozøízneme ST v~písmenkové hloubce~$k$ a spoèítáme,
-kolik pùvodních listù je pod ka¾dým novým.
-
-\:{\I Nejdel¹í spoleèné podslovo} slov~$\alpha$ a $\beta$ -- postavíme ST pro slovo $\alpha\$_1\beta\$_2$,
-jeho listy odpovídají suffixùm slov $\alpha$ a $\beta$. Tak¾e staèí pomocí DFS najít nejhlub¹í vnitøní
-vrchol, pod kterým se vyskytují listy pro~$\alpha$ i $\beta$. Podobnì mù¾eme sestrojit ST\$ pro libovolnou
-mno¾inu slov.\foot{Jen si musíme dát pozor, abychom si moc nezvìt¹ili abecedu, ale to bude jasné,
-a¾ pøedvedeme konkrétní konstrukce.}
-
-\:{\I Nejdel¹í palindromické podslovo} (tj. takové $\beta\subset\alpha$, pro nì¾ je $\beta_R=\beta$)
--- postavíme spoleèný ST\$ pro slova $\alpha$ a $\alpha_R$. Postupnì procházíme pøes v¹echny mo¾né støedy
-palindromického podslova a v¹imneme si, ¾e takové slovo je pro ka¾dý støed nejdel¹ím spoleèným
-prefixem podslova od~tohoto bodu do~konce a podslova od~tohoto bodu pozpátku k~zaèátku,
-èili nìjakého suffixu $\alpha$ a nìjakého suffixu $\alpha_R$. Tyto suffixy ov¹em odpovídají
-listùm sestrojeného ST a jejich nejdel¹í spoleèný prefix je nejbli¾¹ím spoleèným pøedchùdcem
-ve~stromu, tak¾e staèí pro strom vybudovat datovou strukturu pro spoleèné pøedchùdce
-a s~její pomocí doká¾eme jeden støed prozkoumat v~konstantním èase.
-
-\:{\I Burrows-Wheelerova Transformace} \cite{burrows:bwt} -- jejím základem je lexikografické setøídìní v¹ech
-rotací slova~$\sigma$, co¾ zvládneme sestrojením ST pro slovo~$\sigma\sigma$, jeho
-uøíznutím v~písmenkové hloubce~$\vert\sigma\vert$ a vypsáním novì vzniklých listù v~poøadí
-\uv{zleva doprava}.
-\endlist
-
-\s{Cvièení:} Zkuste vymyslet co nejlep¹í algoritmy pro tyto problémy bez pou¾ití~ST.
-
-\h{Suffix Array}
-
-\>V~nìkterých pøípadech se hodí místo suffixového stromu pou¾ívat kompaktnìj¹í datové struktury.
-
-\s{Notace:} Pro slovo $\sigma$ bude $\sigma[i]$ znaèit jeho $i$-tý znak (èíslujeme od~jednièky),
-$\sigma[i:j]$ pak podslovo slo¾ené z~$i$-tého a¾ $j$-tého znaku. Libovolnou z~mezí mù¾eme vynechat, proto
-$\sigma[i:{}]$ bude suffix od~$i$ do~konce a $\sigma[{}:j]$ prefix od~zaèátku do~$j$.
-Pokud $j<i$, definujeme $\sigma[i:j]$ jako prázdné slovo, tak¾e prázdný suffix mù¾eme
-napøíklad zapsat jako $\sigma[\vert\sigma\vert+1:{}].$
-
-${\rm LCP}(\alpha,\beta)$ bude znaèit délku nejdel¹ího spoleèného prefixu slov $\alpha$ a $\beta$,
-èili nejvìt¹í $i\le \vert\alpha\vert,\vert\beta\vert$ takové, ¾e $\alpha[{}:i]=\beta[{}:i]$.
-
-\s{Definice:} {\I Suffix Array} $A_\sigma$ pro slovo $\sigma$ délky~$n$ je posloupnost v¹ech suffixù
-slova~$\sigma$ v~lexikografickém poøadí. Mù¾eme ho reprezentovat napøíklad jako permutaci $A$ èísel
-$1,\ldots,n+1$, pro ní¾ $\sigma[A[1]:{}] < \sigma[A[2]:{}] < \ldots < \sigma[A[n+1]:]$.
-
-\s{Definice:} {\I Longest Common Prefix Array} $L_\sigma$ pro slovo $\sigma$ je posloupnost,
-v~ní¾ $L_\sigma[i]:={\rm LCP}(A_\sigma[i],A_\sigma[i+1])$.
-
-\s{Vìta:} Suffixový strom pro slovo $\sigma\$$ je lineárnì ekvivalentní s~dvojicí $(A_\sigma,L_\sigma)$.
-[Jinými slovy, kdy¾ máme jedno, mù¾eme z~toho v~lineárním èase spoèítat druhé, a naopak.]
-
-\proof Kdy¾ projdeme ST($\sigma$) do hloubky, poøadí listù odpovídá $A_\sigma$ a písmenkové hloubky vnitøních
-vrcholù v~inorderu odpovídají $L_\sigma$. Naopak ST($\sigma$) získáme tak, ¾e sestrojíme kartézský strom
-pro~$L_\sigma$ (získáme vnitøní vrcholy ST), doplníme do~nìj listy, pøiøadíme jim suffixy podle~$A_\sigma$
-a nakonec podle listù rekonstruujeme nálepky hran.
-\qed
-
-\h{Rekurzivní konstrukce}
-
-\>Tento algoritmus konstruuje pro slovo $\sigma$ délky~$n$ jeho suffix array a LCP array v~èase $\O(n+{\rm Sort}(n,\Sigma))$,
-kde ${\rm Sort}(\ldots)$ je èas potøebný pro setøídìní $n$ symbolù z~abecedy~$\Sigma$. V~kombinaci s~pøedchozími
-výsledky nám tedy dává lineární konstrukci ST($\sigma$) pro libovolnou fixní abecedu.
-
-\s{Algoritmus:} (Konstrukce $A$ a $L$ podle Kärkkäinena a Sanderse \cite{karkkainen03simple})
-
-\algo
-\:Redukujeme abecedu na~$1\ldots n$: ve~vstupním slovu je nejvý¹e $n$ rùzných znakù,
-tak¾e je staèí setøídit a pøeèíslovat.
-
-\:Definujeme slova $\sigma_0$, $\sigma_1$, $\sigma_2$ následovnì:
-$$\eqalign{
-\sigma_0[i] &:= \left<\sigma[3i],\sigma[3i+1],\sigma[3i+2]\right>\cr
-\sigma_1[i] &:= \left<\sigma[3i+1],\sigma[3i+2],\sigma[3i+3]\right>\cr
-\sigma_2[i] &:= \left<\sigma[3i+2],\sigma[3i+3],\sigma[3i+4]\right>\cr
-}$$
-V¹echna $\sigma_k$ jsou slova délky $\approx n/3$ nad~abecedou velikosti $n^3$. Dovolíme
-si mírnì zneu¾ívat notaci a pou¾ívat symbol $\sigma_k$ i jejich pøepis do~abecedy pùvodní.
-
-\:Zavoláme algoritmus rekurzivnì na slovo $\sigma_0\sigma_1$, èím¾ získáme $A_{01}$ a $L_{01}$.
-
-\:Z~$A_{01}$ a $L_{01}$ vydìlíme $A_0=A_{\sigma_0}$, $A_1$, $L_0$ a $L_1$. Také si pro ka¾dý prvek
-$A_i$ zapamatujeme, kde se v~$A_{01}$ vyskytoval.
-
-\:Dopoèítáme $A_2$: Jeliko¾ $\sigma_2[i:{}] = \sigma[3i+2:{}] = \sigma[3i+2]\sigma[3i+3:\nobreak{}\nobreak] = \sigma[3i+2]\sigma_0[i+1:{}]$
-a v¹echna $\sigma_0[i:{}]$ u¾ máme setøídìná, mù¾eme v¹echna $\sigma_2[i:{}]$ setøídit dvìma prùchody pøihrádkového tøídìní.
-
-\:Dopoèítáme $L_2$: Stejným trikem jako $A_2$ -- pokud jsou první písmena rùzná, je spoleèný prefix prázdný, jinak
-má délku $1+{\rm LCP}(\sigma_0[i+1:{}],\sigma_0[j+1:{}]) = 1+\min_{i+1\le k< j+1} L_0[k]$. Minimum zvládneme pro ka¾dou
-dvojici $i,j$ spoèítat v~konstantním èase pomocí datové struktury pro intervalová minima.
-
-\:$A_0,A_1,A_2\buildrel merge\over\longrightarrow A$ -- sléváme tøi setøídìné posloupnosti,
-tak¾e staèí umìt prvky libovolných dvou posloupností v~konstantním èase porovnat:
-$$\eqalign{
-\sigma_0[i:{}] < \sigma_1[j:{}] &~\hbox{podle zapamatovaných pozic v~$A_{01}$} \cr
-\sigma_0[i:{}] < \sigma_2[k:{}] &\equiv \sigma[3i]\,\sigma_1[i:{}] < \sigma[3k+2]\,\sigma_0[k+1:{}]\cr
-&\Leftrightarrow (\sigma[3i] < \sigma[3k+2]) \vee {} \cr&\hphantom{{}\Leftrightarrow{}} (\sigma[3i] = \sigma[3k+2] \wedge \sigma_1[i:{}] < \sigma_0[k+1:{}])\cr
-\sigma_1[j:{}]<\sigma_2[k:{}] &\equiv \sigma[3j+1]\,\sigma[3j+2]\,\sigma_0[j+1:{}] < \cr&\hphantom{{}\equiv{}} \sigma[3k+2]\,\sigma[3k+3]\,\sigma_1[k+1:{}]
-}$$
-
-\:Dopoèítáme $L$ -- pokud sousedí suffix ze~$\sigma_{0,1}$ se suffixem ze~$\sigma_{0,1}$,
-vyèteme výsledek pøímo z~$L_{01}$. Pokud sousedí $\sigma_2$ se $\sigma_2$, staèí pou¾ít
-u¾ spoèítané $L_2$. Pokud sousedí $\sigma_{0,1}$ se $\sigma_2$, odebereme první jeden
-nebo dva znaky, ty porovnáme samostatnì a v~pøípadì shody zbude suffix ze~$\sigma_0$
-a suffix ze~$\sigma_1$ (stejnì jako pøi slévání) a pro ty doká¾eme $L$ dopoèítat
-pomocí struktury pro intervalová minima v~$L_{01}$.
-
-\endalgo
-
-\s{Analýza èasové slo¾itosti:} Tøídìní v~prvním volání trvá ${\rm Sort}(n,\Sigma)$, ve~v¹ech
-ostatních voláních je lineární (trojice èísel velikosti $\O(n)$ mù¾eme tøídit tøíprùchodovým
-pøihrádkovým tøídìním s~$\O(n)$ pøihrádkami). Z~toho dostáváme:
-$$T(n) = T(2/3\cdot n) + \O(n),~\hbox{a tedy}~T(n)=\O(n).$$
-\qed
-
-\h{Ukkonenova inkrementální konstrukce}
-
-\>Ukkonenùv algoritmus \cite{ukkonen95line} konstruuje suffixový strom bez dolarù inkrementálnì: zaène se stromem
-pro prázdné slovo (ten má jediný vrchol, a to koøen) a postupnì pøidává dal¹í znaky na~konec
-slova. To zvládne v~èase $\O(1)$ amortizovanì na~pøidání jednoho znaku.
-Pro slovo~$\sigma$ tedy doká¾e sestrojit ST v~èase $\O(\vert\sigma\vert)$.
-
-Budeme pøedpokládat, ¾e hrany vedoucí z~jednoho vrcholu je mo¾né indexovat jejich
-prvními písmeny -- to bezpeènì platí, pokud je abeceda pevná; není-li, mù¾eme
-si pomoci hashováním.
-
-\s{Pozorování:} Kdy¾ slovo~$\sigma$ roz¹íøíme na~$\sigma a$, ST se zmìní následovnì:
-
-\numlist\ndotted
-\:Pokud $\beta$ byl nevnoøený suffix slova~$\sigma$, je i $\beta a$ nevnoøený suffix~$\sigma a$. Z~toho víme, ¾e listy
-zùstanou listy, pouze jim potøebujeme prodlou¾it nálepky. Pomù¾eme si snadno: zavedeme
-{\I otevøené hrany,} jejich¾ nálepka je \uv{od~pozice~$i$ do konce}. Listy se tak
-o~sebe postarají samy.
-\:Pokud $\beta$ bylo vìtvící slovo, zùstane nadále vìtvící -- tedy vnitøní vrcholy ve~stromu zùstanou.
-\:Pokud $\beta$ byl vnoøený suffix (tj. vnitøní èi skrytý vrchol), pak se $\beta a$ buïto
-vyskytuje v~$\sigma$, a tím pádem je to vnoøený suffix nového slova a strom není nutné
-upravovat, nebo se v~$\sigma$ nevyskytuje a tehdy pro nìj musíme zalo¾it novou odboèku
-a nový list s~otevøenou hranou.
-\endlist
-
-Víme tedy, co v¹echno musí algoritmus ve~stromu pøí roz¹íøení slova upravit, zbývá
-vyøe¹it, jak to udìlat efektivnì. K~tomu se hodí pár definic a lemmat:
-
-\s{Definice:} {\I Aktivní suffix} $\alpha(\sigma)$ øíkáme nejdel¹ímu vnoøenému suffixu slova~$\sigma$.
-
-\s{Lemma:} Suffix $\beta$ slova $\sigma$ je vnoøený $\Leftrightarrow$ $\vert\beta\vert \le \vert\alpha(\sigma)\vert.$
-
-\proof Ka¾dý suffix vnoøeného suffixu je opìt vnoøený. \qed
-
-\s{Lemma:} Pro ka¾dé $\sigma$, $a$ platí: $\alpha(\sigma a)$ je suffixem $\alpha(\sigma)a.$
-
-\proof $\alpha(\sigma a)$ i $\alpha(\sigma)a$ jsou suffixy slova $\sigma a$, a~proto staèí porovnat jejich délky.
-Slovo $\beta := \hbox{\uv{$\alpha(\sigma a)$ bez koncového~$a$}}$ je vnoøeným suffixem v~$\sigma$, tak¾e
-$\vert\beta\vert \le \vert\alpha(\sigma)\vert$, a~tedy také $\vert\alpha(\sigma a)\vert = \vert\beta a\vert \le \vert\alpha(\sigma)a\vert$.
-\qed
-
-\s{Definice:} Suffix $\beta a$ je {\I zralý} $\equiv$ $\beta$ je vnoøený suffix~$\sigma$, ale $\beta a$ není podslovem~$\sigma$
-(tedy musíme pro~nìj pøi pøidávání znaku~$a$ k~aktuálnímu slovu~$\sigma$ zakládat nový vrchol).
-
-\s{Lemma:} Suffix $\beta$ je zralý $\Leftrightarrow$ $\vert\alpha(\sigma)a\vert \ge \vert\beta a\vert > \vert\alpha(\sigma a)\vert$.
-
-\proof Jeliko¾ $\beta$ je vnoøeným suffixem $\sigma$, musí platit první nerovnost. Aby byl zralý,
-musí také nebýt vnoøeným suffixem $\sigma a$, a~tomu odpovídá druhá nerovnost.
-\qed
-
-\s{Idea algoritmu:} Udr¾ujeme si $\alpha=\alpha(\sigma)$ a pøi pøidání znaku $a$ zkontrolujeme, zda $\alpha a$ je
-stále vnoøený suffix. Pokud ano, nic se nemìní, pokud ne, pøidáme vnitøní vrchol, $\alpha$ zkrátíme
-zleva o~znak a testujeme dál.
-
-\s{Analýza:} Úprav stromu provedeme $\O(1)$ amortizovanì (ka¾dá úprava slovo $\alpha$ zkrátí,
-ka¾dé pøidání znaku ho~prodlou¾í o~znak, tak¾e v¹ech zkrácení je $\O(\vert\sigma\vert)$). Staèí
-tedy ukázat, jak provést úpravu v~(amortizovanì) konstantním èase, k~èemu¾ potøebujeme
-$\alpha$ reprezentovat ¹ikovnì a také si udr¾ovat pomocné informace (zpìtné hrany),
-abychom umìli rychle zkracovat.
-
-\s{Definice:} {\I Referenèní pár} je dvojice $(\pi,\tau)$, v~ní¾ $\pi$ je vrchol
-stromu a $\tau$ libovolné slovo. Tento pár popisuje slovo $\pi\tau$. Referenèní
-pár je {\I kanonický,} pokud neexistuje hrana vedoucí z~vrcholu $\pi$ s~nálepkou,
-která by byla prefixem slova~$\tau$.
-
-\s{Pozorování:} Ke~ka¾dému slovu existuje právì jeden kanonický referenèní pár,
-který ho popisuje. V¹imnìte si, ¾e je to ze~v¹ech referenèních párù pro toto slovo
-ten s~nejdel¹ím~$\pi$ (nejhlub¹ím vrcholem).
-
-\s{Definice:} Zpìtná hrana $\<back>[\pi]$ vede z~vrcholu $\pi$ do~vrcholu,
-který je ze~v¹ech vrcholù nejdel¹ím vlastním suffixem slova~$\pi$.
-
-\s{Pozorování:} Zpìtné hrany jsme sice zavedli stejnì obecnì, jako se to dìlá
-pøi konstrukci vyhledávacích automatù podle Aha a McCorasickové, ale v~na¹em
-pøípadì se \<back> pro vnitøní vrcholy chová daleko jednodu¹eji (a~na ¾ádné
-jiné ho potøebovat nebudeme): pokud je $\pi$ vnitøní vrchol, musí to být
-vìtvící podslovo, a~tím pádem ka¾dé jeho zkrácení zleva musí být také vìtvící
-podslovo. Tedy $\<back>(\pi)$ dá~$\pi$ bez prvního znaku, a~to se nám
-bude hodit pøi zkracování suffixù.
-
-\s{Algoritmus podrobnìji:} (Doplnili jsme detaily do~pøedchozího algoritmu.)
-
-\algo
-\:Vstup: $\alpha=\alpha(\sigma)$ reprezentovaný jako kanonický referenèní pár $(\pi,\tau)$, $T$ suffixový strom pro~$\sigma$ a jeho funkce \<back>, nový znak~$a$.
-\:Zjistíme, jestli $\alpha a$ je pøítomen ve~stromu, a pøípadnì ho zalo¾íme:
-\::Pokud $\tau=\varepsilon$: ($\alpha=\pi$ je vnitøní vrchol)
-\:::Vede-li z~vrcholu $\pi$ hrana s~nálepkou zaèínající znakem $a$, pak je pøítomen.
-\:::Nevede-li, není pøítomen, a~tak pøidáme novou otevøenou hranu vedoucí z~$\pi$ do~nového listu.
-\::Pokud $\tau\ne\varepsilon$: ($\alpha$ je skrytý vrchol)
-\:::Najdeme hranu, po~ní¾ z~$\pi$ pokraèuje slovo $\tau$ (která to je, poznáme podle prvního znaku slova~$\tau$).
-\:::Pokud v~popisce této hrany po~$\tau$ následuje znak~$a$, pak je $\alpha a$ pøítomen.
-\:::Pokud nenásleduje, tak nebyl pøítomen, èili tuto hranu rozdìlíme: pøidáme na~ni nový vnitøní vrchol,
- do~nìj¾ povede hrana s~popiskou~$\tau$ a z~nìj zbytek pùvodní hrany a otevøená hrana do~nového listu.
-\:Pokud $\alpha a$ byl pøítomen, tak $\alpha$ zkrátíme a test opakujeme:
-\::Je-li $\pi\ne\varepsilon$, nastavíme $\pi := \<back>(\pi)$. V~opaèném pøípadì (jsme v~koøeni) zkrátíme $\tau$ o~znak zleva.
-\::Pár $(\pi,\tau)$ u¾ popisuje zkrácené slovo, ale nemusí být kanonický, tak¾e to je¹tì napravíme:
-\:::Dokud existuje hrana vedoucí z~$\pi$, její¾ popiska je prefixem slova $\tau$, tak se
- po~této hranì posuneme, èili prodlou¾íme $\pi$ o~tuto popisku a zkrátíme o~ni~$\tau$.
-\::Zpìt na~krok 2.
-\:Pokud $\alpha a$ u¾ je pøítomen, zbývá pøidat $a$ k~$\alpha$ a zastavit se:
-\::$\tau := \tau a$.
-\::Kanonikalizace stejnì jako v~bodech 12--13.\foot{Dokonce jednodu¹¹í, proto¾e projdeme nejvý¹e jednu hranu.}
-\:Dopoèítáme zpìtné hrany (viz ní¾e).
-\:Výstup: $\alpha=\alpha(\sigma a)$ coby kanonický referenèní pár $(\pi,\tau)$, $T$ suffixový strom pro~$\sigma a$
- a jeho funkce \<back>.
-\endalgo
-
-\s{Èasová slo¾itost:}
-
-Kanonikalizace pracuje v~amortizovanì konstantním èase, proto¾e ka¾dá její iterace
-zkrátí~$\tau$ a za~ka¾dé spu¹tìní algoritmu se~$\tau$ prodlou¾í jen jednou, a~to o~jeden znak.
-
-Prùchodù hlavním cyklem je, jak u¾ víme, amortizovanì konstantní poèet a ka¾dý prùchod
-zvládneme v~konstantním èase.
-
-Zbývá dodat, jak nastavovat novým vrcholùm jejich \<back>. To potøebujeme
-jen pro vnitøní vrcholy (na~zpìtné hrany z~listù se algoritmus nikdy neodkazuje)
-a v¹imneme si, ¾e pokud jsme zalo¾ili vrchol, odpovídá tento vrchol v¾dy souèasnému~$\alpha$
-a zpìtná hrana z~nìj povede do~zkrácení slova~$\alpha$ o~znak zleva, co¾ je
-pøesnì vrchol, který zalo¾íme (nebo zjistíme, ¾e u¾ existuje) v~pøí¹tí iteraci
-hlavního cyklu. V~dal¹í iteraci urèitì je¹tì nebudeme tuto hranu potøebovat,
-proto¾e $\pi$ v¾dy jen zkracujeme, a~tak mù¾eme vznik zpìtné hrany o~iteraci
-zpozdit a zvládnout to tak také v~èase $\O(1)$.
-
-Celkovì je tedy èasová slo¾itost inkrementálního udr¾ování suffixového
-stromu amortizovanì konstantní.
-\qed
-
-\references
-\bye
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+++ /dev/null
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-include ../Makerules
+++ /dev/null
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-0 0 0 setrgbcolor
-newpath
-0 0 moveto
-( B) show
-grestore
-grestore
-gsave [1 0 0 -1 41.721359 101.02974] concat
-gsave
-/BitstreamVeraSans-Roman-ISOLatin1 findfont
-12 scalefont
-setfont
-0 0 0 setrgbcolor
-newpath
-0 0 moveto
-( U) show
-grestore
-grestore
-gsave [1 0 0 -1 41.721359 116.02974] concat
-gsave
-/BitstreamVeraSans-Roman-ISOLatin1 findfont
-12 scalefont
-setfont
-0 0 0 setrgbcolor
-newpath
-0 0 moveto
-( S) show
-grestore
-grestore
-grestore
-showpage
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