From: Martin Mares Date: Tue, 18 Oct 2011 18:07:15 +0000 (+0200) Subject: Vyhledavani v textu: Oziveno X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=386ab660024df019beb542d34789dae5db7edbf4;p=ads2.git Vyhledavani v textu: Oziveno --- diff --git a/1-kmp/1-kmp.tex b/1-kmp/1-kmp.tex new file mode 100644 index 0000000..041c06a --- /dev/null +++ b/1-kmp/1-kmp.tex @@ -0,0 +1,278 @@ +\input lecnotes.tex + +\prednaska{6}{Vyhledávání v~textu}{(zapsal: Petr Jankovský)} + +Nyní se budeme vìnovat následujícímu problému: v~textu délky $S$ (senì) budeme chtít najít v¹echny výskyty hledaného slova délky $J$ (jehly). Nejprve se podívejme na~jeden primitivní algoritmus, který nefunguje. Je ale zajímavé rozmyslet si, proè. + +\h{Hloupý algoritmus} +Zaèneme prvním písmenkem hledaného slova a~budeme postupnì procházet text, a¾ najdeme první výskyt poèáteèního písmenka. Poté budeme testovat, zda souhlasí i~písmenka dal¹í. Pokud nastane neshoda, v~hledaném slovì se vrátíme na~zaèátek a~v~textu pokraèujeme znakem, ve~kterém neshoda nastala. Podívejme se na~pøíklad. + +\s{Pøíklad:} Budeme hledat slovo |jehla| v~textu |jevkupcejejehla|. Vezmeme si tedy první písmenko |j| v~hledaném slovì a~zjistíme, ¾e v~textu se nachází hned na~zaèátku. Vezmeme tedy dal¹í písmenko |e|, které se vyskytuje jako druhé i~v~textu. Pøi tøetím písmenku ale narazíme na~neshodu. V~tuto chvíli tedy zresetujeme a~opìt hledáme výskyt písmenka |j|, tentokrát v¹ak a¾ od~tøetího písmene v~textu. Takto postupujeme postupnì dál, a¾ narazíme na~dal¹í |je|, které ov¹em není následováno písmenem~|h|, tudí¾ opìt zresetujeme a~nakonec najdeme shodu s~celým hledaným øetìzcem. V~tomto pøípadì tedy algoritmus na¹el hledané slovo. + +Tento algoritmus v¹ak zjevnì mù¾e hanebnì selhat. Mù¾e se stát, ¾e zaèneme porovnávat, a¾ v~jednu chvíli narazíme na~neshodu. Celý tento kus tedy pøeskoèíme. Pøi tom se ale v~tomto kusu textu mohl vyskytovat nìjaký pøekrývající se výskyt hledané \uv{jehly}. Hledejme napøíklad øetìzec |kokos| v~textu |clanekokokosu|. Algoritmus tedy zaène porovnávat. Ve~chvíli kdy najde prefix |koko| a~na~vstupu dostane~|k|, dochází k~neshodì. Proto algoritmus zresetuje a~pokraèuje v~hledání od~tohoto znaku. Najde sice je¹tì výskyt |ko|, ov¹em s~dal¹ím písmenkem |s| ji¾ dochází k~neshodì a~algotimus sel¾e. Nesprávnì se toti¾ \uv{upnul} na~první nalezené |koko| a~s~dal¹ím~|k| pak \uv{zahodil} i~správný zaèátek. + +Máme tedy algoritmus, který i~kdy¾ je ¹patnì, tak funguje urèitì kdykoli se první písmenko hledaného slova v~tomto slovì u¾ nikde jinde nevyskytuje -- co¾ |jehla| splòovala, ale |kokos| u¾ ne. + +{\I Hloupý algoritmus} se na~ka¾dé písmenko textu podívá jednou, tudí¾ èasová slo¾itost bude lineární s~délkou textu ve~kterém hledáme -- tedy $\O(S)$. + +\h{Pomalý algoritmus} +Zkusíme algoritmus vylep¹it tak, aby fungoval správnì: pokud nastane nìjaká neshoda, vrátíme se zpátky tìsnì za~zaèátek toho, kdy se nám to zaèalo shodovat. To je ov¹em vlastnì skoro toté¾, jako brát postupnì v¹echny mo¾né zaèátky v~\uv{senì} a~pro ka¾dý z~nìj ovìøit, jestli se tam \uv{jehla} nachází èi nikoliv. + +Tento algoritmus evidentnì funguje. Bì¾í v¹ak v~èase: $S$ mo¾ných zaèátkù, krát èas potøebný na~jedno porovnání (zda se na~dané pozici nenachází \uv{jehla}), co¾ nám mù¾e trvat a¾ $J$. Proto je èasová slo¾itost $\O(SJ)$. V praxi bude algoritmus èasto rychlej¹í, proto¾e typicky velmi brzo zjistíme, ¾e se øetìzce neshodují, ale je mo¾né vymyslet vstup, kde bude potøeba porovnání opravdu tolik. + +Nyní se pokusme najít takový algoritmus, který by byl tak rychlý, jako {\I Hloupý algoritmus}, ale chytrý, jako ten {\I Pomalý}. + +\h{Chytrý algoritmus} +Ne¾ vlastní algoritmus vybudujeme, zkusíme se cestou nauèit pøemý¹let o~øetìzcích obèas trochu pøekrouceným zpùsobem. Podívejme se na~je¹tì jeden pøíklad. + +\s{Pøíklad:} Vezmìme si napøíklad staré italské pøízvisko |barbarossa|, které znamená Rudovous. Pøedstavme si, ¾e takovéto slovo hledáme v~nìjakém textu, který zaèíná |barbar|. Víme, ¾e a¾ sem se nám hledaný øetìzec shodoval. Øeknìme, ¾e dal¹í písmenko textu se shodovat pøestane -- místo |o| naèteme napøíklad opìt |b|. {\I Hloupý algoritmus} by velil vrátit se k~|a| a~od~nìj hledat dál. Uvìdomme si ale, ¾e kdy¾ se vracíme z~|barbar| do~|arbar| (tedy øetìzce, který ji¾ známe), mù¾eme si pøedpoèítat, jak dopadne hledání, kdy¾ ho pustíme na~nìj. V~pøedpoèítaném bychom tedy chtìli ukládat, ¾e kdy¾ máme øetìzec |arbar|, tak |ar| a~|r| nám do~hledaného nepasuje a~a¾~|bar| se bude shodovat. Tedy místo toho, abychom spustili nové hledání od~|a|, mù¾eme ho spustit a¾~od~|b|. Co víc, my dokonce víme, jak dopadne to -- pokud toti¾ nastane neshoda po~pøeètení |barbar|, je to stejné, jako kdybychom pøeèetli pouze |bar|, na~které se (pùvodne neshodující se) |b| u¾ navázat dá. Kdyby se nedalo navázat ani tam, tak bychom opìt zkracovali... Nejen, ¾e tedy víme, kam se máme vrátit, ale víme dokonce i~to, co tam najdeme. + +My¹lenka, ke které míøíme, je pøedpoèítat si nìjakou tabulku, která nám bude øíkat, jak se máme pøi hledání vracet a~jak to dopadne, a~pak u¾ jenom prohlédávat s~pou¾itím této tabulky. + +Aby se nám o~tìchto algoritmech lépe mluvilo a~pøedev¹ím psalo, pojïme si povìdìt nìkolik definic. + +\s{Definice:} +\itemize\ibull +\:{\I Abeceda $\Sigma$} je koneèná mno¾ina znakù\foot{Mù¾eme pøi tom jít a¾~do~extrémù. Pøíkladem extrémních abeced je binární abeceda slo¾ená pouze z~nul a~jednièek. Pøíklad z~druhého konce (který rádi dìlají lingvisti) je abeceda, která má jako abecedu v¹echna èeská slova. V¹echny èeské vìty, pak nejsou nic jiného, ne¾ slova nad touto abecedou. Pou¾itá abeceda tedy mù¾e být i~relativnì obrovská. Dal¹ím takovým pøíkladem mù¾e být Unicode. Pro na¹e potøeby ale zatím budeme pøedpokládat, ¾e abeceda je nejen konstantnì velká, ale i~rozumnì malá. Budeme si moci tedy dovolit napøíklad indexovat pole znakem abecedy (kdybychom nemohli, tak bychom místo pole pou¾ili napøíklad hashovací tabulku, èi nìco podobného\dots).}, ze~kterých tvoøíme text, øetìzce, slova. + + +\:{\I $\Sigma^*$} je mno¾ina v¹ech slov nad abecedou $\Sigma$. Èili mno¾ina v¹ech neprázdných koneèných posloupností znakù ze $\Sigma$. +\endlist +\s{Znaèení:} +Aby se nám nepletlo znaèení, budeme rozli¹ovat promìnné pro slova, promìnné pro písmenka a~promìnné pro èísla. + +\itemize\ibull +\:{\I Slova} budeme znaèit malými písmenky øecké abecedy $\alpha$, $\beta$, ... . +\:$\iota$ bude oznaèovat \uv{jehlu} +\:$\sigma$ bude oznaèovat \uv{seno} +\:{\I Znaky} oznaèíme malými písmeny latinky $a$, $b$, \dots . +\:{\I Èísla} budeme znaèit velkými písmeny $A$, $B$, \dots . +\:{\I Délka slova} $\vert \alpha \vert$ pro $\alpha \in \Sigma^*$ je poèet jeho znakù. +\:{\I Prázdné slovo} znaèíme písmenem $\varepsilon$, $\vert \varepsilon \vert = 0$. +\:{\I Zøetìzení} $\alpha\beta$ vznikne zapsáním slov $\alpha$ a~$\beta$ za sebe. Platí $\vert \alpha\beta \vert=\vert \alpha \vert+\vert \beta \vert$, $\alpha\varepsilon=\varepsilon\alpha=\alpha$. +\:$\alpha[k]$ je $k$-tý znak slova $\alpha$, indexujeme od~$0$. +\:$\alpha[k:l]$ je podslovo zaèínající $k$-tým znakem a~$l$-tý znak je první, který v~nìm není. Jedná se tedy o~podslovo skládající se z~$\alpha[k]$,$\alpha[k+1]$,\dots,$\alpha[l-1]$. Platí tedy: $\alpha[k:k]=\varepsilon$, $\alpha[k:k+1]=\alpha[k]$. Jednu (èi obì) meze mù¾eme i~vynechat, tento zápis pak bude znamenat buï \uv{od zaèátku slova a¾ nìkam}, nebo \uv{odnìkud a¾ do~konce}: +%TODO - zaøadit pod pøedchozí bod +\:$\alpha[:k]$ je {\I prefix} obsahující prvních $k$ znakù slova $\alpha$ ($\alpha[0]$,\dots,$\alpha[k-1]$). +\:$\alpha[k:]$ je {\I suffix} obsahující znaky slova $\alpha$ poèínaje $k$-tým znakem a¾ do~konce. +\:$\alpha[:] = \alpha$ +\endlist + + +V¹imnìme si, ¾e prázdné slovo je prefixem, suffixem i~podslovem jakéhokoliv slova vèetnì sebe sama. +Ka¾dé slovo je také prefixem, suffixem i~podslovem sebe sama. To se ne v¾dy hodí. Nìkdy budeme chtít øíct, ¾e nìjaké slovo je {\I vlastním} prefixem nebo suffixem. To bude znamenat, ¾e to nebude celé slovo. + +\> $\alpha$ je {\I vlastní prefix} slova $\beta \equiv \alpha$ je prefix $\beta~\&~\alpha \neq \beta$. + +\h{Vyhledávací automat (Knuth, Morris, Pratt)} +{\I Vyhledávací automat} bude graf, jeho¾ vrcholùm budeme øíkat {\I stavy}. Jejich jména budou prefixy hledaného slova a~hrany budou odpovídat tomu, jak jeden prefix mù¾eme získat z~pøedchozího prefixu pøidáním jednoho písmene. Poèáteèní stav je prázdné slovo $\varepsilon$ a~koncový je celá $\iota$. Dopøedné hrany grafu budou popisovat pøechod mezi stavy ve~smyslu zvìt¹ení délky jména stavu (dopøedná funkce $h(\alpha)$, urèující znak na~dopøedné hranì z~$\alpha$). Zpìtné hrany grafu budou popisovat pøechod (zpìtná funkce $z(\alpha)$) mezi stavem $\alpha$ a~nejdel¹ím vlastním suffixem $\alpha$, který je prefixem $\iota$, kdy¾ nastane neshoda. + +\figure{barb.eps}{Vyhledávací automat.}{4.1in} + +\s{Hledej($\sigma$):} +\algo +\:$\alpha \leftarrow \varepsilon$. +\:Pro $x\in\sigma$ postupnì: +\:$\indent$Dokud $h(\alpha) \neq x~\&~\alpha \neq \varepsilon : \alpha \leftarrow z(\alpha)$. +\:$\indent$Pokud $h(\alpha) = x: \alpha \leftarrow \alpha x$. +\:$\indent$Pokud $\alpha = \iota$, ohlásíme výskyt. +\endalgo + +\>Vstupem je $\iota$, hledané slovo (jehla) délky $J=\vert \iota \vert$ a~$\sigma$, text (seno) délky $S=\vert \sigma \vert$. +\>Výstupem jsou v¹echny výskyty hledaného slova $\iota$ v~textu $\sigma$, tedy mno¾ina $\left\{ k \mid \sigma[k:k+J]=\iota \right\}$ + +Pojïme nyní dokázat, ¾e tento algoritmus správnì ohlásí v¹echny výskyty. + +\s{Definice}: $\alpha(\tau) := $ stav automatu po~pøeètení $\tau$ + +\s{Invariant:} Pokud algoritmus pøeète nìjaký vstup, nachází se ve~stavu, který je nejdel¹ím suffixem pøeèteného vstupu, který je nìjakým stavem. +$\alpha(\tau) =$ nejdel¹í stav (nejdel¹í prefix jehly), který je suffixem $\tau$ (pøeèteného vstupu). + +Pojïme si rozmyslet, ¾e z~tohoto invariantu ihned plyne, ¾e algoritmus najde to, co má. Kdykoli toti¾ ohlásí nìjaký výskyt, tak tam tento výskyt opravdu je. Kdykoli pak má nìjaký výskyt ohlásit, tak se v~této situaci jako suffix toho právì pøeèteného textu vyskytuje hledané slovo, pøièem¾ hledané slovo je urèitì stav a~zároveò nejdel¹í ze v¹ech existujících stavù. Tak¾e invariant opravdu øíká, ¾e jsme právì v~koncovém stavu a~algoritmus nám tedy ohlásí výskyt. + +\proof {\I (invariantu)} +Indukcí podle kroku algoritmu. Na~zaèátku pro prázdný naètený vstup invariant triviálnì platí, tedy prázdný suffix $\tau$ je prefixem $\iota$. V~kroku $n$ máme naètený vstup $\tau$ a~k~nìmu pøipojíme znak $x$. Invariant nám øíká, ¾e nejdel¹í stav, který je suffixem, je nejdel¹í suffix, který je stavem. Nyní se ptáme, jaký je nejdel¹í stav, který se dá \uv{napasovat} na~konec øetìzce $\tau x$. Kdykoli v¹ak takovýto suffix máme, tak z~nìj mù¾eme $x$ na~konci odebrat, èím¾ dostaneme suffix slova $\tau$. + +\>Tedy: pokud $\beta$ je neprázdným suffixem slova $\tau x$, pak $\beta = \gamma x$, kde $\gamma$ je suffix $\tau$. + +Suffix, který máme sestrojit, tedy vznikne z~nìjakého suffixu slova $\tau$ pøipsáním~$x$. Chceme najít nejdel¹í suffix slova $\tau x$, který je stavem, tak¾e chceme najít i~nejdel¹í suffix pùvodního slova $\tau$, za který se dá pøidat $x$ tak, aby vy¹lo jméno stavu. Staèí tedy u¾ jen \uv{probírat} suffixy slova $\tau$ od~nejdel¹ího po~nejkrat¹í, zkou¹et k~nim pøidávat $x$ a~a¾ to pùjde, tak jsme na¹li nejdel¹í suffix $\tau x$. Pøesnì toto ov¹em algoritmus dìlá, nebo» zpìtná funkce mu v¾dy øekne nejbli¾¹í krat¹í suffix, který je stavem. Pokud pak nemù¾eme $x$ pøidat ani do~$\varepsilon$, pak je øe¹ením prázdný suffix. Algoritmus tedy funguje. \qed + +Nyní pojïmì zkoumat to, jak je ve~skuteènosti ná¹ algoritmus rychlý. K tomu bychom si ale nejdøív mìli øíct, jak pøesnì budeme automat reprezentovat. V~algoritmu vystupují nìjaká porovnávání stavù, pøièem¾ není úplnì jasné, jak zaøídit, aby v¹e trvalo konstantnì dlouho. Vyjde nám to ale docela snadno. K reprezentaci automatu nám toti¾ budou staèit pouze dvì pole. + +\s{Reprezentace automatu:} +Oèíslujeme si stavy délkami pøíslu¹ných prefixù, tedy èísly $0 \dots J$. Poté je¹tì potøebujeme nìjakým zpùsobem zakódovat dopøedné a~zpìtné hrany. Vzhledem k~tomu, ¾e z~ka¾dého vrcholu vede v¾dy nejvý¹e jedna dopøedná a~nejvý¹e jedna zpìtná, tak nám evidentnì staèí pamatovat si pro ka¾dý typ hran pouze jedno èíslo na~vrchol. Budeme mít tedy nìjaké pole dopøedných hran, které nám pro ka¾dý stav øekne, jakým písmenkem je nadepsaná dopøedná hrana ze stavu $I$ do~$I+1$. To jsou ale pøesnì písmenka jehly, tak¾e si staèí pamatovat jehlu samotnou. Èili z~$I$ do~$I+1$ vede hrana nadepsaná $\iota [I]$. Pro zpìtné hrany pak budeme potøebovat pole $Z$, které nám pro stav $I$ øekne èíslo stavu, do~kterého vede zpìtná hrana. Tedy $Z[I]$ je cíl zpìtné hrany ze stavu $I$. +S~touto reprezentací ji¾ doká¾eme na¹i hledací proceduru pøímoèaøe pøepsat tak, aby sahala pouze do~tìchto dvou polí: +\algo +\:$I \leftarrow 0$. +\:Pro znaky $x$ z~textu: +\:$\indent$Dokud $\iota[I] \neq x~\&~I \neq 0: I \leftarrow Z[I]$. +\:$\indent$Pokud $\iota[I] = x$, pak $I \leftarrow I + 1$. +\:$\indent$Pokud $I = J$, ohlásíme výskyt. +\endalgo + +Zatím se v~algoritmu je¹tì skrývá drobná chyba -- toti¾ algoritmus se obèas zeptá na~dopøednou hranu z~posledního stavu. Pokud jsme právì ohlásili výskyt (jsme tedy v~posledním stavu) a~pøijde nìjaký dal¹í znak, algoritmus se ptá, zda je roven tomu, co je na~dopøedné hranì z~posledního stavu. Ta ale ov¹em neexistuje. Jednodu¹e to ale napravíme tak, ¾e si pøidáme fiktivní hranu, na~které se vyskytuje nìjaké \uv{nepísmenko} -- nìco, co se nerovná ¾ádnému jinému písmenku. Zajistíme tak, ¾e se po~této hranì nikdy nevydáme. Dodefinujeme tedy $\iota[J]$ odli¹nì od~v¹ech znakù.\foot{V jazyce C se toto dodefinování provede vlastnì zadarmo, nebo» ka¾dý øetìzec je v~nìm ukonèen znakem s~kódem nula, který se ve~vstupu nevyskytne\dots Algoritmus bude tedy fungovat i~bez tohoto dodefinování. V jiných jazycích je ale tøeba na~nìj nezapomenout!} + +\s{Lemma:} Funkce {\I Hledej} bì¾í v~èase $\O(S)$. + +\proof +Funkce {\I Hledej} chodí po~dopøedných a~zpìtných hranách. Dopøedných hran projdeme urèitì maximálnì tolik, kolik je délka sena. Pro ka¾dý znak pøeètený ze sena toti¾ jdeme nejvý¹e jednou po~dopøedné hranì. Se zpìtnými hranami se to má tak, ¾e na~jeden pøeètený znak z~textu se mù¾eme po~zpìtné hranì vracet maximálnì $J$-krát. Z~tohoto by nám v¹ak vy¹la slo¾itost $\O(JS)$, èím¾ bychom si nepomohli. Zachrání nás ale pøímoèarý potenciál. Uvìdomme si, ¾e chùze po~dopøedné hranì zvý¹í $I$ o~jedna a~chùze po~zpìtné hranì $I$ sní¾í alespoò o~jedna. Vzhledem k~tomu, ¾e $I$ není nikdy záporné a~na~zaèátku je nulové, zjistíme, ¾e krokù zpìt mù¾e být maximálnì tolik, kolik krokù dopøedu. Èasová slo¾itost hledání je tedy lineární vzhledem k~délce sena. \qed + +Nyní nám zbývá na~první pohled malièkost -- toti¾ zkonstruovat automat. Zkonstruovat dopøedné hrany zvládneme zjevnì snadno, jsou toti¾ explicitnì popsané hledaným slovem. Tì¾¹í u¾ to bude pro hrany zpìtné. Vyu¾ijeme k~tomu následující pozorování: + +\s{Pozorování:} +Pøedstavme si, ¾e automat u¾ máme hotový a~tím, ¾e budeme sledovat jeho chování, chceme zjistit, jak v~nìm vedou zpìtné hrany. +Vezmìme si nìjaký stav~$\beta$. To, kam z~nìj vede zpìtná hrana zjistíme tak, ¾e spustíme automat na~øetìzec $\beta$~bez prvního písmenka a~stav, ve~kterém se automat zastaví, je pøesnì ten, kam má vést i~zpìtná hrana z~$\beta$. Jinými slovy víme, ¾e $z(\beta) = \alpha (\beta[1:])$. +Proè takováto vìc funguje? V¹imìme si, ¾e definice $z$ a~to, co nám o~$\alpha$ øíká invariant, je témìø toto¾né -- $z(\beta)$ je nejdel¹í vlastní suffix $\beta$, který je stavem, $\alpha(\beta)$ je nejdel¹í suffix $\beta$, který je stavem. Jediná odli¹nost je v~tom, ¾e definice $z$ narozdíl od~definice $\alpha$ zakazuje nevlastní suffixy. Jak nyní vylouèit suffix $\beta$, který by byl roven $\beta$ samotné? Zkrátíme $\beta$ o~první znak. Tím pádem v¹echny suffixy $\beta$ bez prvního znaku jsou stejné jako v¹echny vlastní suffixy $\beta$. + +K èemu je toto pozorování dobré? Rozmysleme si, ¾e pomocí nìj u¾ doká¾eme zkonstruovat zpìtné hrany. Není to ale trochu divné, kdy¾ pøi simulování automatu na~øetìzec bez prvního znaku u¾ zpìtné hrany potøebujeme? Není. Za chvíli zjistíme, ¾e takto mù¾eme zji¹»ovat zpìtné hrany postupnì -- a~to tak, ¾e pou¾íváme v¾dy jenom ty, které jsme u¾ sestrojili. + +Takovémuhle pøístupu, kdy pøi konstruování chtìného u¾ pou¾íváme to, co chceme sestrojit, ale pouze ten kousek, který ji¾ máme hotový, se v~angliètinì øíká {\I bootstrapping}\foot{Z~tohoto slova vzniklo i~{\I bootování} poèítaèù, kdy operaèní systém v~podstatì zavádí sám sebe. Bootstrap znamená èesky ¹truple -- tedy oèko na~konci boty, které slou¾í k~usnadnìní nazouvání. A~jak souvisí ¹truple s~algoritmem? To se zase musíme vrátit k~pøíbìhùm o~baronu Prá¹ilovi, mezi nimi¾ je i~ten, ve~kterém baron Prá¹il vypráví o~tom, jak sám sebe vytáhl z~ba¾iny za ¹truple. Stejnì tak i~my budeme algoritmus konstruovat tím, ¾e se budeme sami vytahovat za ¹truple, tedy bootstrappovat.}. +V¹imnìme si, ¾e pøi výpoètu se vstupem $\beta$ projde automat jenom prvních $\vert \beta \vert$ stavù. Automat se evidentnì nemù¾e dostat dál, proto¾e na~ka¾dý krok dopøedu (doprava) spotøebuje písmenko $\beta$. Tak¾e krokù doprava je maximálnì tolik, kolik je $\vert \beta \vert$. Jinými slovy kdybychom ji¾ mìli zkonstruované zpìtné hrany pro prvních $\vert \beta \vert$ stavù (tedy $0 \dots \vert \beta \vert - 1$), tak pøi tomto výpoètu, který potøebujeme na~zkonstruování zpìtné hrany z~$\beta$, je¹tì tuto zpìtnou hranu nemù¾eme potøebovat. Vystaèíme si s~tìmi, které ji¾ máme zkonstruované. + +Nabízí se tedy zaèít zpìtnou hranou z~prvního znaku (která vede evidentnì do~$\varepsilon$), pak postupnì brát dal¹í stavy a~pro ka¾dý z~nich si spoèítat, kdy spustíme automat na~jméno stavu bez prvního znaku a~tím získáme dal¹í zpìtnou hranu. Toto funguje, ale je to kvadratické \dots. Máme toti¾ $J$ stavù a~pro ka¾dý z~nich nám automat bì¾í v~èase a¾ lineárním s~$J$. Jak z~toho ven? + +Z~prvního stavu povede zpìtná funkce do~$\varepsilon$. Pro dal¹í stavy chceme spoèítat zpìtnou funkci. Z~druhého stavu $\iota[0:2]$ tedy automat spustíme na~$\iota[1:2]$, dále pak na~$\iota[1:3]$, $\iota[1:4]$, atd. Ty øetìzce, pro které potøebujeme spo¹tìt automat, abychom dostali zpìtné hrany, jsou tedy ve~skuteènosti takové, ¾e ka¾dý dal¹í dostaneme roz¹íøením pøedchozího o~jeden znak. To jsou ale pøesnì ty stavy, kterými projde automat pøi zpracovávání øetìzce $\iota$ od~prvního znaku dál. Jedním prùchodem automatu nad jehlou bez prvního písmenka se tím pádem rovnou dozvíme v¹echny údaje, které potøebujeme. +Z~pøedchozího pozorování plyne, ¾e nikdy nebudeme potøebovat zpìtnou hranu, kterou jsme je¹tì nezkonstruovali a~jeliko¾ víme, ¾e jedno prohledání trvá lineárnì s~délkou toho, v~èem hledáme, tak toto celé pobì¾í v~lineárním èase. Dostaneme tedy následující algoritmus: + +\s{Konstrukce zpìtné funkce:} +\algo +\:$Z[0] \leftarrow ?$, $Z[1] \leftarrow 0$. +\:$I \leftarrow 0$. +\:Pro $k = 2 \dots J$: +\::$I \leftarrow \( I , \iota [k])$. +\::$Z[k] \leftarrow I$. +\endalgo + +Zaèínáme tím, ¾e nastavíme zpìtnou hranu z~prvních dvou stavù, pøièem¾ $z[0]$ je nedefinované, proto¾e tuto zpìtnou hranu nikdy nepou¾íváme. Dále postupnì simulujeme výpoèet automatu nad slovem bez prvního znaku a~po ka¾dém kroku se dozvíme novou zpìtnou hranu. {\I Krokem} automatu pak není nic jiného ne¾ vnitøek (3. a~4. bod) na¹í hledací procedury. To, kam jsme se dostali, pak zaznamenáme jako zpìtnou funkci z~$k$. +Èili pou¹tíme automat na~jehlu bez prvního písmenka, provedeme v¾dy jeden krok automatu (pøes dal¹í písmenko jehly) a~zapamatujeme si, jakou zpìtnou funkci jsme zrovna dostali. Díky pozorováním navíc víme, ¾e zpìtné hrany konstruujeme správnì, nikdy nepou¾ijeme zpìtnou hranu, kterou jsme je¹tì nesestrojili a~víme i~to, ¾e celou konstrukci zvládneme v~lineárním èase s~délkou jehly. + +\s{Vìta:} Algoritmus KMP najde v¹echny výskyty v~èase $O(J+S)$. + +\proof +Lineární èas s~délkou jehly potøebujeme na~postavení automatu, lineární èas s~délkou sena pak potøebujeme na~samotné vyhledání. + +\h{Rabinùv-Karpùv algoritmus} + +Nyní si uká¾eme je¹tì jeden algoritmus na~hledání jedné jehly, který nebude mít v~nejhor¹ím pøípadì lineární slo¾itost, ale bude ji mít prùmìrnì. Bude daleko jednodu¹¹í a~uká¾e se, ¾e je v~praxi daleko rychlej¹í. Bude to algoritmus zalo¾ený na~hashování. + + +Pøedstavme si, ¾e máme seno délky $S$ a~jehlu délky $J$, a~vezmìme si nìjakou hashovací funkci, které dáme na~vstup $J$-tici znakù (tedy podslova dlouhá jako jehla). Tato hashovací funkce nám je pak zobrazí do~mno¾iny $\{0,\ldots,N-1\}$ pro nìjaké dost velké~$N$. Jak nám toto pomù¾e pøi hledání jehly? Vezmeme si libovolné \uv{okénko} délky $J$ a~ne¾ budeme zji¹»ovat, zda se v~nìm jehla vyskytuje, tak si spoèítáme hashovací funkci a~porovnáme ji s~hashem jehly. Èili ptáme se, jestli je hash ze sena od~nìjaké pozice $I$ do~pozice $I+J$ roven hashi jehly -- formálnì: $h(\sigma [I: I+J ]) = h(\iota)$. Teprve tehdy, kdy¾ zjistíme, ¾e se hodnota hashovací fce shoduje, zaèneme doopravdy porovnávat øetìzce. + +Není to ale nìjaká hloupost? Mù¾e nám vùbec takováto konstrukce pomoci? Není to tak, ¾e na~spoèítání hashovací funkce z~$J$ znakù, potøebujeme tìch $J$ znakù pøeèíst, co¾ je stejnì rychlé, jako rovnou øetìzce porovnávat? Pou¾ijeme trik, který bude spoèívat v~tom, ¾e si zvolíme ¹ikovnou hashovací funkci. Udìláme to tak, abychom ji mohli pøi posunutí \uv {okénka} o~jeden znak doprava v~konstantním èase pøepoèítat. Chceme umìt z~$h(x_1 \dots x_j)$ spoèítat $h(x_2 \dots x_{j+1})$. +Na~zaèátku si tedy spoèítáme hash jehly a~první $J$-tice znakù sena. Pak ji¾ jenom posouváme \uv {okénko} o~jedna, pøepoèítáme hashovací funkci a~kdy¾ se shoduje s~hashem jehly, tak porovnáme. Budeme pøitom vìøit tomu, ¾e pokud se tam jehla nevyskytuje, pak máme hashovací funkci natolik rovnomìrnou, ¾e pravdìpodobnost toho, ¾e se pøesto strefíme do~hashe jehly, je $1/N$. Jinými slovy jenom v~jednom z~øádovì $N$ pøípadù budeme porovnávat fale¹nì -- tedy provedeme porovnání a~vyjde nám, ¾e výsledek je neshoda. V~prùmìrném pøípadì tedy mù¾eme stlaèit slo¾itost a¾ témìø k~lineární. + +Podívejme se teï na~prùmìrnou èasovou slo¾itost. Budeme urèitì potøebovat èas na~projití jehly a~sena. Navíc strávíme nìjaký èas nad fale¹nými porovnáními, kterých bude v~prùmìru na~ka¾dý $N$-tý znak sena jedno porovnání s~jehlou -- tedy $SJ / N$, pøièem¾ $N$ mù¾eme zvolit dost velké na~to, abychom tento èlen dostali pod nìjakou rozumnou konstantu... Nakonec budeme potøebovat jedno porovnání na~ka¾dý opravdový výskyt, èemu¾ se nevyhneme. Pøipoèteme tedy je¹tì $J \cdot$ {\I $\sharp$výskytù}. Dostáváme tedy: $ \O(J+S+SJ/N+J \cdot$ {\I $\sharp$výskytù}). + +Zbývá malièkost -- toti¾ kde vzít hashovací funkci, která toto v¹e splòuje. Jednu si uká¾eme. Bude to vlastnì takový hezký polynom: +$$h(x_1 \dots x_j) := \left(\sum_{I=1}^{J} x_I \cdot p^{J-I}\right) \bmod N.$$ +Jinak zapsáno se tedy jedná o: +$$(x_1 \cdot p^{J-1} + x_2 \cdot p^{J-2} + \dots + x_J \cdot p^0 ) \bmod N.$$ +Po posunutí okénka o~jedna chceme dostat: +$$(x_2 \cdot p^{J-1} + x_3 \cdot p^{J-2} + \dots + x_J \cdot p^1 + x_{J+1} \cdot p^0 ) \bmod N.$$ +Kdy¾ se ale podíváme na~èleny tìchto dvou polynomù, zjistíme, ¾e se li¹í jen o~málo. Pùvodní polynom staèí pøenásobit~$p$, odeèíst první èlen s~$x_1$ a~naopak pøièíst chybìjící èlen $x_{J+1}$. Dostáváme tedy: +$$h(x_2 \dots x_{J+1}) = (p \cdot h(x_1 \dots x_J) - x_1 \cdot p^J + x_{J+1}) \bmod N.$$ +Pøepoèítání hashovací funkce tedy není nic jiného, ne¾ pøenásobení té minulé~$p$, odeètení nìjakého násobku toho znaku, který vypadl z~okénka, a~pøiètení toho znaku, o~který se okénko posunulo. Pokud tedy máme k~dispozici aritmetické operace v~konstantním èase, zvládneme konstantnì pøepoèítávat i~hashovací funkci. + +Tato hashovací funkce se dokonce nejen hezky poèítá, ale dokonce se i~opravdu \uv{hezky} chová (tedy \uv{rozumnì} náhodnì), pokud zvolíme vhodné~$p$. To bychom mìli zvolit tak, aby bylo rozhodnì nesoudìlné s~$N$ -- tedy $\(p, N) = 1$. Aby se nám navíc dobøe projevilo modulo obsa¾ené v~hashovací funkci, mìlo by být~$p$ relativnì velké (lze dopoèítat, ¾e optimum je mezi $2/3 \cdot N$ a~$3/4 \cdot N$). S~takto zvoleným~$p$ se tato hashovací funkce chová velmi pøíznivì a~v~praxi má celý algoritmus takøka lineární èasovou slo¾itost (prùmìrnou). + +\h{Hledání více øetìzcù najednou} +Nyní si zahrajeme tuté¾ hru, ov¹em v~trochu slo¾itìj¹ích kulisách. Podíváme se na~algoritmus, který si poradí i~s více ne¾ jednou jehlou. +Mìjme tedy jehly $\iota_1 \dots \iota_n$, a~jejich délky $J_i = \vert \iota_i \vert $. Dále budeme potøebovat seno $\sigma$ délky $S=\vert \sigma \vert$. + +Pøedtím, ne¾ se pustíme do~vlastního vyhledávacího algoritmu, mo¾ná bychom si mìli ujasnit, co vlastnì bude jeho výstupem. U problému hledání jedné jehly to bylo jasné -- byla to nìjaká mno¾ina pozic v~senì, na~kterých zaèínaly výskyty jehly. Jak tomu ale bude zde? Sice bychom také mohli vrátit pouze mno¾inu pozic, ale my budeme chtít malièko víc. Budeme toti¾ chtít vìdìt i~to, která jehla se na~které pozici vyskytuje. Výstup tedy bude vypadat následovnì: $V = \{(i,j)~\vert~\sigma[i:i+J_j]= \iota_j \}$. + +Zde se v¹ak skrývá jedna drobná zrada. Budeme se asi muset vzdát nadìje, ¾e najdeme algoritmus, jeho¾ slo¾itost je lineární v~celkové délce v¹ech jehel a~sena. Výstup toti¾ mù¾e být del¹í ne¾ lineární. Mù¾e se nám klidnì stát, ¾e na~jedné pozici v~senì se bude vyskytovat více rùzných jehel -- pokud bude jedna jehla prefixem jiné (co¾ jsme nikde nezakázali), tak máme povinnost ohlásit oba výskyty. Vzhledem k~tomu budeme hledat takový algoritmus, který bude lineární v~délce vstupu plus délce výstupu, co¾ je evidentnì to nejlep¹í, èeho mù¾eme dosáhnout. + +Algoritmus, který si nyní uká¾eme, vymysleli nìkdy v~roce 1975 pan Aho a~paní Corasicková. Bude to takové zobecnìní Knuthova-Morrisova-Prattova algoritmu. + +\h{Algoritmus Aho-Corasicková} + +Opìt se budeme sna¾it sestrojit nìjaký vyhledávací automat a~nìjakým zpùsobem tento automat pou¾ít k~procházení sena. Podívejme se nejprve na~pøíklad. Budeme chtít vyhledávat tato slova: |ara|, |bar|, |arab|, |baraba|, |barbara|. Mìjme tedy tìchto pìt jehel a~rozmysleme si, jak by vypadal nìjaký automat, který by tato slova umìl zatím jenom rozpoznávat. Pro jedno slovo automat vypadal jako cesta, zde u¾ to bude strom. (viz obrázek). + +\figure{ara_strom_blank.eps}{Vyhledávací automat -- strom.}{1in} + +Navíc budeme muset do~automatu zanést, kde nìjaké slovo konèí. V~pùvodním automatu pro jedno slovo to bylo jednoduché -- ono jedno jediné slovo odpovídalo poslednímu vrcholu cesty. Tady se v¹ak slova mohou vyskytovat vícekrát a~konèit nejenom v~listech ale i~v~nìjakém vnitøním vrcholu (co¾ se stane tehdy, pokud je jedno hledané slovo prefixem jiného hledaného slova). Formálnì to nebudeme dokazovat, ale snadno nahlédneme, ¾e listy stromu odpovídají hledaným slovùm, ale opaènì to neplatí. + +\figure{ara_strom_end.eps}{Vyhledávací automat s~konci slov.}{1in} + +Dále bychom mìli do~automatu pøidat zpìtné hrany. Jejich definice bude úplnì stejná jako u automatu pro hledání jednoho slova. Jinými slovy z~ka¾dého stavu pùjde zpìtná hrana do~nejdel¹ího vlastního suffixu, který je stavem. Èili kdy¾ budeme mít nìjaké jméno stavu, budeme se ho sna¾it co nejménì (ale alespoò o~znak) zkrátit zleva, abychom zase dostali jméno stavu. Z~koøene -- prázdného stavu -- pak evidentnì ¾ádná zpìtná hrana nepovede. + +\figure{ara_strom_final.eps}{Vyhledávací automat se zpìtnými hranami.}{1,25in} + +Zbývá nám je¹tì si rozmyslet, jakým zpùsobem bude ná¹ automat hlásit výstup. Opìt smìøujeme k~tomu, aby se automat po~pøeètení nìjakého kusu textu nacházel ve~stavu odpovídajícímu nejdel¹ímu mo¾nému suffixu toho textu. Zatímco u hledání jediné jehly bylo hlá¹ení výskytù jednoduché -- kdykoliv jsme se dostali na~konec \uv{automatové cestièky} tady to bude opìt slo¾itìj¹í. + +První, co se nabízí, je vyu¾ít toho, ¾e jsme si oznaèili nìjaké vrcholy, kde hledaná slova konèí. Co tedy zkusit hlásit výskyt tohoto slova v¾dy, kdy¾ pøijdeme do~nìjakého oznaèeného vrcholu? Tento zpùsob v¹ak nefunguje, pokud se uvnitø nìkteré jehly skrývá jehla vnoøená. Napøíklad po~pøeètení slova |bara|, nám ná¹ souèasný automat neøíká, ¾e bychom mìli nìjaké slovo ohlásit, a~pøitom tam evidentì konèí podøetìzec |ara|. Stejnì tak pokud pøeèteme |barbara|, u¾ si nev¹imneme toho, ¾e tam konèí zároveò i~|ara|. Pouhé \uv{hlá¹ení teèek} tedy nefunguje. + +Dále si mù¾eme v¹imnout toho, ¾e v¹echna slova, která by se mìla v~daném stavu hlásit, jsou suffixy jména tohoto stavu. Pøitom víme, ¾e zpìtná hrana jméno stavu zkracuje zleva. Tak¾e speciálnì v¹echny suffixy daného stavu, které jsou také stavy, se dají najít tak, ¾e se vydáme po~zpìtných hranách do~koøene. Nabízí se tedy v¾dy projít cestu po~zpìtných hranách a¾ do~koøene a~hlásit v¹echny \uv{teèky}. Tento zpùsob by nám v¹ak celý algoritmus znaènì zpomalil, proto¾e cesta do~koøene mù¾e být relativnì dlouhá, ale teèek na~ní obvykle bude málo. + +Mohli bychom také zkusit si pro ka¾dý stav $\beta$ pøedpoèítat mno¾inu $cache(\beta)$, která by obsahovala v¹echna slova, která máme hlásit, kdy¾ se ve~stavu $\beta$ nacházíme. Pokud pak do~tohoto stavu vstoupíme, podíváme se na~tuto mno¾inu a~vypí¹eme v¹e, co v~ní je. Výpis nám bude evidentnì trvat lineárnì k~velikosti mno¾iny, celkovì tedy lineárnì k~velikosti výstupu. Problém je ale ten, ¾e jednotlivé cache mohou být hodnì velké, tak¾e je nestihneme sestrojit v lineárním èase. (Rozmyslete si pøíklad slovníku, kdy se to stane.) + +To, co nám ale ji¾ opravdu pomù¾e, bude zavedení zkratek. V¹imli jsme si, ¾e po~zpìtných hranách mù¾eme projít do~koøene a~hlásit v¹echny nalezené teèky. Vadilo nám ale, ¾e se mù¾e stát, ¾e budeme dlouho po~cestì chodit a~pøi tom ¾ádné teèky nenalézat. Zavedeme si proto zkratky k~nejbli¾¹í teèce. + +\s{Definice} (zkratková hrana): +Budeme mít tedy nìjakou funkci $slovo(\beta) :=$ slovo, které konèí ve~stavu $\beta$ (nebo $\emptyset$, pokud ¾ádné takové slovo není). Dále pak funkci $out(\beta) :=$ nejbli¾¹í vrchol dosa¾itelný po~zpìtných hranách, èili nejdel¹í vlastní suffix stavu $\beta$, v~nìm¾ je definovaná funkce $slovo$. Trochu lid¹tìji øeèeno, ten nejbli¾¹í dosa¾itelný vrchol, ve~kterém je teèka. + +Po pøidání tìchto zkratkových hran ji¾ máme reprezentaci, ve~které opravdu umíme v~daném stavu vyjmenovat v¹echna slova, která máme vypsat, a~to v~èase lineárním s~tím, kolik tìch slov je. + +\s{Definice:} +Vyhledávací automat sestává ze stromu dopøedných hran (vrcholy jsou prefixy jehel, hrany odpovídají roz¹íøení o~písmenko), zpìtných hran ($z(\beta) :=$ nejdel¹í vlastní suffix slova $\beta$, který je stavem) a~zkratkových hran. + +Automat pak bude na~na¹em pøíkladu vypadat takto (zkratkové hrany jsou znázornìny zelenì): + +\figure{ara_strom_zkr.eps}{Vyhledávací automat se zkratkovými hranami.}{1,25in} + +Nyní u¾ nám zbývá jenom vlastní algoritmus -- nejdøív popí¹eme algoritmus, který bude hledat pomocí takového automatu, a~potom se pustíme do~toho, jak se takový automat staví. + +Nejprve si nadefinujeme, jak vypadá jeden krok automatu. Bude to vlastnì nìjaká funkce, která dostane stav a~písmenko. Ona nás pak pomocí tohoto písmenka posune po~automatu. ($f(\alpha, x)$ bude dopøedná hrana ze stavu $\alpha$ oznaèená písmenem~$x$) + +\s{Krok ($\alpha$, $x$):} +\algo +\:Dokud $f(\alpha, x) = \emptyset~\&~\alpha \neq \~~\alpha \leftarrow z(\alpha)$. +\:Pokud $f(\alpha, x) \neq \emptyset:~~\alpha \leftarrow f(\alpha, x)$. +\:Vrátíme výsledek. +\endalgo + +\s{Hledání:} +\algo +\:$\alpha \leftarrow \$. +\:Pro znaky $x$ ze slova $\sigma$: +\::$\alpha \leftarrow \(\alpha, x)$. +\::$\beta \leftarrow \alpha$ +\::Dokud $\beta \neq \emptyset$: +\:::Je-li $\(\beta) \neq \emptyset$: +\::::Ohlásíme $\(\beta)$. +\:::$\beta \leftarrow \(\beta)$. +\endalgo + +Algoritmus hledání vlastnì není nic jiného, ne¾ prosté projití po~zelených zkratkových hranách ze stavu $\alpha$, ve~kterém právì jsme, a~ohlá¹ení v¹eho, co po~cestì najdeme. + +V ka¾dém okam¾iku se automat nachází ve~stavu, který odpovídá nejdel¹ímu mo¾nému suffixu toho, co jsme u¾ pøeèetli. Dùkaz tohoto invariantu je stejný jako u verze automatu pro hledání pouze jedné jehly, nebo» vychází pouze z~definice zpìtných hran. Podobnì nahlédneme, ¾e èasová slo¾itost vyhledávací procedury je lineární v~délce sena plus to, co spotøebujeme na~hlá¹ení výskytù. Nejprve na~chvíli zapomeneme, ¾e nìjaké výskyty hlásíme a~spoèítáme jenom kroky. Ty mohou vést dopøedu a~zpátky. Krok dopøedu prodlu¾uje jméno stavu o~jedna, krok dozadu zkracuje aspoò o~jedna. Tudí¾ krokù dozadu je maximálnì tolik, co krokù dopøedu a~krokù dopøedu je maximálnì tolik, kolik je délka sena. V¹echny kroky dohromady tedy trvají $\O(S)$. Hlá¹ení výskytù pak trvá $\O(S~+ \vert V \vert)$. Celé hledání tedy trvá lineárnì v~délce vstupu a~výstupu. + +Zbývá nám u¾ jen konstrukce automatu. Opìt vyu¾ijeme faktu, ¾e zpìtná hrana ze stavu $\beta$ vede tam, kam by se dostal automat pøi hledání $\beta$ bez prvního písmenka. Tak¾e zase chceme nìco, jako simulovat výpoèet toho automatu na~slovech bez prvního písmenka a~doufat v~to, ¾e si vystaèíme s~tou èástí automatu, kterou jsme u¾ postavili. Tentokrát to v¹ak nemù¾eme dìlat jedno slovo po~druhém, proto¾e zpìtné hrany mohou vést køí¾em mezi jednotlivými vìtvemi automatu. Mohlo by se nám tedy stát, ¾e pøi hledání nìjakého slova potøebujeme zpìtnou hranu, která vede do~jiného slova, které jsme je¹tì nezkonstruovali. Tak¾e tento postup sel¾e. Mù¾eme v¹ak vyu¾ít toho, ¾e ka¾dá zpìtná hrana vede ve~stromu alespoò o~jednu hladinu vý¹. Mù¾eme tak strom konstruovat po~hladinách. Lze si to tedy pøedstavit tak, ¾e paralelnì spustíme vyhledávání v¹ech slov bez prvních písmenek a~v¾dycky udìláme jeden podkrok ka¾dého z~tìch hledání, co¾ nám dá zpìtné hrany z~dal¹ího patra stromu. + +\s{Konstrukce automatu:} +\algo +\:Zalo¾íme prázdný strom, $r \leftarrow$ jeho koøen. +\:Vlo¾íme do~stromu slova $\iota_1 \dots \iota_n$, nastavíme $slovo(*)$. +\:$z(r) \leftarrow \emptyset$, $out(r) \leftarrow \emptyset$. +\:Zalo¾íme frontu $F$ a~vlo¾íme do~ní syny koøene. +\:$\forall v~\in F:~~z(v) \leftarrow r, \(v) \leftarrow \emptyset$. +\:Dokud $F \neq \emptyset$: +\::Vybereme $u$ z~fronty $F$. +\::Pro v¹echny syny $v$ vrcholu $u$: +\:::$q \leftarrow \(z(u), \)$. +\:::$z(v) \leftarrow q$. +\:::Pokud $slovo(q) \neq \emptyset$, pak $out(v) \leftarrow q$. +\::::Jinak $out(v) \leftarrow out(q)$. +\:::Vlo¾íme $v$ do~fronty $F$. +\endalgo + +To, ¾e tento algoritmus zkonstruuje zpìtné hrany jak má, vyplývá z~toho, ¾e nedìláme nic jiného, ne¾ ¾e spou¹tíme výpoèty po~hladinách na~v¹echna hledaná slova bez prvního písmenka. Stejnì tak to, ¾e dobìhne v~lineárním èase, je takté¾ dùsledkem toho, ¾e efektivnì spou¹tíme v¹echny tyto výpoèty. Jen nìkdy udìláme najednou krok dvou èi více výpoètù (napøíklad |araba| a~|arbara| se poèítají na~zaèátku, dokud jsou stejné, jen jednou). Èasová slo¾itost této konstrukce je tedy men¹í nebo rovna souètu èasových slo¾itostí výpoètù nad v¹emi tìmi slovy. To u¾ ale víme, ¾e je lineární v~celkové délce tìchto slov. Konstrukce automatu tedy trvá nejvý¹e tolik, co hledání v¹ech $\iota_i$, co¾ je $\O(\sum_{i} \iota_i)$. + +\s{Vìta:} Algoritmus Aho-Corasicková najde v¹echny výskyty v~èase +$$\O\left(\sum_i~\iota_i~+~S~+~\sharp\\right).$$ + +Je¹tì se na~závìr zamysleme, jak bychom si takový automat ukládali do~pamìti. Urèitì se nám bude hodit si stavy nìjak oèíslovat (tøeba v~poøadí, v~jakém budou vznikat). Potom funkce pro zpìtné a~zkratkové hrany mohou být reprezentované polem indexovaným èíslem stavu. Funkce {\I Slovo}, která øíká, jaké slovo ve~stavu konèí, zase mù¾e být pole indexované stavem, které nám øekne poøadové èíslo slova ve~slovníku. Pro dopøedné hrany v~ka¾dém vrcholu pak mù¾eme mít pole indexované písmenky abecedy, které nám pro ka¾dé písmenko øekne, buï ¾e taková hrana není, nebo nám øekne, kam tato hrana vede. Je vidìt, ¾e takovéto pole se hodí pro pomìrnì malé abecedy. U¾ pro abecedu A-Z~bude velikosti 26 a~z~vìt¹iny bude prázdné, tak¾e bychom plýtvali pamìtí. V praxi se proto èasto pou¾ívá hashovací tabulka. Pøípadnì bychom mohli mít i~jen jednu velkou spoleènou hashovací tabulku, která bude reprezentovat funkci celou, ve~které budou zahashované dvojice (stav, písmenko). Tìchto dvojic je evidentnì tolik, kolik hran stromu, èili lineárnì s~velikostí slovníku, a~je to asi nejkompaktnìj¹í reprezentace. + + +\bye diff --git a/1-kmp/Makefile b/1-kmp/Makefile new file mode 100644 index 0000000..3f18d9f --- /dev/null +++ b/1-kmp/Makefile @@ -0,0 +1,3 @@ +P=1-kmp + +include ../Makerules diff --git a/1-kmp/ara_strom.eps b/1-kmp/ara_strom.eps new file mode 100644 index 0000000..7d054ab --- /dev/null +++ b/1-kmp/ara_strom.eps @@ -0,0 +1,1334 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: inkscape 0.46 +%%Pages: 1 +%%Orientation: Portrait +%%BoundingBox: 163 220 443 747 +%%HiResBoundingBox: 163.64601 220.18395 442.55996 746.83829 +%%EndComments +%%Page: 1 1 +0 842 translate +0.8 -0.8 scale +0 0 0 setrgbcolor +[] 0 setdash +1 setlinewidth +0 setlinejoin +0 setlinecap +gsave [1 0 0 1 0 0] concat +gsave [0.5411688 0 0 0.5411688 203.68828 84.001202] concat +0 0 0 setrgbcolor +[] 0 setdash +1.478282 setlinewidth +2 setlinejoin +0 setlinecap +newpath +299.99999 173.79076 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+newpath +711.15927 159.63036 moveto +703.71296 189.73566 643.62165 205.468 570.98179 213.59973 curveto +531.11935 218.06216 487.47795 220.2357 446.68392 221.23953 curveto +366.0505 223.2237 296.54156 220.6379 289.32855 222.12506 curveto +stroke +grestore +showpage +%%EOF diff --git a/1-kmp/barb.svg b/1-kmp/barb.svg new file mode 100644 index 0000000..e2a9bfd --- /dev/null +++ b/1-kmp/barb.svg @@ -0,0 +1,456 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + ε + b + ba + bar + barb + barba + barbar + barbaro + barbarossa + ... + b a r b a r o ... a + + + + + + + + + + + + + + + + + + + + diff --git a/old/6-kmp/6-kmp.tex b/old/6-kmp/6-kmp.tex deleted file mode 100644 index 041c06a..0000000 --- a/old/6-kmp/6-kmp.tex +++ /dev/null @@ -1,278 +0,0 @@ -\input lecnotes.tex - -\prednaska{6}{Vyhledávání v~textu}{(zapsal: Petr Jankovský)} - -Nyní se budeme vìnovat následujícímu problému: v~textu délky $S$ (senì) budeme chtít najít v¹echny výskyty hledaného slova délky $J$ (jehly). Nejprve se podívejme na~jeden primitivní algoritmus, který nefunguje. Je ale zajímavé rozmyslet si, proè. - -\h{Hloupý algoritmus} -Zaèneme prvním písmenkem hledaného slova a~budeme postupnì procházet text, a¾ najdeme první výskyt poèáteèního písmenka. Poté budeme testovat, zda souhlasí i~písmenka dal¹í. Pokud nastane neshoda, v~hledaném slovì se vrátíme na~zaèátek a~v~textu pokraèujeme znakem, ve~kterém neshoda nastala. Podívejme se na~pøíklad. - -\s{Pøíklad:} Budeme hledat slovo |jehla| v~textu |jevkupcejejehla|. Vezmeme si tedy první písmenko |j| v~hledaném slovì a~zjistíme, ¾e v~textu se nachází hned na~zaèátku. Vezmeme tedy dal¹í písmenko |e|, které se vyskytuje jako druhé i~v~textu. Pøi tøetím písmenku ale narazíme na~neshodu. V~tuto chvíli tedy zresetujeme a~opìt hledáme výskyt písmenka |j|, tentokrát v¹ak a¾ od~tøetího písmene v~textu. Takto postupujeme postupnì dál, a¾ narazíme na~dal¹í |je|, které ov¹em není následováno písmenem~|h|, tudí¾ opìt zresetujeme a~nakonec najdeme shodu s~celým hledaným øetìzcem. V~tomto pøípadì tedy algoritmus na¹el hledané slovo. - -Tento algoritmus v¹ak zjevnì mù¾e hanebnì selhat. Mù¾e se stát, ¾e zaèneme porovnávat, a¾ v~jednu chvíli narazíme na~neshodu. Celý tento kus tedy pøeskoèíme. Pøi tom se ale v~tomto kusu textu mohl vyskytovat nìjaký pøekrývající se výskyt hledané \uv{jehly}. Hledejme napøíklad øetìzec |kokos| v~textu |clanekokokosu|. Algoritmus tedy zaène porovnávat. Ve~chvíli kdy najde prefix |koko| a~na~vstupu dostane~|k|, dochází k~neshodì. Proto algoritmus zresetuje a~pokraèuje v~hledání od~tohoto znaku. Najde sice je¹tì výskyt |ko|, ov¹em s~dal¹ím písmenkem |s| ji¾ dochází k~neshodì a~algotimus sel¾e. Nesprávnì se toti¾ \uv{upnul} na~první nalezené |koko| a~s~dal¹ím~|k| pak \uv{zahodil} i~správný zaèátek. - -Máme tedy algoritmus, který i~kdy¾ je ¹patnì, tak funguje urèitì kdykoli se první písmenko hledaného slova v~tomto slovì u¾ nikde jinde nevyskytuje -- co¾ |jehla| splòovala, ale |kokos| u¾ ne. - -{\I Hloupý algoritmus} se na~ka¾dé písmenko textu podívá jednou, tudí¾ èasová slo¾itost bude lineární s~délkou textu ve~kterém hledáme -- tedy $\O(S)$. - -\h{Pomalý algoritmus} -Zkusíme algoritmus vylep¹it tak, aby fungoval správnì: pokud nastane nìjaká neshoda, vrátíme se zpátky tìsnì za~zaèátek toho, kdy se nám to zaèalo shodovat. To je ov¹em vlastnì skoro toté¾, jako brát postupnì v¹echny mo¾né zaèátky v~\uv{senì} a~pro ka¾dý z~nìj ovìøit, jestli se tam \uv{jehla} nachází èi nikoliv. - -Tento algoritmus evidentnì funguje. Bì¾í v¹ak v~èase: $S$ mo¾ných zaèátkù, krát èas potøebný na~jedno porovnání (zda se na~dané pozici nenachází \uv{jehla}), co¾ nám mù¾e trvat a¾ $J$. Proto je èasová slo¾itost $\O(SJ)$. V praxi bude algoritmus èasto rychlej¹í, proto¾e typicky velmi brzo zjistíme, ¾e se øetìzce neshodují, ale je mo¾né vymyslet vstup, kde bude potøeba porovnání opravdu tolik. - -Nyní se pokusme najít takový algoritmus, který by byl tak rychlý, jako {\I Hloupý algoritmus}, ale chytrý, jako ten {\I Pomalý}. - -\h{Chytrý algoritmus} -Ne¾ vlastní algoritmus vybudujeme, zkusíme se cestou nauèit pøemý¹let o~øetìzcích obèas trochu pøekrouceným zpùsobem. Podívejme se na~je¹tì jeden pøíklad. - -\s{Pøíklad:} Vezmìme si napøíklad staré italské pøízvisko |barbarossa|, které znamená Rudovous. Pøedstavme si, ¾e takovéto slovo hledáme v~nìjakém textu, který zaèíná |barbar|. Víme, ¾e a¾ sem se nám hledaný øetìzec shodoval. Øeknìme, ¾e dal¹í písmenko textu se shodovat pøestane -- místo |o| naèteme napøíklad opìt |b|. {\I Hloupý algoritmus} by velil vrátit se k~|a| a~od~nìj hledat dál. Uvìdomme si ale, ¾e kdy¾ se vracíme z~|barbar| do~|arbar| (tedy øetìzce, který ji¾ známe), mù¾eme si pøedpoèítat, jak dopadne hledání, kdy¾ ho pustíme na~nìj. V~pøedpoèítaném bychom tedy chtìli ukládat, ¾e kdy¾ máme øetìzec |arbar|, tak |ar| a~|r| nám do~hledaného nepasuje a~a¾~|bar| se bude shodovat. Tedy místo toho, abychom spustili nové hledání od~|a|, mù¾eme ho spustit a¾~od~|b|. Co víc, my dokonce víme, jak dopadne to -- pokud toti¾ nastane neshoda po~pøeètení |barbar|, je to stejné, jako kdybychom pøeèetli pouze |bar|, na~které se (pùvodne neshodující se) |b| u¾ navázat dá. Kdyby se nedalo navázat ani tam, tak bychom opìt zkracovali... Nejen, ¾e tedy víme, kam se máme vrátit, ale víme dokonce i~to, co tam najdeme. - -My¹lenka, ke které míøíme, je pøedpoèítat si nìjakou tabulku, která nám bude øíkat, jak se máme pøi hledání vracet a~jak to dopadne, a~pak u¾ jenom prohlédávat s~pou¾itím této tabulky. - -Aby se nám o~tìchto algoritmech lépe mluvilo a~pøedev¹ím psalo, pojïme si povìdìt nìkolik definic. - -\s{Definice:} -\itemize\ibull -\:{\I Abeceda $\Sigma$} je koneèná mno¾ina znakù\foot{Mù¾eme pøi tom jít a¾~do~extrémù. Pøíkladem extrémních abeced je binární abeceda slo¾ená pouze z~nul a~jednièek. Pøíklad z~druhého konce (který rádi dìlají lingvisti) je abeceda, která má jako abecedu v¹echna èeská slova. V¹echny èeské vìty, pak nejsou nic jiného, ne¾ slova nad touto abecedou. Pou¾itá abeceda tedy mù¾e být i~relativnì obrovská. Dal¹ím takovým pøíkladem mù¾e být Unicode. Pro na¹e potøeby ale zatím budeme pøedpokládat, ¾e abeceda je nejen konstantnì velká, ale i~rozumnì malá. Budeme si moci tedy dovolit napøíklad indexovat pole znakem abecedy (kdybychom nemohli, tak bychom místo pole pou¾ili napøíklad hashovací tabulku, èi nìco podobného\dots).}, ze~kterých tvoøíme text, øetìzce, slova. - - -\:{\I $\Sigma^*$} je mno¾ina v¹ech slov nad abecedou $\Sigma$. Èili mno¾ina v¹ech neprázdných koneèných posloupností znakù ze $\Sigma$. -\endlist -\s{Znaèení:} -Aby se nám nepletlo znaèení, budeme rozli¹ovat promìnné pro slova, promìnné pro písmenka a~promìnné pro èísla. - -\itemize\ibull -\:{\I Slova} budeme znaèit malými písmenky øecké abecedy $\alpha$, $\beta$, ... . -\:$\iota$ bude oznaèovat \uv{jehlu} -\:$\sigma$ bude oznaèovat \uv{seno} -\:{\I Znaky} oznaèíme malými písmeny latinky $a$, $b$, \dots . -\:{\I Èísla} budeme znaèit velkými písmeny $A$, $B$, \dots . -\:{\I Délka slova} $\vert \alpha \vert$ pro $\alpha \in \Sigma^*$ je poèet jeho znakù. -\:{\I Prázdné slovo} znaèíme písmenem $\varepsilon$, $\vert \varepsilon \vert = 0$. -\:{\I Zøetìzení} $\alpha\beta$ vznikne zapsáním slov $\alpha$ a~$\beta$ za sebe. Platí $\vert \alpha\beta \vert=\vert \alpha \vert+\vert \beta \vert$, $\alpha\varepsilon=\varepsilon\alpha=\alpha$. -\:$\alpha[k]$ je $k$-tý znak slova $\alpha$, indexujeme od~$0$. -\:$\alpha[k:l]$ je podslovo zaèínající $k$-tým znakem a~$l$-tý znak je první, který v~nìm není. Jedná se tedy o~podslovo skládající se z~$\alpha[k]$,$\alpha[k+1]$,\dots,$\alpha[l-1]$. Platí tedy: $\alpha[k:k]=\varepsilon$, $\alpha[k:k+1]=\alpha[k]$. Jednu (èi obì) meze mù¾eme i~vynechat, tento zápis pak bude znamenat buï \uv{od zaèátku slova a¾ nìkam}, nebo \uv{odnìkud a¾ do~konce}: -%TODO - zaøadit pod pøedchozí bod -\:$\alpha[:k]$ je {\I prefix} obsahující prvních $k$ znakù slova $\alpha$ ($\alpha[0]$,\dots,$\alpha[k-1]$). -\:$\alpha[k:]$ je {\I suffix} obsahující znaky slova $\alpha$ poèínaje $k$-tým znakem a¾ do~konce. -\:$\alpha[:] = \alpha$ -\endlist - - -V¹imnìme si, ¾e prázdné slovo je prefixem, suffixem i~podslovem jakéhokoliv slova vèetnì sebe sama. -Ka¾dé slovo je také prefixem, suffixem i~podslovem sebe sama. To se ne v¾dy hodí. Nìkdy budeme chtít øíct, ¾e nìjaké slovo je {\I vlastním} prefixem nebo suffixem. To bude znamenat, ¾e to nebude celé slovo. - -\> $\alpha$ je {\I vlastní prefix} slova $\beta \equiv \alpha$ je prefix $\beta~\&~\alpha \neq \beta$. - -\h{Vyhledávací automat (Knuth, Morris, Pratt)} -{\I Vyhledávací automat} bude graf, jeho¾ vrcholùm budeme øíkat {\I stavy}. Jejich jména budou prefixy hledaného slova a~hrany budou odpovídat tomu, jak jeden prefix mù¾eme získat z~pøedchozího prefixu pøidáním jednoho písmene. Poèáteèní stav je prázdné slovo $\varepsilon$ a~koncový je celá $\iota$. Dopøedné hrany grafu budou popisovat pøechod mezi stavy ve~smyslu zvìt¹ení délky jména stavu (dopøedná funkce $h(\alpha)$, urèující znak na~dopøedné hranì z~$\alpha$). Zpìtné hrany grafu budou popisovat pøechod (zpìtná funkce $z(\alpha)$) mezi stavem $\alpha$ a~nejdel¹ím vlastním suffixem $\alpha$, který je prefixem $\iota$, kdy¾ nastane neshoda. - -\figure{barb.eps}{Vyhledávací automat.}{4.1in} - -\s{Hledej($\sigma$):} -\algo -\:$\alpha \leftarrow \varepsilon$. -\:Pro $x\in\sigma$ postupnì: -\:$\indent$Dokud $h(\alpha) \neq x~\&~\alpha \neq \varepsilon : \alpha \leftarrow z(\alpha)$. -\:$\indent$Pokud $h(\alpha) = x: \alpha \leftarrow \alpha x$. -\:$\indent$Pokud $\alpha = \iota$, ohlásíme výskyt. -\endalgo - -\>Vstupem je $\iota$, hledané slovo (jehla) délky $J=\vert \iota \vert$ a~$\sigma$, text (seno) délky $S=\vert \sigma \vert$. -\>Výstupem jsou v¹echny výskyty hledaného slova $\iota$ v~textu $\sigma$, tedy mno¾ina $\left\{ k \mid \sigma[k:k+J]=\iota \right\}$ - -Pojïme nyní dokázat, ¾e tento algoritmus správnì ohlásí v¹echny výskyty. - -\s{Definice}: $\alpha(\tau) := $ stav automatu po~pøeètení $\tau$ - -\s{Invariant:} Pokud algoritmus pøeète nìjaký vstup, nachází se ve~stavu, který je nejdel¹ím suffixem pøeèteného vstupu, který je nìjakým stavem. -$\alpha(\tau) =$ nejdel¹í stav (nejdel¹í prefix jehly), který je suffixem $\tau$ (pøeèteného vstupu). - -Pojïme si rozmyslet, ¾e z~tohoto invariantu ihned plyne, ¾e algoritmus najde to, co má. Kdykoli toti¾ ohlásí nìjaký výskyt, tak tam tento výskyt opravdu je. Kdykoli pak má nìjaký výskyt ohlásit, tak se v~této situaci jako suffix toho právì pøeèteného textu vyskytuje hledané slovo, pøièem¾ hledané slovo je urèitì stav a~zároveò nejdel¹í ze v¹ech existujících stavù. Tak¾e invariant opravdu øíká, ¾e jsme právì v~koncovém stavu a~algoritmus nám tedy ohlásí výskyt. - -\proof {\I (invariantu)} -Indukcí podle kroku algoritmu. Na~zaèátku pro prázdný naètený vstup invariant triviálnì platí, tedy prázdný suffix $\tau$ je prefixem $\iota$. V~kroku $n$ máme naètený vstup $\tau$ a~k~nìmu pøipojíme znak $x$. Invariant nám øíká, ¾e nejdel¹í stav, který je suffixem, je nejdel¹í suffix, který je stavem. Nyní se ptáme, jaký je nejdel¹í stav, který se dá \uv{napasovat} na~konec øetìzce $\tau x$. Kdykoli v¹ak takovýto suffix máme, tak z~nìj mù¾eme $x$ na~konci odebrat, èím¾ dostaneme suffix slova $\tau$. - -\>Tedy: pokud $\beta$ je neprázdným suffixem slova $\tau x$, pak $\beta = \gamma x$, kde $\gamma$ je suffix $\tau$. - -Suffix, který máme sestrojit, tedy vznikne z~nìjakého suffixu slova $\tau$ pøipsáním~$x$. Chceme najít nejdel¹í suffix slova $\tau x$, který je stavem, tak¾e chceme najít i~nejdel¹í suffix pùvodního slova $\tau$, za který se dá pøidat $x$ tak, aby vy¹lo jméno stavu. Staèí tedy u¾ jen \uv{probírat} suffixy slova $\tau$ od~nejdel¹ího po~nejkrat¹í, zkou¹et k~nim pøidávat $x$ a~a¾ to pùjde, tak jsme na¹li nejdel¹í suffix $\tau x$. Pøesnì toto ov¹em algoritmus dìlá, nebo» zpìtná funkce mu v¾dy øekne nejbli¾¹í krat¹í suffix, který je stavem. Pokud pak nemù¾eme $x$ pøidat ani do~$\varepsilon$, pak je øe¹ením prázdný suffix. Algoritmus tedy funguje. \qed - -Nyní pojïmì zkoumat to, jak je ve~skuteènosti ná¹ algoritmus rychlý. K tomu bychom si ale nejdøív mìli øíct, jak pøesnì budeme automat reprezentovat. V~algoritmu vystupují nìjaká porovnávání stavù, pøièem¾ není úplnì jasné, jak zaøídit, aby v¹e trvalo konstantnì dlouho. Vyjde nám to ale docela snadno. K reprezentaci automatu nám toti¾ budou staèit pouze dvì pole. - -\s{Reprezentace automatu:} -Oèíslujeme si stavy délkami pøíslu¹ných prefixù, tedy èísly $0 \dots J$. Poté je¹tì potøebujeme nìjakým zpùsobem zakódovat dopøedné a~zpìtné hrany. Vzhledem k~tomu, ¾e z~ka¾dého vrcholu vede v¾dy nejvý¹e jedna dopøedná a~nejvý¹e jedna zpìtná, tak nám evidentnì staèí pamatovat si pro ka¾dý typ hran pouze jedno èíslo na~vrchol. Budeme mít tedy nìjaké pole dopøedných hran, které nám pro ka¾dý stav øekne, jakým písmenkem je nadepsaná dopøedná hrana ze stavu $I$ do~$I+1$. To jsou ale pøesnì písmenka jehly, tak¾e si staèí pamatovat jehlu samotnou. Èili z~$I$ do~$I+1$ vede hrana nadepsaná $\iota [I]$. Pro zpìtné hrany pak budeme potøebovat pole $Z$, které nám pro stav $I$ øekne èíslo stavu, do~kterého vede zpìtná hrana. Tedy $Z[I]$ je cíl zpìtné hrany ze stavu $I$. -S~touto reprezentací ji¾ doká¾eme na¹i hledací proceduru pøímoèaøe pøepsat tak, aby sahala pouze do~tìchto dvou polí: -\algo -\:$I \leftarrow 0$. -\:Pro znaky $x$ z~textu: -\:$\indent$Dokud $\iota[I] \neq x~\&~I \neq 0: I \leftarrow Z[I]$. -\:$\indent$Pokud $\iota[I] = x$, pak $I \leftarrow I + 1$. -\:$\indent$Pokud $I = J$, ohlásíme výskyt. -\endalgo - -Zatím se v~algoritmu je¹tì skrývá drobná chyba -- toti¾ algoritmus se obèas zeptá na~dopøednou hranu z~posledního stavu. Pokud jsme právì ohlásili výskyt (jsme tedy v~posledním stavu) a~pøijde nìjaký dal¹í znak, algoritmus se ptá, zda je roven tomu, co je na~dopøedné hranì z~posledního stavu. Ta ale ov¹em neexistuje. Jednodu¹e to ale napravíme tak, ¾e si pøidáme fiktivní hranu, na~které se vyskytuje nìjaké \uv{nepísmenko} -- nìco, co se nerovná ¾ádnému jinému písmenku. Zajistíme tak, ¾e se po~této hranì nikdy nevydáme. Dodefinujeme tedy $\iota[J]$ odli¹nì od~v¹ech znakù.\foot{V jazyce C se toto dodefinování provede vlastnì zadarmo, nebo» ka¾dý øetìzec je v~nìm ukonèen znakem s~kódem nula, který se ve~vstupu nevyskytne\dots Algoritmus bude tedy fungovat i~bez tohoto dodefinování. V jiných jazycích je ale tøeba na~nìj nezapomenout!} - -\s{Lemma:} Funkce {\I Hledej} bì¾í v~èase $\O(S)$. - -\proof -Funkce {\I Hledej} chodí po~dopøedných a~zpìtných hranách. Dopøedných hran projdeme urèitì maximálnì tolik, kolik je délka sena. Pro ka¾dý znak pøeètený ze sena toti¾ jdeme nejvý¹e jednou po~dopøedné hranì. Se zpìtnými hranami se to má tak, ¾e na~jeden pøeètený znak z~textu se mù¾eme po~zpìtné hranì vracet maximálnì $J$-krát. Z~tohoto by nám v¹ak vy¹la slo¾itost $\O(JS)$, èím¾ bychom si nepomohli. Zachrání nás ale pøímoèarý potenciál. Uvìdomme si, ¾e chùze po~dopøedné hranì zvý¹í $I$ o~jedna a~chùze po~zpìtné hranì $I$ sní¾í alespoò o~jedna. Vzhledem k~tomu, ¾e $I$ není nikdy záporné a~na~zaèátku je nulové, zjistíme, ¾e krokù zpìt mù¾e být maximálnì tolik, kolik krokù dopøedu. Èasová slo¾itost hledání je tedy lineární vzhledem k~délce sena. \qed - -Nyní nám zbývá na~první pohled malièkost -- toti¾ zkonstruovat automat. Zkonstruovat dopøedné hrany zvládneme zjevnì snadno, jsou toti¾ explicitnì popsané hledaným slovem. Tì¾¹í u¾ to bude pro hrany zpìtné. Vyu¾ijeme k~tomu následující pozorování: - -\s{Pozorování:} -Pøedstavme si, ¾e automat u¾ máme hotový a~tím, ¾e budeme sledovat jeho chování, chceme zjistit, jak v~nìm vedou zpìtné hrany. -Vezmìme si nìjaký stav~$\beta$. To, kam z~nìj vede zpìtná hrana zjistíme tak, ¾e spustíme automat na~øetìzec $\beta$~bez prvního písmenka a~stav, ve~kterém se automat zastaví, je pøesnì ten, kam má vést i~zpìtná hrana z~$\beta$. Jinými slovy víme, ¾e $z(\beta) = \alpha (\beta[1:])$. -Proè takováto vìc funguje? V¹imìme si, ¾e definice $z$ a~to, co nám o~$\alpha$ øíká invariant, je témìø toto¾né -- $z(\beta)$ je nejdel¹í vlastní suffix $\beta$, který je stavem, $\alpha(\beta)$ je nejdel¹í suffix $\beta$, který je stavem. Jediná odli¹nost je v~tom, ¾e definice $z$ narozdíl od~definice $\alpha$ zakazuje nevlastní suffixy. Jak nyní vylouèit suffix $\beta$, který by byl roven $\beta$ samotné? Zkrátíme $\beta$ o~první znak. Tím pádem v¹echny suffixy $\beta$ bez prvního znaku jsou stejné jako v¹echny vlastní suffixy $\beta$. - -K èemu je toto pozorování dobré? Rozmysleme si, ¾e pomocí nìj u¾ doká¾eme zkonstruovat zpìtné hrany. Není to ale trochu divné, kdy¾ pøi simulování automatu na~øetìzec bez prvního znaku u¾ zpìtné hrany potøebujeme? Není. Za chvíli zjistíme, ¾e takto mù¾eme zji¹»ovat zpìtné hrany postupnì -- a~to tak, ¾e pou¾íváme v¾dy jenom ty, které jsme u¾ sestrojili. - -Takovémuhle pøístupu, kdy pøi konstruování chtìného u¾ pou¾íváme to, co chceme sestrojit, ale pouze ten kousek, který ji¾ máme hotový, se v~angliètinì øíká {\I bootstrapping}\foot{Z~tohoto slova vzniklo i~{\I bootování} poèítaèù, kdy operaèní systém v~podstatì zavádí sám sebe. Bootstrap znamená èesky ¹truple -- tedy oèko na~konci boty, které slou¾í k~usnadnìní nazouvání. A~jak souvisí ¹truple s~algoritmem? To se zase musíme vrátit k~pøíbìhùm o~baronu Prá¹ilovi, mezi nimi¾ je i~ten, ve~kterém baron Prá¹il vypráví o~tom, jak sám sebe vytáhl z~ba¾iny za ¹truple. Stejnì tak i~my budeme algoritmus konstruovat tím, ¾e se budeme sami vytahovat za ¹truple, tedy bootstrappovat.}. -V¹imnìme si, ¾e pøi výpoètu se vstupem $\beta$ projde automat jenom prvních $\vert \beta \vert$ stavù. Automat se evidentnì nemù¾e dostat dál, proto¾e na~ka¾dý krok dopøedu (doprava) spotøebuje písmenko $\beta$. Tak¾e krokù doprava je maximálnì tolik, kolik je $\vert \beta \vert$. Jinými slovy kdybychom ji¾ mìli zkonstruované zpìtné hrany pro prvních $\vert \beta \vert$ stavù (tedy $0 \dots \vert \beta \vert - 1$), tak pøi tomto výpoètu, který potøebujeme na~zkonstruování zpìtné hrany z~$\beta$, je¹tì tuto zpìtnou hranu nemù¾eme potøebovat. Vystaèíme si s~tìmi, které ji¾ máme zkonstruované. - -Nabízí se tedy zaèít zpìtnou hranou z~prvního znaku (která vede evidentnì do~$\varepsilon$), pak postupnì brát dal¹í stavy a~pro ka¾dý z~nich si spoèítat, kdy spustíme automat na~jméno stavu bez prvního znaku a~tím získáme dal¹í zpìtnou hranu. Toto funguje, ale je to kvadratické \dots. Máme toti¾ $J$ stavù a~pro ka¾dý z~nich nám automat bì¾í v~èase a¾ lineárním s~$J$. Jak z~toho ven? - -Z~prvního stavu povede zpìtná funkce do~$\varepsilon$. Pro dal¹í stavy chceme spoèítat zpìtnou funkci. Z~druhého stavu $\iota[0:2]$ tedy automat spustíme na~$\iota[1:2]$, dále pak na~$\iota[1:3]$, $\iota[1:4]$, atd. Ty øetìzce, pro které potøebujeme spo¹tìt automat, abychom dostali zpìtné hrany, jsou tedy ve~skuteènosti takové, ¾e ka¾dý dal¹í dostaneme roz¹íøením pøedchozího o~jeden znak. To jsou ale pøesnì ty stavy, kterými projde automat pøi zpracovávání øetìzce $\iota$ od~prvního znaku dál. Jedním prùchodem automatu nad jehlou bez prvního písmenka se tím pádem rovnou dozvíme v¹echny údaje, které potøebujeme. -Z~pøedchozího pozorování plyne, ¾e nikdy nebudeme potøebovat zpìtnou hranu, kterou jsme je¹tì nezkonstruovali a~jeliko¾ víme, ¾e jedno prohledání trvá lineárnì s~délkou toho, v~èem hledáme, tak toto celé pobì¾í v~lineárním èase. Dostaneme tedy následující algoritmus: - -\s{Konstrukce zpìtné funkce:} -\algo -\:$Z[0] \leftarrow ?$, $Z[1] \leftarrow 0$. -\:$I \leftarrow 0$. -\:Pro $k = 2 \dots J$: -\::$I \leftarrow \( I , \iota [k])$. -\::$Z[k] \leftarrow I$. -\endalgo - -Zaèínáme tím, ¾e nastavíme zpìtnou hranu z~prvních dvou stavù, pøièem¾ $z[0]$ je nedefinované, proto¾e tuto zpìtnou hranu nikdy nepou¾íváme. Dále postupnì simulujeme výpoèet automatu nad slovem bez prvního znaku a~po ka¾dém kroku se dozvíme novou zpìtnou hranu. {\I Krokem} automatu pak není nic jiného ne¾ vnitøek (3. a~4. bod) na¹í hledací procedury. To, kam jsme se dostali, pak zaznamenáme jako zpìtnou funkci z~$k$. -Èili pou¹tíme automat na~jehlu bez prvního písmenka, provedeme v¾dy jeden krok automatu (pøes dal¹í písmenko jehly) a~zapamatujeme si, jakou zpìtnou funkci jsme zrovna dostali. Díky pozorováním navíc víme, ¾e zpìtné hrany konstruujeme správnì, nikdy nepou¾ijeme zpìtnou hranu, kterou jsme je¹tì nesestrojili a~víme i~to, ¾e celou konstrukci zvládneme v~lineárním èase s~délkou jehly. - -\s{Vìta:} Algoritmus KMP najde v¹echny výskyty v~èase $O(J+S)$. - -\proof -Lineární èas s~délkou jehly potøebujeme na~postavení automatu, lineární èas s~délkou sena pak potøebujeme na~samotné vyhledání. - -\h{Rabinùv-Karpùv algoritmus} - -Nyní si uká¾eme je¹tì jeden algoritmus na~hledání jedné jehly, který nebude mít v~nejhor¹ím pøípadì lineární slo¾itost, ale bude ji mít prùmìrnì. Bude daleko jednodu¹¹í a~uká¾e se, ¾e je v~praxi daleko rychlej¹í. Bude to algoritmus zalo¾ený na~hashování. - - -Pøedstavme si, ¾e máme seno délky $S$ a~jehlu délky $J$, a~vezmìme si nìjakou hashovací funkci, které dáme na~vstup $J$-tici znakù (tedy podslova dlouhá jako jehla). Tato hashovací funkce nám je pak zobrazí do~mno¾iny $\{0,\ldots,N-1\}$ pro nìjaké dost velké~$N$. Jak nám toto pomù¾e pøi hledání jehly? Vezmeme si libovolné \uv{okénko} délky $J$ a~ne¾ budeme zji¹»ovat, zda se v~nìm jehla vyskytuje, tak si spoèítáme hashovací funkci a~porovnáme ji s~hashem jehly. Èili ptáme se, jestli je hash ze sena od~nìjaké pozice $I$ do~pozice $I+J$ roven hashi jehly -- formálnì: $h(\sigma [I: I+J ]) = h(\iota)$. Teprve tehdy, kdy¾ zjistíme, ¾e se hodnota hashovací fce shoduje, zaèneme doopravdy porovnávat øetìzce. - -Není to ale nìjaká hloupost? Mù¾e nám vùbec takováto konstrukce pomoci? Není to tak, ¾e na~spoèítání hashovací funkce z~$J$ znakù, potøebujeme tìch $J$ znakù pøeèíst, co¾ je stejnì rychlé, jako rovnou øetìzce porovnávat? Pou¾ijeme trik, který bude spoèívat v~tom, ¾e si zvolíme ¹ikovnou hashovací funkci. Udìláme to tak, abychom ji mohli pøi posunutí \uv {okénka} o~jeden znak doprava v~konstantním èase pøepoèítat. Chceme umìt z~$h(x_1 \dots x_j)$ spoèítat $h(x_2 \dots x_{j+1})$. -Na~zaèátku si tedy spoèítáme hash jehly a~první $J$-tice znakù sena. Pak ji¾ jenom posouváme \uv {okénko} o~jedna, pøepoèítáme hashovací funkci a~kdy¾ se shoduje s~hashem jehly, tak porovnáme. Budeme pøitom vìøit tomu, ¾e pokud se tam jehla nevyskytuje, pak máme hashovací funkci natolik rovnomìrnou, ¾e pravdìpodobnost toho, ¾e se pøesto strefíme do~hashe jehly, je $1/N$. Jinými slovy jenom v~jednom z~øádovì $N$ pøípadù budeme porovnávat fale¹nì -- tedy provedeme porovnání a~vyjde nám, ¾e výsledek je neshoda. V~prùmìrném pøípadì tedy mù¾eme stlaèit slo¾itost a¾ témìø k~lineární. - -Podívejme se teï na~prùmìrnou èasovou slo¾itost. Budeme urèitì potøebovat èas na~projití jehly a~sena. Navíc strávíme nìjaký èas nad fale¹nými porovnáními, kterých bude v~prùmìru na~ka¾dý $N$-tý znak sena jedno porovnání s~jehlou -- tedy $SJ / N$, pøièem¾ $N$ mù¾eme zvolit dost velké na~to, abychom tento èlen dostali pod nìjakou rozumnou konstantu... Nakonec budeme potøebovat jedno porovnání na~ka¾dý opravdový výskyt, èemu¾ se nevyhneme. Pøipoèteme tedy je¹tì $J \cdot$ {\I $\sharp$výskytù}. Dostáváme tedy: $ \O(J+S+SJ/N+J \cdot$ {\I $\sharp$výskytù}). - -Zbývá malièkost -- toti¾ kde vzít hashovací funkci, která toto v¹e splòuje. Jednu si uká¾eme. Bude to vlastnì takový hezký polynom: -$$h(x_1 \dots x_j) := \left(\sum_{I=1}^{J} x_I \cdot p^{J-I}\right) \bmod N.$$ -Jinak zapsáno se tedy jedná o: -$$(x_1 \cdot p^{J-1} + x_2 \cdot p^{J-2} + \dots + x_J \cdot p^0 ) \bmod N.$$ -Po posunutí okénka o~jedna chceme dostat: -$$(x_2 \cdot p^{J-1} + x_3 \cdot p^{J-2} + \dots + x_J \cdot p^1 + x_{J+1} \cdot p^0 ) \bmod N.$$ -Kdy¾ se ale podíváme na~èleny tìchto dvou polynomù, zjistíme, ¾e se li¹í jen o~málo. Pùvodní polynom staèí pøenásobit~$p$, odeèíst první èlen s~$x_1$ a~naopak pøièíst chybìjící èlen $x_{J+1}$. Dostáváme tedy: -$$h(x_2 \dots x_{J+1}) = (p \cdot h(x_1 \dots x_J) - x_1 \cdot p^J + x_{J+1}) \bmod N.$$ -Pøepoèítání hashovací funkce tedy není nic jiného, ne¾ pøenásobení té minulé~$p$, odeètení nìjakého násobku toho znaku, který vypadl z~okénka, a~pøiètení toho znaku, o~který se okénko posunulo. Pokud tedy máme k~dispozici aritmetické operace v~konstantním èase, zvládneme konstantnì pøepoèítávat i~hashovací funkci. - -Tato hashovací funkce se dokonce nejen hezky poèítá, ale dokonce se i~opravdu \uv{hezky} chová (tedy \uv{rozumnì} náhodnì), pokud zvolíme vhodné~$p$. To bychom mìli zvolit tak, aby bylo rozhodnì nesoudìlné s~$N$ -- tedy $\(p, N) = 1$. Aby se nám navíc dobøe projevilo modulo obsa¾ené v~hashovací funkci, mìlo by být~$p$ relativnì velké (lze dopoèítat, ¾e optimum je mezi $2/3 \cdot N$ a~$3/4 \cdot N$). S~takto zvoleným~$p$ se tato hashovací funkce chová velmi pøíznivì a~v~praxi má celý algoritmus takøka lineární èasovou slo¾itost (prùmìrnou). - -\h{Hledání více øetìzcù najednou} -Nyní si zahrajeme tuté¾ hru, ov¹em v~trochu slo¾itìj¹ích kulisách. Podíváme se na~algoritmus, který si poradí i~s více ne¾ jednou jehlou. -Mìjme tedy jehly $\iota_1 \dots \iota_n$, a~jejich délky $J_i = \vert \iota_i \vert $. Dále budeme potøebovat seno $\sigma$ délky $S=\vert \sigma \vert$. - -Pøedtím, ne¾ se pustíme do~vlastního vyhledávacího algoritmu, mo¾ná bychom si mìli ujasnit, co vlastnì bude jeho výstupem. U problému hledání jedné jehly to bylo jasné -- byla to nìjaká mno¾ina pozic v~senì, na~kterých zaèínaly výskyty jehly. Jak tomu ale bude zde? Sice bychom také mohli vrátit pouze mno¾inu pozic, ale my budeme chtít malièko víc. Budeme toti¾ chtít vìdìt i~to, která jehla se na~které pozici vyskytuje. Výstup tedy bude vypadat následovnì: $V = \{(i,j)~\vert~\sigma[i:i+J_j]= \iota_j \}$. - -Zde se v¹ak skrývá jedna drobná zrada. Budeme se asi muset vzdát nadìje, ¾e najdeme algoritmus, jeho¾ slo¾itost je lineární v~celkové délce v¹ech jehel a~sena. Výstup toti¾ mù¾e být del¹í ne¾ lineární. Mù¾e se nám klidnì stát, ¾e na~jedné pozici v~senì se bude vyskytovat více rùzných jehel -- pokud bude jedna jehla prefixem jiné (co¾ jsme nikde nezakázali), tak máme povinnost ohlásit oba výskyty. Vzhledem k~tomu budeme hledat takový algoritmus, který bude lineární v~délce vstupu plus délce výstupu, co¾ je evidentnì to nejlep¹í, èeho mù¾eme dosáhnout. - -Algoritmus, který si nyní uká¾eme, vymysleli nìkdy v~roce 1975 pan Aho a~paní Corasicková. Bude to takové zobecnìní Knuthova-Morrisova-Prattova algoritmu. - -\h{Algoritmus Aho-Corasicková} - -Opìt se budeme sna¾it sestrojit nìjaký vyhledávací automat a~nìjakým zpùsobem tento automat pou¾ít k~procházení sena. Podívejme se nejprve na~pøíklad. Budeme chtít vyhledávat tato slova: |ara|, |bar|, |arab|, |baraba|, |barbara|. Mìjme tedy tìchto pìt jehel a~rozmysleme si, jak by vypadal nìjaký automat, který by tato slova umìl zatím jenom rozpoznávat. Pro jedno slovo automat vypadal jako cesta, zde u¾ to bude strom. (viz obrázek). - -\figure{ara_strom_blank.eps}{Vyhledávací automat -- strom.}{1in} - -Navíc budeme muset do~automatu zanést, kde nìjaké slovo konèí. V~pùvodním automatu pro jedno slovo to bylo jednoduché -- ono jedno jediné slovo odpovídalo poslednímu vrcholu cesty. Tady se v¹ak slova mohou vyskytovat vícekrát a~konèit nejenom v~listech ale i~v~nìjakém vnitøním vrcholu (co¾ se stane tehdy, pokud je jedno hledané slovo prefixem jiného hledaného slova). Formálnì to nebudeme dokazovat, ale snadno nahlédneme, ¾e listy stromu odpovídají hledaným slovùm, ale opaènì to neplatí. - -\figure{ara_strom_end.eps}{Vyhledávací automat s~konci slov.}{1in} - -Dále bychom mìli do~automatu pøidat zpìtné hrany. Jejich definice bude úplnì stejná jako u automatu pro hledání jednoho slova. Jinými slovy z~ka¾dého stavu pùjde zpìtná hrana do~nejdel¹ího vlastního suffixu, který je stavem. Èili kdy¾ budeme mít nìjaké jméno stavu, budeme se ho sna¾it co nejménì (ale alespoò o~znak) zkrátit zleva, abychom zase dostali jméno stavu. Z~koøene -- prázdného stavu -- pak evidentnì ¾ádná zpìtná hrana nepovede. - -\figure{ara_strom_final.eps}{Vyhledávací automat se zpìtnými hranami.}{1,25in} - -Zbývá nám je¹tì si rozmyslet, jakým zpùsobem bude ná¹ automat hlásit výstup. Opìt smìøujeme k~tomu, aby se automat po~pøeètení nìjakého kusu textu nacházel ve~stavu odpovídajícímu nejdel¹ímu mo¾nému suffixu toho textu. Zatímco u hledání jediné jehly bylo hlá¹ení výskytù jednoduché -- kdykoliv jsme se dostali na~konec \uv{automatové cestièky} tady to bude opìt slo¾itìj¹í. - -První, co se nabízí, je vyu¾ít toho, ¾e jsme si oznaèili nìjaké vrcholy, kde hledaná slova konèí. Co tedy zkusit hlásit výskyt tohoto slova v¾dy, kdy¾ pøijdeme do~nìjakého oznaèeného vrcholu? Tento zpùsob v¹ak nefunguje, pokud se uvnitø nìkteré jehly skrývá jehla vnoøená. Napøíklad po~pøeètení slova |bara|, nám ná¹ souèasný automat neøíká, ¾e bychom mìli nìjaké slovo ohlásit, a~pøitom tam evidentì konèí podøetìzec |ara|. Stejnì tak pokud pøeèteme |barbara|, u¾ si nev¹imneme toho, ¾e tam konèí zároveò i~|ara|. Pouhé \uv{hlá¹ení teèek} tedy nefunguje. - -Dále si mù¾eme v¹imnout toho, ¾e v¹echna slova, která by se mìla v~daném stavu hlásit, jsou suffixy jména tohoto stavu. Pøitom víme, ¾e zpìtná hrana jméno stavu zkracuje zleva. Tak¾e speciálnì v¹echny suffixy daného stavu, které jsou také stavy, se dají najít tak, ¾e se vydáme po~zpìtných hranách do~koøene. Nabízí se tedy v¾dy projít cestu po~zpìtných hranách a¾ do~koøene a~hlásit v¹echny \uv{teèky}. Tento zpùsob by nám v¹ak celý algoritmus znaènì zpomalil, proto¾e cesta do~koøene mù¾e být relativnì dlouhá, ale teèek na~ní obvykle bude málo. - -Mohli bychom také zkusit si pro ka¾dý stav $\beta$ pøedpoèítat mno¾inu $cache(\beta)$, která by obsahovala v¹echna slova, která máme hlásit, kdy¾ se ve~stavu $\beta$ nacházíme. Pokud pak do~tohoto stavu vstoupíme, podíváme se na~tuto mno¾inu a~vypí¹eme v¹e, co v~ní je. Výpis nám bude evidentnì trvat lineárnì k~velikosti mno¾iny, celkovì tedy lineárnì k~velikosti výstupu. Problém je ale ten, ¾e jednotlivé cache mohou být hodnì velké, tak¾e je nestihneme sestrojit v lineárním èase. (Rozmyslete si pøíklad slovníku, kdy se to stane.) - -To, co nám ale ji¾ opravdu pomù¾e, bude zavedení zkratek. V¹imli jsme si, ¾e po~zpìtných hranách mù¾eme projít do~koøene a~hlásit v¹echny nalezené teèky. Vadilo nám ale, ¾e se mù¾e stát, ¾e budeme dlouho po~cestì chodit a~pøi tom ¾ádné teèky nenalézat. Zavedeme si proto zkratky k~nejbli¾¹í teèce. - -\s{Definice} (zkratková hrana): -Budeme mít tedy nìjakou funkci $slovo(\beta) :=$ slovo, které konèí ve~stavu $\beta$ (nebo $\emptyset$, pokud ¾ádné takové slovo není). Dále pak funkci $out(\beta) :=$ nejbli¾¹í vrchol dosa¾itelný po~zpìtných hranách, èili nejdel¹í vlastní suffix stavu $\beta$, v~nìm¾ je definovaná funkce $slovo$. Trochu lid¹tìji øeèeno, ten nejbli¾¹í dosa¾itelný vrchol, ve~kterém je teèka. - -Po pøidání tìchto zkratkových hran ji¾ máme reprezentaci, ve~které opravdu umíme v~daném stavu vyjmenovat v¹echna slova, která máme vypsat, a~to v~èase lineárním s~tím, kolik tìch slov je. - -\s{Definice:} -Vyhledávací automat sestává ze stromu dopøedných hran (vrcholy jsou prefixy jehel, hrany odpovídají roz¹íøení o~písmenko), zpìtných hran ($z(\beta) :=$ nejdel¹í vlastní suffix slova $\beta$, který je stavem) a~zkratkových hran. - -Automat pak bude na~na¹em pøíkladu vypadat takto (zkratkové hrany jsou znázornìny zelenì): - -\figure{ara_strom_zkr.eps}{Vyhledávací automat se zkratkovými hranami.}{1,25in} - -Nyní u¾ nám zbývá jenom vlastní algoritmus -- nejdøív popí¹eme algoritmus, který bude hledat pomocí takového automatu, a~potom se pustíme do~toho, jak se takový automat staví. - -Nejprve si nadefinujeme, jak vypadá jeden krok automatu. Bude to vlastnì nìjaká funkce, která dostane stav a~písmenko. Ona nás pak pomocí tohoto písmenka posune po~automatu. ($f(\alpha, x)$ bude dopøedná hrana ze stavu $\alpha$ oznaèená písmenem~$x$) - -\s{Krok ($\alpha$, $x$):} -\algo -\:Dokud $f(\alpha, x) = \emptyset~\&~\alpha \neq \~~\alpha \leftarrow z(\alpha)$. -\:Pokud $f(\alpha, x) \neq \emptyset:~~\alpha \leftarrow f(\alpha, x)$. -\:Vrátíme výsledek. -\endalgo - -\s{Hledání:} -\algo -\:$\alpha \leftarrow \$. -\:Pro znaky $x$ ze slova $\sigma$: -\::$\alpha \leftarrow \(\alpha, x)$. -\::$\beta \leftarrow \alpha$ -\::Dokud $\beta \neq \emptyset$: -\:::Je-li $\(\beta) \neq \emptyset$: -\::::Ohlásíme $\(\beta)$. -\:::$\beta \leftarrow \(\beta)$. -\endalgo - -Algoritmus hledání vlastnì není nic jiného, ne¾ prosté projití po~zelených zkratkových hranách ze stavu $\alpha$, ve~kterém právì jsme, a~ohlá¹ení v¹eho, co po~cestì najdeme. - -V ka¾dém okam¾iku se automat nachází ve~stavu, který odpovídá nejdel¹ímu mo¾nému suffixu toho, co jsme u¾ pøeèetli. Dùkaz tohoto invariantu je stejný jako u verze automatu pro hledání pouze jedné jehly, nebo» vychází pouze z~definice zpìtných hran. Podobnì nahlédneme, ¾e èasová slo¾itost vyhledávací procedury je lineární v~délce sena plus to, co spotøebujeme na~hlá¹ení výskytù. Nejprve na~chvíli zapomeneme, ¾e nìjaké výskyty hlásíme a~spoèítáme jenom kroky. Ty mohou vést dopøedu a~zpátky. Krok dopøedu prodlu¾uje jméno stavu o~jedna, krok dozadu zkracuje aspoò o~jedna. Tudí¾ krokù dozadu je maximálnì tolik, co krokù dopøedu a~krokù dopøedu je maximálnì tolik, kolik je délka sena. V¹echny kroky dohromady tedy trvají $\O(S)$. Hlá¹ení výskytù pak trvá $\O(S~+ \vert V \vert)$. Celé hledání tedy trvá lineárnì v~délce vstupu a~výstupu. - -Zbývá nám u¾ jen konstrukce automatu. Opìt vyu¾ijeme faktu, ¾e zpìtná hrana ze stavu $\beta$ vede tam, kam by se dostal automat pøi hledání $\beta$ bez prvního písmenka. Tak¾e zase chceme nìco, jako simulovat výpoèet toho automatu na~slovech bez prvního písmenka a~doufat v~to, ¾e si vystaèíme s~tou èástí automatu, kterou jsme u¾ postavili. Tentokrát to v¹ak nemù¾eme dìlat jedno slovo po~druhém, proto¾e zpìtné hrany mohou vést køí¾em mezi jednotlivými vìtvemi automatu. Mohlo by se nám tedy stát, ¾e pøi hledání nìjakého slova potøebujeme zpìtnou hranu, která vede do~jiného slova, které jsme je¹tì nezkonstruovali. Tak¾e tento postup sel¾e. Mù¾eme v¹ak vyu¾ít toho, ¾e ka¾dá zpìtná hrana vede ve~stromu alespoò o~jednu hladinu vý¹. Mù¾eme tak strom konstruovat po~hladinách. Lze si to tedy pøedstavit tak, ¾e paralelnì spustíme vyhledávání v¹ech slov bez prvních písmenek a~v¾dycky udìláme jeden podkrok ka¾dého z~tìch hledání, co¾ nám dá zpìtné hrany z~dal¹ího patra stromu. - -\s{Konstrukce automatu:} -\algo -\:Zalo¾íme prázdný strom, $r \leftarrow$ jeho koøen. -\:Vlo¾íme do~stromu slova $\iota_1 \dots \iota_n$, nastavíme $slovo(*)$. -\:$z(r) \leftarrow \emptyset$, $out(r) \leftarrow \emptyset$. -\:Zalo¾íme frontu $F$ a~vlo¾íme do~ní syny koøene. -\:$\forall v~\in F:~~z(v) \leftarrow r, \(v) \leftarrow \emptyset$. -\:Dokud $F \neq \emptyset$: -\::Vybereme $u$ z~fronty $F$. -\::Pro v¹echny syny $v$ vrcholu $u$: -\:::$q \leftarrow \(z(u), \)$. -\:::$z(v) \leftarrow q$. -\:::Pokud $slovo(q) \neq \emptyset$, pak $out(v) \leftarrow q$. -\::::Jinak $out(v) \leftarrow out(q)$. -\:::Vlo¾íme $v$ do~fronty $F$. -\endalgo - -To, ¾e tento algoritmus zkonstruuje zpìtné hrany jak má, vyplývá z~toho, ¾e nedìláme nic jiného, ne¾ ¾e spou¹tíme výpoèty po~hladinách na~v¹echna hledaná slova bez prvního písmenka. Stejnì tak to, ¾e dobìhne v~lineárním èase, je takté¾ dùsledkem toho, ¾e efektivnì spou¹tíme v¹echny tyto výpoèty. Jen nìkdy udìláme najednou krok dvou èi více výpoètù (napøíklad |araba| a~|arbara| se poèítají na~zaèátku, dokud jsou stejné, jen jednou). Èasová slo¾itost této konstrukce je tedy men¹í nebo rovna souètu èasových slo¾itostí výpoètù nad v¹emi tìmi slovy. To u¾ ale víme, ¾e je lineární v~celkové délce tìchto slov. Konstrukce automatu tedy trvá nejvý¹e tolik, co hledání v¹ech $\iota_i$, co¾ je $\O(\sum_{i} \iota_i)$. - -\s{Vìta:} Algoritmus Aho-Corasicková najde v¹echny výskyty v~èase -$$\O\left(\sum_i~\iota_i~+~S~+~\sharp\\right).$$ - -Je¹tì se na~závìr zamysleme, jak bychom si takový automat ukládali do~pamìti. Urèitì se nám bude hodit si stavy nìjak oèíslovat (tøeba v~poøadí, v~jakém budou vznikat). Potom funkce pro zpìtné a~zkratkové hrany mohou být reprezentované polem indexovaným èíslem stavu. Funkce {\I Slovo}, která øíká, jaké slovo ve~stavu konèí, zase mù¾e být pole indexované stavem, které nám øekne poøadové èíslo slova ve~slovníku. Pro dopøedné hrany v~ka¾dém vrcholu pak mù¾eme mít pole indexované písmenky abecedy, které nám pro ka¾dé písmenko øekne, buï ¾e taková hrana není, nebo nám øekne, kam tato hrana vede. Je vidìt, ¾e takovéto pole se hodí pro pomìrnì malé abecedy. U¾ pro abecedu A-Z~bude velikosti 26 a~z~vìt¹iny bude prázdné, tak¾e bychom plýtvali pamìtí. V praxi se proto èasto pou¾ívá hashovací tabulka. Pøípadnì bychom mohli mít i~jen jednu velkou spoleènou hashovací tabulku, která bude reprezentovat funkci celou, ve~které budou zahashované dvojice (stav, písmenko). Tìchto dvojic je evidentnì tolik, kolik hran stromu, èili lineárnì s~velikostí slovníku, a~je to asi nejkompaktnìj¹í reprezentace. - - -\bye diff --git a/old/6-kmp/Makefile b/old/6-kmp/Makefile deleted file mode 100644 index 1831d14..0000000 --- a/old/6-kmp/Makefile +++ /dev/null @@ -1,3 +0,0 @@ -P=6-kmp - -include ../Makerules diff --git a/old/6-kmp/ara_strom.eps b/old/6-kmp/ara_strom.eps deleted file mode 100644 index 7d054ab..0000000 --- a/old/6-kmp/ara_strom.eps +++ /dev/null @@ -1,1334 +0,0 @@ -%!PS-Adobe-3.0 EPSF-3.0 -%%Creator: inkscape 0.46 -%%Pages: 1 -%%Orientation: Portrait -%%BoundingBox: 163 220 443 747 -%%HiResBoundingBox: 163.64601 220.18395 442.55996 746.83829 -%%EndComments -%%Page: 1 1 -0 842 translate -0.8 -0.8 scale -0 0 0 setrgbcolor -[] 0 setdash -1 setlinewidth -0 setlinejoin -0 setlinecap -gsave [1 0 0 1 0 0] concat -gsave [0.5411688 0 0 0.5411688 203.68828 84.001202] concat -0 0 0 setrgbcolor -[] 0 setdash -1.478282 setlinewidth -2 setlinejoin -0 setlinecap 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