]> mj.ucw.cz Git - ads2.git/commitdiff
Nulta verze prednasky o vyhledavani v textu.
authorMartin Mares <mj@ucw.cz>
Mon, 19 Nov 2007 09:53:38 +0000 (10:53 +0100)
committerMartin Mares <mj@ucw.cz>
Mon, 19 Nov 2007 09:53:38 +0000 (10:53 +0100)
6-kmp/6-kmp.tex [new file with mode: 0644]
6-kmp/Makefile [new file with mode: 0644]
6-kmp/vautomat.eps [new file with mode: 0644]
all/Makefile

diff --git a/6-kmp/6-kmp.tex b/6-kmp/6-kmp.tex
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+\input lecnotes.tex
+
+\prednaska{6}{Vyhledávání v textu}{(zapsal K. Ka¹èák, M. Klauèo, M. Vachna)}
+
+\s{Úkol:} V textu najít v¹echny výskyty hledaného slova(hledaných slov).
+
+\h{Hloupý algoritmus}
+
+Algoritmus prochází sekvenènì textem a hledaným vzorovým slovem. Pøi neshodì se ve vzorovem slovì vrací na zaèátek a v textu pokraèuje znakem, v kterém nastala neshoda. Èasová slo¾itost je $\O(S)$, kde $S$ je délka textu. Tento algoritmus funguje pouze jen pro vzorové slová bez opakujících se znakù.
+
+\s{Pøíklad:} Hledání vzorového slova $JEHLA$ v textu $VKUPCEJEJEHLA$. Ve chvíli kdy máme prefix $JE$ a na vstupu dostaneme $J$, dochází k neshodì a pokraèujeme v hledání od tohoto znaku.
+
+\h{Neefektivní algoritmus}
+
+Algoritmus prochází text od zaèátku a¾ do konce a pro ka¾dou pozici v textu zkontroluje, zda na této pozici nezaèíná hledané slovo. Tak pro ka¾dou pozici provede a¾ $S$ porovnání znakù, èili celkem a¾ $SJ$ porovnání. Proto je èasová slo¾itost $\O(SJ)$, kde $S$ je délka textu a $J$ délka vzorového slova.
+
+\h{Chytrý algoritmus}
+
+Algoritmus je vylep¹ením Neefektivního algoritmu, konkretnì zpùsobu jakým sa vrací v textu pøi neshodì mezi znakem textu a
+znakem vzorového slova.
+
+\s{Pøíklad:} Pro vzorové slovo $AJAAJAK$ jsme na¹li v textu prefix $AJAAJA$. Oèakávame $K$.
+\itemize\ibull
+\:Kdy¾ ale dostaneme $A$ a budeme mít prefix $AJAAJAA$, vracíme se v textu za první $AJA$, tedy prefix zkrátíme na $AJAA$ a pokraèujeme v hledání.
+\:Kdy¾ je nasledující znak $J$ a budeme mít prefix $AJAAJAJ$, vracíme se v textu za $AJAAJ$, tedy prefix zkrátíme na $AJ$ a pokraèujeme v hledání.
+\:V pøípadì, ¾e dostaneme jiný znak, v textu se nevracíme a pokraèujeme dal¹ím znakem v textu.
+\endlist
+
+\s{Definice a znaèení pro øetìzce(slová):}
+\itemize\ibull
+\s{Definice:}
+\itemize\ibull
+\:Abeceda $\sum$ je koneèná mno¾ina znakù, s kterých tvoøíme text, øetìzece, slová jako koneèné posloupnosti znakù z $\sum$. Pøíkladem extrémních abeced je lineární abeceda slo¾ená s nul a jednièek. Pøíklad s druhého konce je abecade, která má jako znaky slova èeského jazyka. V algoritmech nebudeme uva¾ovat velikost abecedy (poèet znakù).
+\:$\sum$* je mno¾ina v¹ech slov nad abecedou $\sum$.
+\endlist
+\s{Znaèení:}
+\itemize\ibull
+\:Slová budeme znaèit malými písmenami øecké abecedy $\alpha$,$\beta$... a znaky velkými písmenami latinky $A$,$B$... .
+\:Prázdné slovo znaèíme písmenem $\epsilon$.
+\:Délka slova $\vert \alpha  \vert$ pro $\alpha \in \sum*$ je poèet znakù.
+\:Zøetìzení $\alpha\beta$ vznikne zapsáním slov $\alpha$ a $\beta$ za sebe. Platí $\alpha\epsilon=\epsilon\alpha=\alpha$, $\vert \alpha\beta  \vert=\vert \alpha \vert+\vert \beta \vert$.
+\:$\alpha[i]$ je $i$-té písmeno slova $\alpha$, indexuje se od $0$.
+\:$\alpha[i:j]$ je podslovo tvoøené písmenami $\alpha[i]$,...,$\alpha[j-1]$. Pøíklady: $\alpha[i:i+1]=\alpha[i]$, $\alpha[i:i]=\epsilon$, $\alpha[:]=\alpha$.
+\:$\alpha[:j]$ je prefix obsahující prvních $j$ znakù slova $\alpha$.
+\:$\alpha[i:]$ je suffix obsahující znaky slova $\alpha$ poèínaje $i$-tým znakem.
+\:Ka¾dé slovo je prefixem i suffixem sebe sama, takovému prefixu/suffixu øíkáme vlastní. V¹em ostatním nevlastní.
+\:Prázdné slovo je podslovem, prefixem i suffixem ka¾dého slova vèetnì prázdného slova.
+\endlist
+\endlist
+
+\s{Problém:}
+\itemize\ibull
+\s{IN:}
+\itemize\ibull
+\:$\iota$ slovo (jehla) délky $J=\vert \iota \vert$
+\:$\sigma$ text (seno) délky $S=\vert \sigma \vert$
+\endlist
+\s{OUT:}
+\itemize\ibull
+\:$\left\{ i\vert \sigma[i:i+J]=\iota \right\}$
+\endlist
+\endlist
+
+\h{Vyhledávací automat (Knuth, Morris, Pratt)}
+Vyhledávací automat bude vlastnì graf jeho¾ vrcholy reprezentují stavy. Jména stavù budou v¹echny prefixy slova $\iota$. Poèáteèný stav je prázdny slovo $\epsilon$ a koneèný je samotná $\iota$. Dopøední hrany grafu budú popisovat pøechod mezi stavy v smysle zvìt¹ení délky jména stavu (dopøedná funkce $d(\alpha , X)$), tedy ka¾dá taková hrana bude oznaèena písmenem $X$  a bude popisovat dané zvìt¹ení délky jména stavu, tedy $\alpha \rightarrow \alpha X$. Zpìtné hrany grafu budú popisovat pøechod ( zpìtná funkce $z(\alpha)$) mezi stavem $\alpha$ a nejdel¹ím vlastním suffixem $\alpha$, který je prefixem $\iota$, kdy¾ nastane neshoda.
+
+\figure{vautomat.eps}{Vyhledávací automat}{5.5in}
+
+\s{Vyhledávaní:}
+\algo
+\:$\alpha \leftarrow \epsilon$.
+\:pro $C\in\sigma$ postupnì:
+\:$\indent$dokud $\neg \exists d(\alpha , C) \wedge \alpha\neq\epsilon : \alpha \leftarrow z(\alpha)$
+\:$\indent$dokud $\exists d(\alpha , C) \Rightarrow \alpha \leftarrow d(\alpha , C)$
+\:$\indent$kdy¾ $\alpha = \iota \Rightarrow$ hledané slovo je v textu
+\endalgo
+
+\s{Alternatíva:}
+\algo
+\:$k \leftarrow 0$.
+\:pro $C\in\sigma$ postupnì:
+\:$\indent$dokud $C\neq \iota[k] \wedge k>0: k \leftarrow z(k)$
+\:$\indent$dokud $C=\iota[k] \Rightarrow k++$
+\:$\indent$kdy¾ $k = J \Rightarrow$ hledané slovo je v textu
+\endalgo
+
+\s{Invariant:} Stav po pøeètení vstupu $\beta$. $\alpha(\beta)$ $=$ nejdel¹í suffix $\beta$, který je prefixem $\iota$.
+S invariantu vyplýva korektnost vyhledávací èásti KMP algoritmu.
+
+\proof
+Dùkaz indukcí. Na zaèátku pro prázdny naètený vstup platí invariant, tedy prázdny suffix $\beta$ je prefixem $\iota$. V kroku $n$ máme naètený vstup $\beta$ a k nìmu naèteme znak $C$. Jestli si odmyslíme $C$, tedy kdy¾ si od jména stavu odmyslíme posledné písmenko, dostaneme znovu jméno stavu. Tak stav, který pasuje na konec vstupu bez toho $C$ je stav, který pasuje na konec pùvodního vstupu, toho o jeden znak krat¹ího. Tím pádem to musí být nìco, co je maximálnì tak dlouhé jako pùvodní stav, u kterého jsme byli, proto¾e to byl nejdel¹í, který pasoval. Staèí procházet postupnì v¹echny stavy, které pasují na konec toho vstupu od nejdel¹ího k nejkrat¹ímu a vzít první, který se dá roz¹íøit o $C$. To je pøesnì to, co algoritmus dìlá. Preto¾e zpìtná funkce øekne nejbli¾¹í krat¹í jméné stavu. Tak¾e algoritmus iteruje pøes stavy, které tam pasují, a¾ najde jeden, který se dá roz¹íøit o $C$ a jeliko¾ iteroval od ty nejdel¹í, tak to je logicky ten nejdel¹í, který tam pasuje.
+\qed
+
+\s{Lemma:} Vyhledávaní dobìhne v èase $\O(S)$.
+
+\proof
+Pro ka¾dý znak vstupního textu mohou nastat dva pøípady. Znak roz¹iruje aktuální prefix, nebo musíme pou¾ít zpìtnou funkci(ypìtnou hranu). Roz¹irování trvá konstantnì mnoho èasu, zatímco zpìtná funkce mu¾e být pro jeden znak volána a¾ $J$-krát. Pøi ka¾dém volání klesne délka aktuálního stavu minimálne o jedna a zároven platí, ¾e kdykoliv stav prodlu¾ujeme, roste právì o jeden znak. Proto v¹ech zkrácení dohromady mu¾e být nejvý¹e tolik, kolik bylo v¹ech prodlou¾ení, t.j. kolik jsme pøeèetli znaku textu. Celkem je tedy poèet krokù lineární vzhledem k délce textu.
+\qed
+
+\s{Konstrukce zpìtné funkce:}
+\algo
+\:sestrojíme dopøedné hrany
+\:$z( \epsilon ) \leftarrow 0$, $z( \iota [0]) \leftarrow \epsilon $
+\:$\indent$ $\alpha \leftarrow \epsilon$
+\:pro $i = 1$ do $J$
+\:$\indent$$\alpha \leftarrow krok( \alpha , \iota [i])$
+\:$\indent$$z( \iota [0:i+1]) \leftarrow \alpha$
+\endalgo
+
+\s{Vysvìtlení:} V¹imnìte si, ¾e $z(i)$ je pøesnì stav, do nej¾ se dostaneme pøi spu¹tìní na¹eho vyhledávacího algoritmu na øetìzec $\iota [2:i]$, èili na $i$-tý prefix bez prvního písmenka. Proè to tak je? Zpìtná funkce øíká, jaký je nejdel¹í vlastní suffix daného stavu, který je také stavem, zatímco $\alpha$ oznaèuje nejdel¹í suffix textu, který je stavem. Tyto dvì vìci se pøeci li¹í jen v tom, ¾e ta druhá pøipou¹tí i nevlastní suffixy, a právì tomu zabráníme odstranìním prvního znaku. Tak¾e $z()$ získáme tak, ¾e spustíme vyhledávání na èást samotného slova $\iota$. Jen¾e k vyhledávání zase potøebujeme zpìtnou funkci $z$. Proto budeme zpìtnou funkci vytváøet postupne od nejkrat¹ích prefixu. Zøejmì $z(1) = \epsilon$. Pokud ji¾ máme $z(i)$, pak výpoèet $z(i+1)$ odpovídá spu¹tení automatu na slovo délky i a pritom budeme zpìtnou funkci potøebovat jen pro stavy délky $i$ nebo men¹í, pro které ji ji¾ máme hotovou.
+
+Navíc nemusíme pro jednotlivé prefixy spou¹tìt výpoèet v¾dy znovu od zaèátku, proto¾e $(i+1)$-ní prefix
+je prodlou¾ením $i$-tého prefixu o jeden znak. Staèí tedy spustit algoritmus na celý øetìzec $\iota[1:J]$ a sledovat, jakými stavy bude procházet. A to budou pøesnì hodnoty zpìtné funkce. Vytvoøení zpìtné funkce se tak nakonec zredukovalo na jediné vyhledávání v textu o délce $J-1$, a proto pobe¾í v case $\O(J)$. Èasová slo¾itost celého algoritmu tedy bude $\O(S+J)$.
+
+\h{Algoritmus Rabin \& Karp}
+Tenhle algoritmus funguje tak, ¾e porovnává hash hledaného øetìzce s hashem aktuálního podøetìzce v textu a aktuální podøetìzec porovná se vzorkem pouze v pøípadì, kdy¾ mají shodný hash. Kdy¾ si zvolíme tu správnou hashovací funkci, budeme moci vypoèítat hash následujíciho podøetìzce na základe hashe toho aktuálního. Jako hashovací funkci $h: \sum^J \rightarrow Z$ pou¾ijeme následující: $h(x_{0},...,x_{j-1}) = ( \sum_{i=1}^J x_{i}.p^{J-i})$ $mod$ $N$, kde $N$ je velikost prostoru do kterého hashujeme. Jak zjistíme hash následujícího podøetìzce?
+\itemize\ibull
+\:$h = x_{0}.p^{J} + x_{1}.p^{J-1} + ... + x_{J-1}.p^{1}$
+\:$h1 = x_{1}.p^{J} + x_{2}.p^{J-1} + ...      + x_{J}.p^{1}$
+\:$h1 = (h - x_{0}.p^{J}).p + x_{J}.p^{1}$
+\endlist
+Èasová slo¾itost je v nejlep¹ím pøípadì lineární vzhledem k délce textu, zatímco nejhor¹í pøípad mú¾e trvat a¾ $\O(JS)$.
+
+\bye
diff --git a/6-kmp/Makefile b/6-kmp/Makefile
new file mode 100644 (file)
index 0000000..1831d14
--- /dev/null
@@ -0,0 +1,3 @@
+P=6-kmp
+
+include ../Makerules
diff --git a/6-kmp/vautomat.eps b/6-kmp/vautomat.eps
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index 0000000..a2f4fa3
--- /dev/null
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index 881976e1cf671668ee2dcb7c6a97c01bc65995e8..c32fc18f66e03394e0af1291e9a872de7a216a95 100644 (file)
@@ -1,5 +1,5 @@
 P=ads2
-X:=$(shell for a in 1 2 3 4 ; do echo ../$$a-*/$$a-*.tex ; done)
+X:=$(shell for a in 1 2 3 4 ; do echo ../$$a-*/$$a-*.tex ; done)
 
 %universe: all ChangeLog