Definitions and trie
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Antti H S Laaksonen
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luku26.tex
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luku26.tex
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\chapter{String algorithms}
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\index{merkkijono@merkkijono}
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\index{aakkosto@aakkosto}
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\index{string}
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\index{alphabet}
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Merkkijonon $s$ merkit ovat $s[1],s[2],\ldots,s[n]$,
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missä $n$ on merkkijonon pituus.
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A string $s$ of length $n$
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is a sequence of characters
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$s[1],s[2],\ldots,s[n]$.
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\key{Aakkosto} sisältää ne merkit,
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joita merkkijonossa voi esiintyä.
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Esimerkiksi aakkosto $\{\texttt{A},\texttt{B},\ldots,\texttt{Z}\}$
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sisältää englannin kielen suuret kirjaimet.
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An \key{alphabet} is a set of characters
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that may appear in strings.
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For example, the alphabet
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$\{\texttt{A},\texttt{B},\ldots,\texttt{Z}\}$
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consists of the capital letters of English.
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\index{osajono@osajono}
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\index{substring}
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\key{Osajono}
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sisältää merkkijonon merkit
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yhtenäiseltä väliltä.
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Merkkijonon osa\-jonojen määrä on $n(n+1)/2$.
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Esimerkiksi merkkijonon \texttt{ALGORITMI}
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yksi osajono on \texttt{ORITM},
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joka muodostuu valitsemalla välin \texttt{ALG\underline{ORITM}I}.
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A \key{substring} consists of consecutive
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characters in a string.
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The number of substrings in a string is $n(n+1)/2$.
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For example, \texttt{ORITH} is a substring
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in \texttt{ALGORITHM}, and it corresponds
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to \texttt{ALG\underline{ORITH}M}.
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\index{alijono@alijono}
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\index{subsequence}
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\key{Alijono}
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on osajoukko merkkijonon merkeistä.
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Merkkijonon alijonojen määrä on $2^n-1$.
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Esimerkiksi merkkijonon \texttt{ALGORITMI}
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yksi alijono on \texttt{LGRMI}, joka muodostuu
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valitsemalla merkit \texttt{A\underline{LG}O\underline{R}IT\underline{MI}}.
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A \key{subsequence} is a subset of characters
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in a string in their original order.
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The number of subsequences in a string is $2^n-1$.
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For example, \texttt{LGRHM} is a subsequece
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in \texttt{ALGORITHM}, and it corresponds
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to \texttt{A\underline{LG}O\underline{R}IT\underline{HM}}.
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\index{alkuosa@alkuosa}
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\index{loppuosa@loppuosa}
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\index{prefiksi@prefiksi}
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\index{suffiksi@suffiksi}
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\index{prefix}
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\index{suffix}
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\key{Alkuosa} on merkkijonon
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alusta alkava osajono,
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ja \key{loppuosa} on merkkijonon
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loppuun päättyvä osajono.
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Esimerkiksi merkkijonon \texttt{KISSA}
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alkuosat ovat \texttt{K}, \texttt{KI},
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\texttt{KIS}, \texttt{KISS} ja \texttt{KISSA}
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ja loppuosat ovat \texttt{A}, \texttt{SA},
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\texttt{SSA}, \texttt{ISSA} ja \texttt{KISSA}.
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Alkuosa tai loppuosa on \key{aito},
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jos se ei ole koko merkkijono.
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A \key{prefix} is a subtring that contains the first
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character of a string,
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and a \key{suffix} is a substring that contains the last character.
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For example, the prefixes of
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\texttt{STORY} are \texttt{S}, \texttt{ST},
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\texttt{STO}, \texttt{STOR} and \texttt{STORY},
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and the suffixes are \texttt{Y}, \texttt{RY},
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\texttt{ORY}, \texttt{TORY} and \texttt{STORY}.
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A prefix or a suffix is \key{proper}
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if it is not the whole string.
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\index{kierto@kierto}
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\index{rotation}
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\key{Kierto} syntyy
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siirtämällä jokin alkuosa merkkijonon loppuun
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tai jokin loppuosa merkkijonon alkuun.
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Esimerkiksi merkkijonon \texttt{APILA}
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kierrot ovat
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\texttt{APILA},
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\texttt{PILAA},
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\texttt{ILAAP},
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\texttt{LAAPI} ja
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\texttt{AAPIL}.
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A \key{rotation} can be generated by moving
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characters one by one from the beginning to the end
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in a string (or vice versa).
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For example, the rotations of \texttt{STORY} are
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\texttt{STORY},
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\texttt{TORYS},
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\texttt{ORYST},
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\texttt{RYSTO} and
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\texttt{YSTOR}.
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\index{jakso@jakso}
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\index{period}
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\key{Jakso} on alkuosa,
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jota toistamalla merkkijono muodostuu.
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Jakson viimeinen toistokerta voi olla osittainen
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niin, että siinä on vain jakson alkuosa.
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Usein on kiinnostavaa selvittää, mikä on merkkijonon
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\key{lyhin jakso}.
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Esimerkiksi merkkijonon \texttt{ABCABCA} lyhin jakso on \texttt{ABC}.
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Tässä tapauksessa merkkijono syntyy toistamalla jaksoa ensin kahdesti kokonaan
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ja sitten kerran osittain.
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A \key{period} is a prefix of a string such that
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we can construct the string by repeating the period.
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The last repetition may be partial and contain
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only a prefix of the period.
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Often it is interesting to find the \key{shortest period}
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of a string.
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For example, the shortest period of
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\texttt{ABCABCA} is \texttt{ABC}.
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In this case, we first repeat the period twice
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and then partially.
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\key{Reuna} on
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merkkijono, joka on sekä
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alkuosa että loppuosa.
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Esimerkiksi merkkijonon \texttt{ABADABA}
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reunat ovat \texttt{A}, \texttt{ABA} ja
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\texttt{ABADABA}.
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Usein halutaan etsiä \key{pisin reuna},
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joka ei ole koko merkkijono.
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\index{border}
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\index{leksikografinen jxrjestys@leksikografinen järjestys}
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A \key{border} is a string that is both
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a prefix and a suffix of a string.
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For example, the borders for \texttt{ABADABA}
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are \texttt{A}, \texttt{ABA} and \texttt{ABADABA}.
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Often we want to find the \key{longest border}
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that is not the whole string.
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Merkkijonojen vertailussa käytössä on yleensä
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\key{leksikografinen järjestys}, joka vastaa aakkosjärjestystä.
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Siinä $x<y$, jos joko $x$ on $y$:n aito alkuosa
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tai on olemassa kohta $k$ niin,
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että $x[i]=y[i]$, kun $i<k$, ja $x[k]<y[k]$.
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\index{lexicographical order}
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\section{Trie-rakenne}
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Usually we compare string using the \key{lexicographical order}
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that corresponds to the alphabetical order.
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It means that $x<y$ if either $x$ is a proper prefix of $y$,
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or there is an index $k$ such that
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$x[i]=y[i]$ when $i<k$ and $x[k]<y[k]$.
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\index{trie@trie}
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\section{Trie structure}
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\key{Trie} on puurakenne,
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joka pitää yllä joukkoa merkkijonoja.
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Merkkijonot tallennetaan puuhun
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juuresta lähtevinä merkkien ketjuina.
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Jos useammalla merkkijonolla on sama alkuosa,
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niiden ketjun alkuosa on yhteinen.
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\index{trie}
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Esimerkiksi joukkoa
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$\{\texttt{APILA},\texttt{APINA},\texttt{SUU},\texttt{SUURI}\}$
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vastaa seuraava trie:
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A \key{trie} is a tree structure that
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maintains a set of strings.
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Strings are stored in a trie as chains
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of characters that start at the root
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of the tree.
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If two strings have a common prefix,
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they also share a chain in the tree.
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For example, the following trie corresponds
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to the set
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$\{\texttt{CANAL},\texttt{CANDY},\texttt{THE},\texttt{THERE}\}$:
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\begin{center}
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\begin{tikzpicture}[scale=0.9]
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@ -120,50 +119,51 @@ vastaa seuraava trie:
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\node[draw, circle] (12) at (1.5,14.5) {$\phantom{1}$};
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\node[draw, circle] (13) at (1.5,13) {*};
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\path[draw,thick,->] (1) -- node[font=\small,label=\texttt{A}] {} (2);
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\path[draw,thick,->] (1) -- node[font=\small,label=\texttt{S}] {} (3);
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\path[draw,thick,->] (2) -- node[font=\small,label=left:\texttt{P}] {} (4);
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\path[draw,thick,->] (4) -- node[font=\small,label=left:\texttt{I}] {} (5);
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\path[draw,thick,->] (5) -- node[font=\small,label=left:\texttt{L}] {} (6);
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\path[draw,thick,->] (5) -- node[font=\small,label=right:\texttt{N}] {} (7);
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\path[draw,thick,->] (6) -- node[font=\small,label=left:\texttt{A}] {}(8);
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\path[draw,thick,->] (7) -- node[font=\small,label=right:\texttt{A}] {} (9);
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\path[draw,thick,->] (3) -- node[font=\small,label=right:\texttt{U}] {} (10);
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\path[draw,thick,->] (10) -- node[font=\small,label=right:\texttt{U}] {} (11);
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\path[draw,thick,->] (1) -- node[font=\small,label=\texttt{C}] {} (2);
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\path[draw,thick,->] (1) -- node[font=\small,label=\texttt{T}] {} (3);
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\path[draw,thick,->] (2) -- node[font=\small,label=left:\texttt{A}] {} (4);
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\path[draw,thick,->] (4) -- node[font=\small,label=left:\texttt{N}] {} (5);
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\path[draw,thick,->] (5) -- node[font=\small,label=left:\texttt{A}] {} (6);
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\path[draw,thick,->] (5) -- node[font=\small,label=right:\texttt{D}] {} (7);
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\path[draw,thick,->] (6) -- node[font=\small,label=left:\texttt{L}] {}(8);
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\path[draw,thick,->] (7) -- node[font=\small,label=right:\texttt{Y}] {} (9);
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\path[draw,thick,->] (3) -- node[font=\small,label=right:\texttt{H}] {} (10);
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\path[draw,thick,->] (10) -- node[font=\small,label=right:\texttt{E}] {} (11);
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\path[draw,thick,->] (11) -- node[font=\small,label=right:\texttt{R}] {} (12);
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\path[draw,thick,->] (12) -- node[font=\small,label=right:\texttt{I}] {} (13);
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\path[draw,thick,->] (12) -- node[font=\small,label=right:\texttt{E}] {} (13);
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\end{tikzpicture}
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\end{center}
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Merkki * solmussa tarkoittaa,
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että jokin merkkijono päättyy kyseiseen solmuun.
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Tämä merkki on tarpeen,
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koska merkkijono voi olla toisen merkkijonon alkuosa,
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kuten tässä puussa merkkijono \texttt{SUU}
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on merkkijonon \texttt{SUURI} alkuosa.
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The character * in a node means that
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a string ends at the node.
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This character is needed because a string
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may be a prefix of another string.
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For example, in this trie, \texttt{THE}
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is a suffix of \texttt{THERE}.
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Triessä merkkijonon lisääminen ja hakeminen
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vievät aikaa $O(n)$, kun $n$ on merkkijonon pituus.
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Molemmat operaatiot voi toteuttaa lähtemällä liikkeelle juuresta
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ja kulkemalla alaspäin ketjua merkkien mukaisesti.
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Tarvittaessa puuhun lisätään uusia solmuja.
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Inserting and searching a string in a trie take $O(n)$ time
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where $n$ is the length of the string.
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Both operations can be implemented by
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starting at the root node and following the
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chain of characters that appear in the string.
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If needed, new nodes will be added to the trie.
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Triestä on mahdollista etsiä
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sekä merkkijonoja että merkkijonojen alkuosia.
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Lisäksi puun solmuissa voi pitää kirjaa,
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monessako merkkijonossa on solmua vastaava alkuosa,
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mikä lisää trien käyttömahdollisuuksia.
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Trie can be used for searching both strings
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and prefixes of strings.
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In addition, we can keep track of the number
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of strings that have each prefix,
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that can be useful in some applications.
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Trie on kätevää tallentaa taulukkona
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A trie can be stored as an array
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\begin{lstlisting}
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int t[N][A];
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\end{lstlisting}
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missä $N$ on solmujen suurin mahdollinen määrä
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(eli tallennettavien merkkijonojen yhteispituus)
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ja $A$ on aakkoston koko.
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Trien solmut numeroidaan $1,2,3,\ldots$ niin,
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että juuren numero on 1,
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ja taulukon kohta $\texttt{t}[s][c]$ kertoo,
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mihin solmuun solmusta $s$ pääsee merkillä $c$.
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where $N$ is the maximum number of nodes
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(the total length of the string to be stored)
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and $A$ is the size of the alphabet.
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The nodes of a trie are numbered
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$1,2,3,\ldots$ so that the number of the root is 1,
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and $\texttt{t}[s][c]$ is the next node in chain
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from node $s$ using character $c$.
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\section{Merkkijonohajautus}
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