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2009.04.25

部分文字列の話

ここしばらく、部分文字列の統計量を利用した機械学習やデータマイニングをやっている。そこの話からちょっと抜粋。

長さnの文字列T[1,...,n]が与えられた時、T中に出現する部分文字列T[i...j] (1≦i≦j≦n)の数はn個の中からiとjの2箇所を選ぶのでO(n^2)個ある。例えば、n=10^6(1MB)だったら、部分文字列の数は約10^12個(1T)と非常に大きい。

しかし、これらの部分文字列の出現位置は同じである場合が多い。例えばT="abracadabra"であれば、"abra"と"abr"の出現場所は1番目と8番目であり、全く同じである。

では出現位置(部分文字列の左端を出現位置とする)が全く同じであるような部分文字列をまとめてグループにした場合、グループの数はいくつになるのだろうか。

これは接尾辞木(wikipedia 授業の資料)を知っているなら簡単に説明できる。

Tに対しT[i,...,n]、つまり頭の文字を削った文字列、をTの接尾辞と呼ぶ。この接尾辞から構築したtrieに対し、子が一つであるような節点を文字を残したまま取り除いて枝にしてしまったものが接尾辞木である(正確には接尾辞木のスケルトン。実際にはsuffix linkとかつける)。各葉には、接尾辞の出現位置を記録しておく。

この接尾辞木において、節点に対応する部分文字列の出現位置は、その節点の子孫の葉の集合に対応する。枝の途中に対応する部分文字列の出現位置も、その先の節点の出現位置と同じであり、同じ枝に対応する部分文字列の出現位置は同じである。

この接尾辞木の節点の数は、接尾辞を一個ずつ加えて接尾辞木を構築する過程を考えると、各追加で、高々一つしか節点は増えないので、高々n-1個である。

一回しか出現していない部分文字列のグループは、接尾辞木中の葉に隣接する枝に対応し、二回以上出現している部分文字列のグループは節点と節点の間の枝に対応している。葉の数は接尾辞の数と同じなのでn個。よって、グループの数は(葉の数)+(節点の数) = 2n-1、二回以上出現しているグループの数はn-1個と分かる。

最初の問いに答えると、部分文字列の数はn^2なのに、それらをまとめると高々n個しかない。ものすごく少ないことがわかる。

さらに、これらの極大部分文字列は左側に伸ばした場合の冗長も考えるとさらにまとめることができる。(たとえば先ほどの例はbraとabraの出現位置もおなじ)。

こうすると例えば、さっきの例のTにおける極大部分文字列で二回以上出現しているものはaとabraのみである。

では、実際にこれらのグループに対応する部分文字列(極大部分文字列)を取り出すにはどうすればいいのか。接尾辞木を構築するのが一番直感的だが、接尾辞木のサイズは非常に大きい。

現在では接尾辞配列と共通接頭辞配列を組み合わせて、線形時間で求めるのが一番手っ取り早い。
接尾辞配列の線形時間はSAISとか使えばできて、線形時間での共通接頭辞配列構築と、それを利用した極大部分文字列の列挙は[pdf]中の論文の疑似コードをそのまま書けばできる。左側の冗長性に関してはBWT配列に関するrankを使えば全部トータルでも線形時間で極大な部分文字列を求めることができる(ちょっと面倒なのでやらなくてもいいが)。

これでテキスト長が10億文字ぐらいになっても10分程度で全ての極大な部分文字列を抜き出すことができる。

出現位置がまとめられたらいろいろ使いどころはある。例えば、私が今やっている全ての部分文字列を利用した文書分類では、全ての文書をつなげた文字列から極大部分文字列を抽出し、そこから各部分文字列に関する統計量(尤度の微分とか)を効率良く計算することができている。

ちょっと毛色の違う例として、例えば、アクセスログのパターンを解析をする場合を考えてみる。ログデータとして、アクセスしたサイトの記録がセッション毎に残されているとする。各セッション中に含まれるサイト数の平均が約百で、サイトの総種類数が1万、セッションが1000万回分記録されているとする。ここから10回以上出現しているアクセスパターン(サイト列)を、そのパターンの長さを問わずに抽出しろというお仕事があったとする。

見積もろうとすると、まず全通りのパターンを力づくで記録するのは、長さ2のアクセスパターンで1万×1万=1億通りのパターンがでる可能があるので長さ10とかはまず無理である。
ちょっと考えて、実際に出現したパターンをハッシュとかで記録して求めようとすると、各セッション毎に一万個(百×百)のアクセスパターンを記録する必要があり、合計で1000億個のアクセスパターンを記録する必要がある。とてもメモリに載らないので10台ぐらい必要、もしくはDBとかでガリガリで1日ぐらいかかるのではないかということになる。まず長さを切ってしまうだろう。

この場合のスマートな回答は、全てのセッションをつなげた長い配列を作り、そこから極大部分文字列を抽出し、10回以上出現する極大部分文字列のみを報告すればよい(極大部分文字列に含まれる部分文字列は全て頻出)。このときセッションを跨いだパターンを抽出しないようにセッション間には特殊文字をいれておけばよい。これらの特殊文字は全て違うものにしといてもよい。こうすると文字種類数も数千万となるが今の接尾辞配列構築アルゴリズムの計算量はアルファベット数には依存しない。1台のマシンでしかも10分程度で漏れなくパターンが抽出できますよといえば、驚かれるかもしれない
(実際できる見積もりを、そのまま言っていいかどうかは、また考える必要があるが)。

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