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2006.03.24

suffix arraysやいろいろ

suffix arraysの話は半年置きぐらいに書いているのかなぁ。
(ココログ全文検索機能無くて、不便ですね・・以前どこに書いたのか分からない。)

私が以前書いたSuffix Arraysの構築方法の記事が古くなってきたので(分かりにくいし)、近いうちにライブラリと一緒に内容も更新しようかなと。今回は、忘れないうちにメモも兼ねてSuffix Arraysの高速な構築方法について。

構築で今一番速いのは、msufsortimproved two-stage (プログラム名はdivsufsort)(its)法だと思います。これらはデータサイズに対して線形時間で構築できる方法では無いのですが、大抵のデータでは線形時間の方法より高速に構築することが可能です。

msufsortはsuffix arrays:SAを直接構築するのではなく、その逆の値であるinverted suffix arrays:ISA を構築します(SA[i] = k ⇔ ISA[k] = i  が成り立つ)。なぜISAを作るかというと、SA構築時にすることは suffix の順位を決定することなのですが、この決定の際、ISAが(途中まででも)分かっていれば直接文字列同士を比較せずにISA同士を比較して順位が決定できるので、文字列の順位を少ない比較回数で決定することができます。あとはtwo-stage法と同様に、ある条件を満たしたsuffixの順位は他のsuffixの順位から直接求められる性質を利用し高速化をはかっています。
# qsufsortも同じようにISAを構築する方法とみることができますがqsufsortは全体のISAを更新していくのに対し、ISAは一部分のsuffixから順にISAを決定していきます(そしてソートせずに直接順位が決定できるところはソートせずにとっておく)。詳しくはmsufsort1.0の解説を読んでください。

itsはtwo-stage法の変種で最近提案された方法です(もしかしたら前に誰かやったかもしれませんが・・)。Suffix Ti = T[i...n] 集合 i=0...nが与えられた時、SuffixをAとBの二種類に分類します。AはTi > Ti+1 Bは Ti < Ti+1 であり、ここの不等号は辞書式順序です(Tの最後には$という他のどの文字より小さい文字を置いておき必ず順序が決定できるようにしてある)。AかBのSuffix Arrays中の順位が決定できれば、残り片方の順位は直接(線形時間で)求められるのですが、ITSでは更にBのSuffix Tiの中でTi+1がAであるSuffix を B*とし、B*のSuffixだけをソートします。B*がソートし終わったら残りBの順位を直接求め、そしてAの順位を求めます。B*であるsuffix の割合は全Suffixの大体1/3程度なので、大雑把にいって3倍高速に計算できます。
moriさんによる実装では更に、B*のソートを工夫しています。まずko-alulu法と同様に、まずB*のSuffixの先頭毎にTを分割し、それら各部分列の順位を決定した後に、その順位を新しい文字集合とみなして新しい文字列T'を構成し、それをlarsson - sadakane法(いわゆるqsufsort)でソートします。全体のアルゴリズムでも、テキスト長がnの時5n byteしか消費しないlight-weightな方法です。

これだけいろいろな方法がでてきても、まだまだ速度は要求されます。自分で実装したところではどちらも大体1秒あたり2MBぐらい処理できるぐらいで、これは昔から比べればはるかに速いのですが、巨大なデータを処理したい場合には結構またされます。これ以上の高速化を図ろうとしたら、データの冗長性(k次エントロピーの意味で)を利用するのが一つかなぁとぼんやり考えています。

圧縮されたCSAもしくはwavelet treeを直接構築する方法もでてきており、それらを高速に処理する方法もいろいろ提案されていますが、それはまたの機会に。

---
他に見つけたおもしろそうな論文
"Statistical Encoding of Succinct Data Structure", Rodrigo Gonzalez, Gonzalo Navarro. TR/DCC-2006-1 [ps.gz]

SODA 2006で出た"Squeezing Succinct Data Structures into Entropy Bounds", K. Sadakane and R. Grossi: が実現している任意の位置からの瞬時復元(定数時間)をLZ78を使わずに通常の算術圧縮とかで実現したよという話。私がDCC2005で発表したStatic PPMに似ている。ちゃんと書けばこう化けるのね・・むむむ

同じところでついでにみつけた
"Practical Constructuion of k Nearest Neighbor Graphs in Metric Spaces", Rodrigo Paredes, Gonzalo Navarro, TR/DCC-2005-6 [ps.gz] も面白そう。詳しくは読んでいないが、高次元で高速なk-NNは欲しかったところ。

これらを読んでいて面白いのがtheを使うところをdeと間違っているところ。英語はどこの国でも難しいよね。

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Comments

おひさしぶりです。群大のSです。

ちょうどゼミで suffix array の最近の動向として、MSufSort や ITS の話題で発表しようとしていたところです。
昨年の今ごろでしたか、三分割法についてうんぬんというお話もありましたが、こちらはさっぱり進展がありませぬ。

Posted by: S | 2006.03.31 09:54 AM

お久しぶりです。

私は最近省メモリ(0次エントロピー)なSA構築(またはbwt、もしくはそれのwavelet tree)を中心にやっています。面白い方法が思いついたところでそれを試しているところです。

two-stageから始まったソートが必要な部分を絞る方法の流れも最終的にはmsufsortのように必要な範囲を動的に決定する方法落ち着くのでしょうかね。それともまだ最初に調べるときにITSのよりうまい方法があるのでしょうか。

個人的にはうまいwavelet treeの更新の方法があればインクリメンタルにSAを作る方法でも結構高速になるのではないかと思っています。

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