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2006.12.28

疎なbit arrayに対する現実的なRank/Selectデータ構造

久しぶりに研究紹介。

最近書いた論文です。
"Practical Entropy-Compressed Rank/Select Dictionary.", Daisuke Okanohara and Kunihiko Sadakane.
In the Proc. of ALENEX 2007, to appear[paper]

この論文では、疎なbit array B[0...n-1]に対するrank/select操作を高速に実現するデータ構造を4つ提案しています。 rank(x)はB[0...x]中にある1の数を返し、select(x)はx番目の1の位置を返す関数です。
古典的な問題ですが、これを最小(entropyに近い限界)かつ高速(定数時間、もしくはそれに近い時間)で実現する現実的なデータ構造は存在しませんでした。これを4つの違ったデータ構造で実現したというものです。
このデータ構造はsuccinct data structureを構成する上で不可欠なもので、今まで各データ構造毎に、疎なbit arrayを効率よく保存する方法がデータ構造の特徴を利用することで提案されていたのが、これを使えば統一的に解決されます。

実際どう作って、どういう特徴があるかはペーパーを参照してください。

このデータ構造はそれ以外にも使える場面があって、例えば、単調増加列の整数列を保存する場合も、その整数がある場合にはその位置のbitは1、それ以外が0であるようなbit vectorで保存すれば、selectを使って整数をランダムアクセスできます。
具体的に転置ファイルの例では、0番から100万番まで文書番号があって、ある単語が出ている文書番号を小さい順に保存する場合を考えたとき、この単語が6万文書で出現しているとしたら、これを保存するのにはそのままでは60000*32bit = 240KByteかかるのですが、提案したデータ構造(この用途にはsdarrayがいいかな)を使うと60000 * (2 + 4)bit = 45K Byteぐらいで保存できます。さらに二つの整数列のintersectionをとることもできます(i番目の位置のbitを取得する操作はrankの操作と同様にしてできる)。

また、疎な行列(やグラフ)を保存するのにも使えて、今いろいろ試しているところです

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Comments

以前にメッセンジャーで話していた内容だね。僕の研究テーマにもとっても有効に使えそうな気がする。論文読んでみます。

Posted by: れお | 2006.12.30 at 11:20 AM

そうですね。グローバルポインタを置き換えることができるのでいろいろ用途はあるかなと。
更新をサポートするのは、一から作り直してもコストは小さいので、たまに作り直すという戦略で実装上できるとは思います。もっと賢くやりたいけど、効率よくというのは単純ではないですね

Posted by: oka | 2006.12.31 at 10:38 AM

ふふふ。最近、indexを更新しなくてもよくなる手法を編み出したので、更新をサポートしなくても利用価値があると思っているのです。データベース業界でも、ディスク容量が大きくなってきたので、in-place updateをしないで、subversionのようにincrementalにデータを追加する風潮がでてきています。

Posted by: れお | 2007.01.01 at 02:41 PM

gfsみたく追加だけサポートするのが、現実の問題に多いんでしょうねぇ。
インククリメンタルに追加のみサポートする上で興味があるのが、常にメモリ上に展開できる一定のサイズのみをのせられるように要約した形でインデクスを保持する方法ですかね。そのためには、インデクスにも確率的な手法を利用する必要があると思っています。それを定式化して性能を保証できるとかがかっこよさそうだな

Posted by: oka | 2007.01.03 at 01:04 AM

あけましておめでとうございます。

私の方は、いろいろ紆余曲折した結果、オンライン構築可能な FM-index と CSA のあいのこのようなものを考えるに至りました。
これには rank/select だけでなく、insert/delete もサポートする動的辞書が必要なのですが、O(n)ビット以下の手法は極端に複雑になってしまいますな…

CPM2007 に出してみたいところですが、データ構造まで作らないと主張が弱いので、締め切りまでに何とかならないかと頭をひねる毎日です。

Posted by: S | 2007.01.05 at 10:39 AM

あけましておめでとうございます。
insert/deleteが可能なrank/select dictionaryは"Dynamic Entropy-Compressed Sequences and Full-Text Indexes",などもありますが実際に効率的な実装ができるかはわからないですね

各nodeが1kBぐらいのbit arrayを管理していて全体は木で操作を行うものとかは昔作ったことがありますがupdateの割合が多いとかなり遅いです。bit arrayが疎なら1の位置を保存した普通の2分木とかでもよさそうなきがします。

Posted by: oka | 2007.01.07 at 09:57 PM

私が試しに作ったのは、"Dynamic Entropy-Compressed ..." の Section 3.2 で述べられている n+o(n) ビット法を Lippert にならって 64bits leaf で表現したものです。Lippert の実装と同程度メモリで9倍高速でした。ある種の富豪プログラミングになっているからかなぁ、と思いますが、検証の暇もなく。

ここから nH 対応の解決をはかるために、Makinenらと方向性を変えて、先日はDO++さんの論文やプログラムを求めた次第です。

もう1つの参考文献は、 Gupta, Hon, Shah and Vitter の "Dyamic rank/select dictionaries with applications to XML indexing" で、こちらは B-tree をベースに葉に bit array の辞書を使うといったところでしょうか。XML 対応ということで、実際にも速そうで期待できます。

ALENEX の面白そうな話も面白そうなものばかりですネ。

Posted by: S | 2007.01.10 at 09:25 PM

後者のは初耳です。読んでみます。
速度に関して言えばキャッシュミスをいかに減らせるかと書き込みをいかに局所的に抑えるかが計算回数よりも重要ですね。読み込みだとプリフェッチが効果的であれば、キャッシュミスはほとんどおきないという話で、(alenexでもそのような話がありました)必ずしも局所的でなくてもいいようです。

動的なデータ構造だと(完全)ハッシュの話とかランダマイズドアルゴリズムとかが重要だと思うのですが、succinct data structureでそういう話があまりでてこないので、そういうところにもうちょっとつっこんでみてもいいかなと最近思ってます

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