Paweł Gawrychowski

A faster grammar-based self-index

To store and search genomic databases efficiently, researchers have recently started building compressed self-indexes based on straight-line programs and LZ77. In this paper we show how, given a balanced straight-line program for a string S[1..n] whose LZ77 parse consists of z phrases, we can add O(z log log z) words and obtain a compressed self-index for S such that, given a pattern P [1..m], we can list the occ occurrences of P in S in O(m2 + (m + occ) log log n) time. All previous self-indexes are either larger or slower in the worst case.

LZ77‐based Self‐indexing with Faster Pattern Matching

To store and search genomic databases efficiently, researchers have recently started building self-indexes based on LZ77. As the name suggests, a self-index for a string supports both random access and pattern matching queries. In this paper we show how, given a string S [1..n] whose LZ77 parse consists of z phrases, we can store a self-index for S in \(\mathcal{O}({z \log (n / z)})\) space such that later, first, given a position i and a length l, we can extract S [i..i + l − 1] in \(\mathcal{O}({\ell + \log n})\) time; second, given a pattern P [1..m], we can list the occ occurrences of P in S in \(\mathcal{O}({m \log m + occ \log \log n})\) time.

Pattern matching in lempel-Ziv compressed strings: fast, simple, and deterministic

Countless variants of the Lempel-Ziv compression are widely used in many real-life applications. This paper is concerned with a natural modification of the classical pattern matching problem inspired by the popularity of such compression methods: given an uncompressed pattern p[1 .. m] and a Lempel-Ziv representation of a string t[1 .. N], does p occur in t? Farach and Thorup [5] gave a randomized O(nlog2 N/n +m) time solution for this problem, where n is the size of the compressed representation of t. Building on the methods of [3] and [6], we improve their result by developing a faster and fully deterministic O(n log N/n +m) time algorithm with the same space complexity. Note that for highly compressible texts, log N/n might be of order n, so for such inputs the improvement is very significant. A small fragment of our method can be used to give an asymptotically optimal solution for the substring hashing problem considered by Farach and Muthukrishnan [4].

Optimal pattern matching in LZW compressed strings

We consider the following variant of the classical pattern matching problem: given an uncompressed pattern <i>p</i>[1..<i>m</i>] and a compressed representation of a string <i>t</i>[1..<i>N</i>], does <i>p</i> occur in <i>t</i>? When <i>t</i> is compressed using the LZW method, we are able to detect the occurrence in optimal linear time, thus answering a question of Amir et al. [1994]. Previous results implied solutions with complexities <i>O</i>(<i>n</i> log <i>m</i> + <i>m</i>) Amir et al. [1994], <i>O</i>(<i>n</i> + <i>m</i>^1+ε) [Kosaraju 1995], or (randomized) <i>O</i>(<i>n</i> log <i>Nn</i> + <i>m</i>) [Farach and Thorup 1995], where <i>n</i> is the size of the compressed representation of <i>t</i>. Our algorithm is conceptually simple and fully deterministic.

Faster approximate pattern matching in compressed repetitive texts

Motivated by the imminent growth of massive, highly redundant genomic databases we study the problem of compressing a string database while simultaneously supporting fast random access, substring extraction and pattern matching to the underlying string(s). Bille et al. (2011) recently showed how, given a straight-line program with r rules for a string s of length n, we can build an

Hyper-minimisation Made Efficient

\ensuremath{\mathcal{O}\!\left( {r} \right)}

Alphabet-Dependent String Searching with Wexponential Search Trees

-word data structure that allows us to extract any substring s [i..j] in

Strong Inapproximability of the Shortest Reset Word

\ensuremath{\mathcal{O}\!\left( {\log n + j - i} \right)}

Queries on LZ-Bounded Encodings

time. They also showed how, given a pattern p of length m and an edit distance k≤m, their data structure supports finding all occ approximate matches to p in s in

Weighted Ancestors in Suffix Trees

\ensuremath{\mathcal{O}\!\left( {r (\min (m k, k^4 + m) + \log n) + \ensuremath{\mathsf{occ}}} \right)}