When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CPM.2019.13
URN: urn:nbn:de:0030-drops-104847
URL: https://drops.dagstuhl.de/opus/volltexte/2019/10484/
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### Compressed Multiple Pattern Matching

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### Abstract

Given d strings over the alphabet {0,1,...,sigma{-}1}, the classical Aho - Corasick data structure allows us to find all occ occurrences of the strings in any text T in O(|T| + occ) time using O(m log m) bits of space, where m is the number of edges in the trie containing the strings. Fix any constant epsilon in (0, 2). We describe a compressed solution for the problem that, provided sigma <=m^delta for a constant delta < 1, works in O(|T| 1/epsilon log(1/epsilon) + occ) time, which is O(|T| + occ) since epsilon is constant, and occupies mH_k + 1.443 m + epsilon m + O(d log m/d) bits of space, for all 0 <= k <= max{0,alpha log_sigma m - 2} simultaneously, where alpha in (0,1) is an arbitrary constant and H_k is the kth-order empirical entropy of the trie. Hence, we reduce the 3.443m term in the space bounds of previously best succinct solutions to (1.443 + epsilon)m, thus solving an open problem posed by Belazzougui. Further, we notice that L = log binom{sigma (m+1)}{m} - O(log(sigma m)) is a worst-case space lower bound for any solution of the problem and, for d = o(m) and constant epsilon, our approach allows to achieve L + epsilon m bits of space, which gives an evidence that, for d = o(m), the space of our data structure is theoretically optimal up to the epsilon m additive term and it is hardly possible to eliminate the term 1.443m. In addition, we refine the space analysis of previous works by proposing a more appropriate definition for H_k. We also simplify the construction for practice adapting the fixed block compression boosting technique, then implement our data structure, and conduct a number of experiments showing that it is comparable to the state of the art in terms of time and is superior in space.

### BibTeX - Entry

```@InProceedings{kosolobov_et_al:LIPIcs:2019:10484,
author =	{Dmitry Kosolobov and Nikita Sivukhin},
title =	{{Compressed Multiple Pattern Matching}},
booktitle =	{30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)},
pages =	{13:1--13:14},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-103-0},
ISSN =	{1868-8969},
year =	{2019},
volume =	{128},
editor =	{Nadia Pisanti and Solon P. Pissis},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},