Papers › Compressed Multiple Pattern Matching
Compressed Multiple Pattern Matching
Dmitry Kosolobov, Nikita Sivukhin
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Given d strings over the alphabet {0,1,…,σ-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(mlogm) bits of space, where m is the number of edges in the trie containing the strings. Fix any constant ε∈(0, 2). We describe a compressed solution for the problem that, provided σ≤m^δ for a constant δ< 1, works in O(|T| 1/ε log1/ε + occ) time, which is O(|T| + occ) since ε is constant, and occupies mHₖ + 1.443 m + εm + O(dlogm/d) bits of space, for all 0 ≤k ≤max{0,αlog_σ m - 2} simultaneously, where α∈(0,1) is an arbitrary constant and Hₖ is the $k$th-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 + ε)m, thus solving an open problem posed by Belazzougui. Further, we notice that L = logσ(m+1)m - O(log(σm)) is a worst-case space lower bound for any solution of the problem and, for d = o(m) and constant ε, our approach allows to achieve L + εm bits of space, which gives an evidence that, for d = o(m), the space of our data structure is theoretically optimal up to the εm 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 Hₖ. 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.
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