Compressed Multiple Pattern Matching
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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(m m) 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/ε1/ε + occ) time, which is O(|T| + occ) since ε is constant, and occupies mH_k + 1.443 m + ε m + O(dm/d) bits of space, for all 0 < k <{0,α_σ m - 2} simultaneously, where α∈ (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 + ε)m, thus solving an open problem posed by Belazzougui. Further, we notice that L = σ (m+1)m - O((σ m)) 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 + ε 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 ε 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.
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