Generalised Pattern Matching Revisited
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In the problem of Generalised Pattern Matching (GPM) [STOC'94, Muthukrishnan and Palem], we are given a text T of length n over an alphabet Σ_T, a pattern P of length m over an alphabet Σ_P, and a matching relationship ⊆Σ_T ×Σ_P, and must return all substrings of T that match P (reporting) or the number of mismatches between each substring of T of length m and P (counting). In this work, we improve over all previously known algorithms for this problem for various parameters describing the input instance: * D being the maximum number of characters that match a fixed character, * S being the number of pairs of matching characters, * I being the total number of disjoint intervals of characters that match the m characters of the pattern P. At the heart of our new deterministic upper bounds for D and S lies a faster construction of superimposed codes, which solves an open problem posed in [FOCS'97, Indyk] and can be of independent interest. To conclude, we demonstrate first lower bounds for GPM. We start by showing that any deterministic or Monte Carlo algorithm for GPM must use Ω(S) time, and then proceed to show higher lower bounds for combinatorial algorithms. These bounds show that our algorithms are almost optimal, unless a radically new approach is developed.
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