Approximate Online Pattern Matching in Sub-linear Time
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We consider the approximate pattern matching problem under edit distance. In this problem we are given a pattern P of length w and a text T of length n over some alphabet Σ, and a positive integer k. The goal is to find all the positions j in T such that there is a substring of T ending at j which has edit distance at most k from the pattern P. Recall, the edit distance between two strings is the minimum number of character insertions, deletions, and substitutions required to transform one string into the other. For a position t in {1,...,n}, let k_t be the smallest edit distance between P and any substring of T ending at t. In this paper we give a constant factor approximation to the sequence k_1,k_2,...,k_n. We consider both offline and online settings. In the offline setting, where both P and T are available, we present an algorithm that for all t in {1,...,n}, computes the value of k_t approximately within a constant factor. The worst case running time of our algorithm is O(n w^3/4). As a consequence we break the O(nw)-time barrier for this problem. In the online setting, we are given P and then T arrives one symbol at a time. We design an algorithm that upon arrival of the t-th symbol of T computes k_t approximately within O(1)-multiplicative factor and w^8/9-additive error. Our algorithm takes O(w^1-(7/54)) amortized time per symbol arrival and takes O(w^1-(1/54)) additional space apart from storing the pattern P. Both of our algorithms are randomized and produce correct answer with high probability. To the best of our knowledge this is the first worst-case sub-linear (in the length of the pattern) time and sub-linear succinct space algorithm for online approximate pattern matching problem.
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