In this paper we present a coarse-grained parallel algorithm for solving the string edit distance problem for a string A and all substrings of a string C. Our method is based on a novel CGM/BSP parallel dynamic programming technique for computing all highest scoring paths in a weighted grid graph. The algorithm requires log p rounds/supersteps and O( n 2 p log m) local computation, where p is the number of processors, p 2 ≤ m ≤ n. To our knowledge, this is the first efficient CGM/BSP algorithm for the alignment of all substrings of C with A. Furthermore, the CGM/BSP parallel dynamic programming technique presented is of interest in its own right and we expect it to lead to other parallel dynamic programming methods for the CGM/BSP.
Given two strings A and B of lengths n a and n b , respectively, the All-substrings Longest Common Subsequence (ALCS) problem obtains, for any substring B of B, the length of the longest string that is a subsequence of both A and B . The sequential algorithm for this problem takes O(n a n b ) time and O(n b ) space. We present a parallel algorithm for the ALCS problem on the Coarse-Grained Multicomputer (BSP/CGM) model with p < √ n a processors, that takes O(n a n b / p) time, O(log p) communication rounds and O(n b √ n a ) space per processor. The proposed algorithm also solves the basic Longest Common Subsequence (LCS) problem that finds the longest string (and not only its length) that is a subsequence of both A and B. To our knowledge, this is the best BSP/CGM algorithm in the literature for the LCS and ALCS problems.
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