Parallel Algorithm for Pattern Matching Problems Under Substring Consistent Equivalence Relations
Davaajav Jargalsaikhan, Diptarama Hendrian, Ryo Yoshinaka, Ayumi Shinohara · arXiv (Cornell University) · 2022
Given a text and a pattern over an alphabet, the pattern matching problem searches for all occurrences of the pattern in the text. An equivalence relation ≈ is a substring consistent equivalence relation (SCER), if for two strings X and Y, X ≈ Y implies |X| = |Y| and X[i:j] ≈ Y[i:j] for all 1 ≤ i ≤ j ≤ |X|. In this paper, we propose an efficient parallel algorithm for pattern matching under any SCER using the "duel-and-sweep" paradigm. For a pattern of length m and a text of length n, our algorithm runs in O(ξ_m^t log³ m) time and O(ξ_m^w ⋅ n log² m) work, with O(τ_n^t + ξ_m^t log² m) time and O(τ_n^w + ξ_m^w ⋅ m log² m) work preprocessing on the Priority Concurrent Read Concurrent Write Parallel Random-Access Machines (P-CRCW PRAM), where τ_n^t, τ_n^w, ξ_m^t, and ξ_m^w are parameters dependent on SCERs, which are often linear in n and m, respectively.