Parallel algorithm for pattern matching problems under substring consistent equivalence relations

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 $\approx$ is called a substring consistent equivalence relation (SCER), if for two strings $X$ and $Y$, $X \approx Y$ implies $|X| = |Y|$ and $X[i:j] \approx Y[i:j]$ for all $1 \le i \le j \le |X|$. 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 $O(\xim^\mathrm{t} \log² m)$ time and $O(\xim^\mathrm{w} \cdot n \log² m)$ work, with $O(\taun^\mathrm{t} + \xim^\mathrm{t} \log² m)$ time and $O(\taun^\mathrm{w} + \xim^\mathrm{w} \cdot m \log² m)$ work preprocessing on the Priority Concurrent Read Concurrent Write Parallel Random-Access Machines (P-CRCW PRAM).