Parallel algorithm for pattern matching problems under substring consistent equivalence relations

TL;DR

This paper introduces a parallel algorithm for pattern matching under substring consistent equivalence relations (SCER), improving efficiency for complex pattern matching problems in parallel computing environments.

Contribution

It presents the first efficient parallel algorithm for pattern matching under any SCER using the duel-and-sweep paradigm, with proven time and work complexity bounds.

Findings

Runs in $O(\xi_m^t \, ext{log}^2 m)$ time

Uses $O( au_n^t + \xi_m^t \, ext{log}^2 m)$ time for preprocessing

Achieves $O( au_n^w + \xi_m^w \, m ext{log}^2 m)$ work complexity

Abstract

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 $O(\xi_m^\mathrm{t} \log^2 m)$ time and $O(\xi_m^\mathrm{w} \cdot n \log^2 m)$ work, with $O(\tau_n^\mathrm{t} + \xi_m^\mathrm{t} \log^2 m)$ time and $O(\tau_n^\mathrm{w} + \xi_m^\mathrm{w} \cdot m \log^2 m)$ work preprocessing on the Priority Concurrent Read Concurrent Write Parallel Random-Access Machines (P-CRCW PRAM).