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).
翻译：根据文字和字母表的图案和图案图案,图案匹配对文本中图案的所有发生情况进行匹配。 等值 $\ approx$ 被称为子字符一致等值关系( SCER ) 。 如果对于两个字符串 $X美元和$Y, $X\ approx Y$ 意味着$X = ⁇ ⁇ Y 美元 和 $X [i:j]\ approx Y[i:j] = procrox Y[i:j] $, 所有 $ 1\ le i\ j\ j\ j\ j\ j\ ⁇ j\ \ ⁇ Xxxxxx 美元。 在本文中, 我们建议使用“ duel- acrox- sweep” 模式在任何 SCER 下为模式匹配设置一个高效的并行算法 。 对于长度 $mum 美元和 美元文本, 我们的算法运行 $O(x_ m) mattlex2\ m_ m) riom_ riom} prox_ m) rix_ m) rial_ m) rixxxxxxxxxxxxxxxxxxxxxx
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