Elastic-Degenerate String Matching with 1 Error
Elastic-Degenerate String Matching with 1 Error
摘要：
弹性简并弦是一个总长度为$N$的$n$有限弦集合的序列，用来表示一组相关的DNA序列，也称为泛基因组。ED字符串匹配（EDSM）问题在于报告ED文本中长度为$m$的模式的所有出现。这个问题最近受到了组合模式匹配团体的一些关注，最终产生了$\tilde{\mathcal{O}}(nm^{ω-1})+\mathcal{O}(N)$时间算法[Bernardini等人，SIAM J.计算2022]，其中$ω$表示矩阵乘法指数，并且$\tilde{\mathcal{O}}(\cdot)$表示法抑制多对数因子。在$k$ -EDSM问题中，EDSM的近似版本，我们被要求报告所有模式出现最多$k$错误。$k$-EDSM可以在编辑距离下以$\mathcal{O}(k^2mG+kN)$时间求解，或者在汉明距离下以$\mathcal{O}(kmG+kN)$时间求解，其中$G$表示ED文本中的字符串的总数[Bernardini等人，Theor. Comput. Sci. 2020年]。不幸的是，$G$仅受$N$约束，因此即使对于$k=1$，现有算法在最坏情况下也在$Ω(mN)$时间内运行。本文证明了$1$-EDSM在编辑距离下可以在$\mathcal{O}((nm^2 + N)\log m)$或$\mathcal{O}(nm^3 + N)$时间内求解.对于决策版本，我们提出了一个更快的$\mathcal{O}(nm^2\sqrt{\log m} + N\log\log m)$时间算法。我们还证明了在Hamming距离下$1$ -EDSM可以在$\mathcal{O}(nm^2 + N\log m)$时间内求解.我们的编辑距离算法依赖于从$1$ -EDSM到经典计算几何问题的特殊实例（2D矩形刺伤或2D范围空虚）的非平凡减少，我们展示了如何有效地解决这些问题。为了获得更快的汉明距离算法，我们依赖于采用和调整$k$-errata树来进行错误索引[Cole等人，STOC 2004]。
英文摘要：
An elastic-degenerate string is a sequence of $n$ finite sets of strings of total length $N$, introduced to represent a set of related DNA sequences, also known as a pangenome. The ED string matching (EDSM) problem consists in reporting all occurrences of a pattern of length $m$ in an ED text. This problem has recently received some attention by the combinatorial pattern matching community, culminating in an $\tilde{\mathcal{O}}(nm^{ω-1})+\mathcal{O}(N)$-time algorithm [Bernardini et al., SIAM J. Comput. 2022], where $ω$ denotes the matrix multiplication exponent and the $\tilde{\mathcal{O}}(\cdot)$ notation suppresses polylog factors. In the $k$-EDSM problem, the approximate version of EDSM, we are asked to report all pattern occurrences with at most $k$ errors. $k$-EDSM can be solved in $\mathcal{O}(k^2mG+kN)$ time, under edit distance, or $\mathcal{O}(kmG+kN)$ time, under Hamming distance, where $G$ denotes the total number of strings in the ED text [Bernardini et al., Theor. Comput. Sci. 2020]. Unfortunately, $G$ is only bounded by $N$, and so even for $k=1$, the existing algorithms run in $Ω(mN)$ time in the worst case. In this paper we show that $1$-EDSM can be solved in $\mathcal{O}((nm^2 + N)\log m)$ or $\mathcal{O}(nm^3 + N)$ time under edit distance. For the decision version, we present a faster $\mathcal{O}(nm^2\sqrt{\log m} + N\log\log m)$-time algorithm. We also show that $1$-EDSM can be solved in $\mathcal{O}(nm^2 + N\log m)$ time under Hamming distance. Our algorithms for edit distance rely on non-trivial reductions from $1$-EDSM to special instances of classic computational geometry problems (2d rectangle stabbing or 2d range emptiness), which we show how to solve efficiently. In order to obtain an even faster algorithm for Hamming distance, we rely on employing and adapting the $k$-errata trees for indexing with errors [Cole et al., STOC 2004].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
