Elastic-Degenerate String Matching with 1 Error
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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 ๐ชฬ(nm^ฯ-1)+๐ช(N)-time algorithm [Bernardini et al., SIAM J. Comput. 2022], where ฯ denotes the matrix multiplication exponent and the ๐ชฬ(ยท) 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 ๐ช(k^2mG+kN) time, under edit distance, or ๐ช(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 ฮฉ(mN) time in the worst case. In this paper we show that 1-EDSM can be solved in ๐ช((nm^2 + N)log m) or ๐ช(nm^3 + N) time under edit distance. For the decision version, we present a faster ๐ช(nm^2โ(log m) + Nloglog m)-time algorithm. We also show that 1-EDSM can be solved in ๐ช(nm^2 + Nlog 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].
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