Approximate Circular Pattern Matching under Edit Distance

Charalampopoulos, Panagiotis and Pissis, Solon P. and Radoszewski, Jakub and Rytter, Wojciech and Walen, Tomasz and Zuba, Wiktor (2024) Approximate Circular Pattern Matching under Edit Distance. Leibniz International Proceedings in Informatics (LIPIcs) 289 , 24:1-24:22. ISSN 1868-8969.

Abstract

In the k-Edit Circular Pattern Matching (k-Edit CPM) problem, we are given a length-n text T, a length-m pattern P, and a positive integer threshold k, and we are to report all starting positions of the substrings of T that are at edit distance at most k from some cyclic rotation of P. In the decision version of the problem, we are to check if any such substring exists. Very recently, Charalampopoulos et al. [ESA 2022] presented O(nk^2)-time and O(nk \log^3 k)-time solutions for the reporting and decision versions of k-Edit CPM, respectively. Here, we show that the reporting and decision versions of k-Edit CPM can be solved in O(n+(n/m)k^6) time and O(n+(n/m)k^5 \log^3 k) time, respectively, thus obtaining the first algorithms with a complexity of the type O(n+(n/m) poly(k)) for this problem. Notably, our algorithms run in O(n) time when m=Ω(k^6) and are superior to the previous respective solutions when m=ω(k^4). We provide a meta-algorithm that yields efficient algorithms in several other interesting settings, such as when the strings are given in a compressed form (as straight-line programs), when the strings are dynamic, or when we have a quantum computer. We obtain our solutions by exploiting the structure of approximate circular occurrences of P in T, when T is relatively short w.r.t. P. Roughly speaking, either the starting positions of approximate occurrences of rotations of P form O(k^4) intervals that can be computed efficiently, or some rotation of P is almost periodic (is at a small edit distance from a string with small period). Dealing with the almost periodic case is the most technically demanding part of this work; we tackle it using properties of locked fragments (originating from [Cole and Hariharan, SICOMP 2002]).

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