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Longest Common Substring with Gaps and Related Problems

Authors Aranya Banerjee , Daniel Gibney , Sharma V. Thankachan

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Author Details

Aranya Banerjee
  • Georgia Institute of Technology, Atlanta, GA, USA
Daniel Gibney
  • University of Texas at Dallas, Richardson, TX, USA
Sharma V. Thankachan
  • North Carolina State University, Raleigh, NC, USA

Funding

  • Thankachan, Sharma V.: Supported by the U.S. National Science Foundation (NSF) award CCF-2316691.

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Aranya Banerjee, Daniel Gibney, and Sharma V. Thankachan. Longest Common Substring with Gaps and Related Problems. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.16

Abstract

The longest common substring (also known as longest common factor) and longest common subsequence problems are two well-studied classical string problems. The former is solvable in optimal 𝒪(n) time for two strings of length m and n with m ≤ n, and the latter is solvable in 𝒪(nm) time, which is conditionally optimal under the Strong Exponential Time Hypothesis. In this work, we study the problem of longest common factor with gaps, that is, finding a set of at most k matching substrings obeying precedence conditions with maximum total length. For k = 1, this is equivalent to the longest common factor problem, and for k = m, this is equivalent to the longest common subsequence problem. Our work demonstrates that, for constant k, this problem can be solved in strongly subquadratic time, i.e., nm^{1 - Θ(1)}. Motivated by co-linear chaining applications in Computational Biology, we further demonstrate that the longest common factor with gaps results can be extended to the case where the matches are restricted to maximal exact matches (MEMs). To further demonstrate the applicability of our techniques, we show that a similar approach can be used for a restricted version of the episode matching problem where one seeks an ordered set of at most k matches whose concatenation equals a query pattern P and the length of the substring of T containing the matches is minimized. These solutions all run in strongly subquadratic time for constant k.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
Keywords
  • Pattern Matching
  • Longest Common Subsequence
  • Episode Matching

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