A Sub-Quadratic Algorithm for the Longest Common Increasing Subsequence Problem

Author Lech Duraj

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Lech Duraj
  • Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland


I would like to thank Grzegorz Guśpiel, Grzegorz Herman and Adam Polak for helpful discussions and proofreading the paper, as well as the reviewers for many insightful comments and suggestions.

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Lech Duraj. A Sub-Quadratic Algorithm for the Longest Common Increasing Subsequence Problem. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 41:1-41:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


The Longest Common Increasing Subsequence problem (LCIS) is a natural variant of the celebrated Longest Common Subsequence (LCS) problem. For LCIS, as well as for LCS, there is an ?(n²)-time algorithm and a SETH-based conditional lower bound of ?(n^{2-ε}). For LCS, there is also the Masek-Paterson ?(n²/log n)-time algorithm, which does not seem to adapt to LCIS in any obvious way. Hence, a natural question arises: does any (slightly) sub-quadratic algorithm exist for the Longest Common Increasing Subsequence problem? We answer this question positively, presenting a ?(n²/log^a n)-time algorithm for a = 1/6-o(1). The algorithm is not based on memorizing small chunks of data (often used for logarithmic speedups, including the "Four Russians Trick" in LCS), but rather utilizes a new technique, bounding the number of significant symbol matches between the two sequences.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • longest common increasing subsequence
  • log-shaving
  • matching pairs


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