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Linear-Time Algorithm for Long LCF with k Mismatches

Authors Panagiotis Charalampopoulos , Maxime Crochemore , Costas S. Iliopoulos, Tomasz Kociumaka , Solon P. Pissis , Jakub Radoszewski , Wojciech Rytter, Tomasz Walen

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Panagiotis Charalampopoulos
  • Department of Informatics, King’s College London, London, UK
Maxime Crochemore
  • Department of Informatics, King’s College London, London, UK
Costas S. Iliopoulos
  • Department of Informatics, King’s College London, London, UK
Tomasz Kociumaka
  • Institute of Informatics, University of Warsaw, Warsaw, Poland
Solon P. Pissis
  • Department of Informatics, King’s College London, London, UK
Jakub Radoszewski
  • Institute of Informatics, University of Warsaw, Warsaw, Poland
Wojciech Rytter
  • Institute of Informatics, University of Warsaw, Warsaw, Poland
Tomasz Walen
  • Institute of Informatics, University of Warsaw, Warsaw, Poland

Funding

Jakub Radoszewski is supported by the "Algorithms for text processing with errors and uncertainties" project carried out within the HOMING programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund. Wojciech Rytter and Tomasz Waleń are supported by the Polish National Science Center, grant no. 2014/13/B/ST6/00770.

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Panagiotis Charalampopoulos, Maxime Crochemore, Costas S. Iliopoulos, Tomasz Kociumaka, Solon P. Pissis, Jakub Radoszewski, Wojciech Rytter, and Tomasz Walen. Linear-Time Algorithm for Long LCF with k Mismatches. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.CPM.2018.23

Abstract

In the Longest Common Factor with k Mismatches (LCF_k) problem, we are given two strings X and Y of total length n, and we are asked to find a pair of maximal-length factors, one of X and the other of Y, such that their Hamming distance is at most k. Thankachan et al. [Thankachan et al. 2016] show that this problem can be solved in O(n log^k n) time and O(n) space for constant k. We consider the LCF_k(l) problem in which we assume that the sought factors have length at least l. We use difference covers to reduce the LCF_k(l) problem with l=Omega(log^{2k+2}n) to a task involving m=O(n/log^{k+1}n) synchronized factors. The latter can be solved in O(m log^{k+1}m) time, which results in a linear-time algorithm for LCF_k(l) with l=Omega(log^{2k+2}n). In general, our solution to the LCF_k(l) problem for arbitrary l takes O(n + n log^{k+1} n/sqrt{l}) time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pattern matching
Keywords
  • longest common factor
  • longest common substring
  • Hamming distance
  • heavy-light decomposition
  • difference cover

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