Faster algorithms for computing longest common increasing subsequences

  • Authors:

  • Gerth Stølting Brodal;Kanela Kaligosi;Irit Katriel;Martin Kutz

  • Affiliations:

  • BRICS, University of Aarhus, Århus, Denmark;Max-Plank-Institut für Informatik, Saarbrücken, Germany;BRICS, University of Aarhus, Århus, Denmark;Max-Plank-Institut für Informatik, Saarbrücken, Germany

  • Venue:
  • CPM'06 Proceedings of the 17th Annual conference on Combinatorial Pattern Matching
  • Year:
  • 2006

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Abstract

We present algorithms for finding a longest common increasing subsequence of two or more input sequences. For two sequences of lengths m and n, where m≥n, we present an algorithm with an output-dependent expected running time of $O((m+n\ell) \log\log \sigma + {\ensuremath{\mathit{Sort}}})$ and O(m) space, where ℓ is the length of an LCIS, σ is the size of the alphabet, and ${\ensuremath{\mathit{Sort}}}$ is the time to sort each input sequence. For k≥3 length-n sequences we present an algorithm which improves the previous best bound by more than a factor k for many inputs. In both cases, our algorithms are conceptually quite simple but rely on existing sophisticated data structures. Finally, we introduce the problem of longest common weakly-increasing (or non-decreasing) subsequences (LCWIS), for which we present an O(m+nlogn)-time algorithm for the 3-letter alphabet case. For the extensively studied longest common subsequence problem, comparable speedups have not been achieved for small alphabets.