Faster algorithms for computing longest common increasing subsequences

Authors:
Martin Kutz;Gerth Stølting Brodal;Kanela Kaligosi;Irit Katriel
Affiliations:
Max-Plank-Institut für Informatik, Saarbrücken, Germany;MADALGO,33Center for Massive Data Algorithmics - a Center of the Danish National Research Foundation. Aarhus University, Aarhus, Denmark;Max-Plank-Institut für Informatik, Saarbrücken, Germany;Aarhus University, Aarhus, Denmark
Venue:
Journal of Discrete Algorithms
Year:
2011

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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 n and m, where m=n, we present an algorithm with an output-dependent expected running time of O((m+n@?)loglog@s+Sort) and O(m) space, where @? is the length of an LCIS, @s is the size of the alphabet, and 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(min{m+nlogn,mloglogm})-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.