Fast computation of a longest increasing subsequence and application

Authors:
Maxime Crochemore;Ely Porat
Affiliations:
King's College London, Strand, London WC2R 2LS, UK and Université Paris-Est, France;Bar-Ilan University, Ramat-Gan 52900, Israel
Venue:
Information and Computation
Year:
2010

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Abstract

We consider the complexity of computing a longest increasing subsequence (LIS) parameterised by the length of the output. Namely, we show that the maximal length k of an increasing subsequence of a permutation of the set of integers {1,2,...,n} can be computed in time O(nloglogk) in the RAM model, improving the previous 30-year bound of O(nlogk). The algorithm also improves on the previous O(nloglogn) bound. The optimality of the new bound is an open question. Reducing the computation of a longest common subsequence (LCS) between two strings to an LIS computation leads to a simple O(rloglogk)-time algorithm for two sequences having r pairs of matching symbols and an LCS of length k.