Faster algorithms for computing longest common increasing subsequences

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 ℓ ) log log σ + Sort ) and O ( m ) space, where ℓ is the length of an LCIS, σ 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 + n log n , m log log m } ) -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.