Faster algorithms for computing longest common increasing subsequences

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+ nl) log log sigma + Sort) and O(m) space, where l is the length of an LCIS, sigma 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 logn, mlog logm})-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. (C) 2011 Elsevier B.V. All rights reserved.