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 $m$ and $n$, where $m\ge n$, we present an algorithm with an output-dependent expected running time of $O((m+n\ell) \log\log \sigma + \cleanSort)$ and $O(m)$ space, where $\ell$ is the length of an LCIS, $\sigma$ is the size of the alphabet, and $\cleanSort$ is the time to sort each input sequence. For $k\ge 3$ 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 $O(m+n\log n)$-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