Efficient Algorithms for Finding a Longest Common Increasing Subsequence

We study the problem of finding a longest common increasing subsequence (LCIS) of multiple sequences of numbers. The LCIS problem is a fundamental issue in various application areas, including the whole genome alignment. In this paper we give an efficient algorithm to find the LCIS of two sequences in O(min(r log `, n`+r) log log n+Sort(n)) time where n is the length of each sequence and r is the number of ordered pairs of positions at which the two sequences match, ` is the length of the LCIS, and Sort(n) is the time to sort n numbers. For m sequences where m ≥ 3, we find the LCIS in O(min(mr2, r log ` log r)+m·Sort(n)) time where r is the total number of m-tuples of positions at which the m sequences match. The previous results find the LCIS of two sequences in O(n2) and O(n` log log n+Sort(n)) time. Our algorithm is faster when r is relatively small, e.g., for r < min(n2/(log ` log log n), n`/ log `).