A Sub-Quadratic Algorithm for the Longest Common Increasing Subsequence Problem

The Longest Common Increasing Subsequence problem (LCIS) is a natural variant of the celebrated Longest Common Subsequence (LCS) problem. For LCIS, as well as for LCS, there is an O (n(2))-time algorithm and a SETH-based conditional lower bound of O(n(2-epsilon)). For LCS, there is also the Masek-Paterson O(n(2)/ log n)-time algorithm, which does not seem to adapt to LCIS in any obvious way. Hence, a natural question arises: does any (slightly) sub-quadratic algorithm exist for the Longest Common Increasing Subsequence problem? We answer this question positively, presenting a O (n(2) / log(a) n)-time algorithm for a = 1/6 o (1). The algorithm is not based on memorizing small chunks of data (often used for logarithmic speedups, including the "Four Russians Trick" in LCS), but rather utilizes a new technique, bounding the number of significant symbol matches between the two sequences.