A sub-quadratic algorithm for the longest common increasing subsequence problem

Abstract

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 $a=\frac{1}{6}-o(1)$. 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.

21 pages; STACS 2020 version -- heavily corrected from the previous one, might actually be readable