A Linear-Time n0.4-Approximation for Longest Common Subsequence

We consider the classic problem of computing the Longest Common Subsequence (LCS) of two strings of length n. The 40-year-old quadratic-time dynamic programming algorithm has recently been shown to be near-optimal by Abboud, Backurs, and Vassilevska Williams [FOCS’15] and Bringmann and Künnemann [FOCS’15] assuming the Strong Exponential Time Hypothesis. This has led the community to look for subquadratic approximation algorithms for the problem.

Yet, unlike the edit distance problem for which a constant-factor approximation in almost-linear time is known, very little progress has been made on LCS, making it a notoriously difficult problem also in the realm of approximation. For the general setting, only a naive O(n^ɛ/2-approximation algorithm with running time OŠ(n^2-ɛ has been known, for any constant 0 < ɛ ≤ 1. Recently, a breakthrough result by Hajiaghayi, Seddighin, Seddighin, and Sun [SODA’19] provided a linear-time algorithm that yields a O(n^0.497956-approximation in expectation; improving upon the naive \(O(\sqrt {n})\)-approximation for the first time.

In this paper, we provide an algorithm that in time O(n_2-ɛ) computes an OŠ(n^2ɛ/5-approximation with high probability, for any 0 < ɛ ≤ 1. Our result (1) gives an OŠ(n^0.4-approximation in linear time, improving upon the bound of Hajiaghayi, Seddighin, Seddighin, and Sun, (2) provides an algorithm whose approximation scales with any subquadratic running time O(n^2-ɛ), improving upon the naive bound of O(n^ɛ/2) for any ɛ, and (3) instead of only in expectation, succeeds with high probability.