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.

我们考虑计算两个长度为n的字符串的最长公共子序列 (LCS)的经典问题。Abboud、Backurs 和 Vassilevska Williams [FOCS'15] 以及 Bringmann 和 Künnemann [FOCS'15] 假设强指数时间假设，最近证明具有 40 年历史的二次时间动态规划算法接近最优。这导致社区寻找问题的次二次近似算法。

然而，与已知几乎线性时间内的常数因子近似的编辑距离问题不同，在 LCS 上取得的进展很少，这使得它在近似领域也是一个众所周知的难题。对于一般设置，只有一个朴素的O ( n ^ɛ /2-approximation algorithm with running time OŠ ( n ^2-ɛ已知，对于任何常数 0 < ɛ ≤ 1。最近，Hajiaghayi，Seddighin，Seddighin 的突破性结果, Sun [SODA'19] 提供了一种线性时间算法，该算法产生O ( n ^0.497956 - 期望近似值；首次改进了朴素的 \(O(\sqrt {n})\) - 近似值。

在本文中，我们提供了一种算法，该算法在时间O ( n _2-ɛ )中以高概率计算OŠ ( n ^2ɛ/5 -近似值，对于任何 0 < ε ≤ 1。我们的结果 (1) 给出了OŠ ( n ^{0.4 - 线性时间的近似值，改进了 Hajiaghayi、Seddighin、Seddighin 和 Sun 的界限，(2) 提供了一种算法，其近似值与任何次二次运行时间}O ( n ^2-ɛ ) 成比例，改进了 O 的朴素界限( n ^ɛ/2 ) 对于任何 ɛ，并且 (3) 而不是仅在预期中，以高概率成功。