Improved Algorithms for Edit Distance and LCS: Beyond Worst Case

Edit distance and longest common subsequence are among the most fundamental problems in combinatorial optimization. Recent developments have proven strong lower bounds against subquadratic time solutions for both problems. Moreover, the best approximation factors for subquadratic time solutions have been limited to 3 for edit distance and super constant for longest common subsequence. Improved approximation algorithms for these problems 1 are some of the biggest open questions in combinatorial optimization. In this work, we present improved algorithms for both edit distance and longest common subsequence. The running times are truly subquadratic, though we obtain 1 + o(1) approximate solutions for both problems if the input satisfies a mild condition. In this setting, first, an adversary chooses one of the input strings. Next, this string is perturbed by a random procedure, and then the adversary chooses the second string after observing the perturbed one.