Polylogarithmic Approximation for Edit Distance and the Asymmetric Query Complexity

We present a near-linear time algorithm that approximates the edit distance between two strings within a polylogarithmic factor. More precisely, for strings of length n and every fixed epsilon > 0, it can compute a (log n)(O(1/epsilon))-approximation in n(1+epsilon) time. This result arises naturally in the study of a new asymmetric query model. In this model, the input consists of two strings x and y, and an algorithm can access y in an unrestricted manner, while being charged for querying every symbol of x. Our query lower bound for this model provides the first rigorous separation between edit distance and Ulam distance, which is edit distance on non-repetitive strings, i.e., permutations.