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Description

We present a near-linear time algorithm that approximates the edit distance between two strings within a polylogarithmic factor; specifically, for strings of length $n$ and every fixed $eps>0$, it can compute a $(log n)^{O(1/eps)}$-approximation in $n^{1+eps}$ time. This is an {em exponential} improvement over the previously known factor, $2^{ ilde O(sqrt{log n})}$, with a comparable running time~cite{OR-edit,AO-edit}. Previously, no efficient polylogarithmic approximation algorithm was known for any computational task involving edit distance (e.g., nearest neighbor search or sketching).

This result arises naturally in the study of a new emph{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 $x$. Indeed, we obtain our main result by designing an algorithm that makes a small number of queries in this model. We also provide a nearly-matching lower bound on the number of queries.

Our lower bound is the first to expose we hardness of edit distance stemming from the input strings being ``repetitive'', which means that many of their substrings are approximately identical. Consequently, our lower bound provides the first rigorous separation between edit distance and Ulam distance, which is edit distance on non-repetitive strings, i.e., permutations.

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