Approximating edit distance efficiently

Edit distance has been extensively studied for the past several years. Nevertheless, no linear-time algorithm is known to compute the edit distance between two strings, or even to approximate it to within a modest factor. Furthermore, for various natural algorithmic problems such as low-distortion embeddings into normed spaces, approximate nearest-neighbor schemes, and sketching algorithms, known results for the edit distance are rather weak. We develop algorithms that solve gap versions of the edit distance problem: given two strings of length n with the promise that their edit distance is either at most k or greater than ℓ, decide which of the two holds. We present two sketching algorithms for gap versions of edit distance. Our first algorithm solves the k vs. (kn)²3/ gap problem, using a constant size sketch. A more involved algorithm solves the stronger k vs. ℓ gap problem, where ℓ can be as small as O(k²) - still with a constant sketch - but works only for strings that are mildly "nonrepetitive". Finally, we develop an n³7/-approximation quasilinear time algorithm for edit distance, improving the previous best factor of n³4/ (Cole and Hariharan, 2002); if the input strings are assumed to be nonrepetitive, then the approximation factor can be strengthened to n¹3/.