Approximating Edit Distance Efficiently

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Abstract
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 \ell, 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)^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} gap problem, using a constant size sketch. A more involved algorithm solves the stronger k vs. \ell gap problem, where \ell can be as small as O(k虏) 驴 still with a constant sketch 驴 but works only for strings that are mildly "non-repetitive". Finally, we develop an n^{{3 \mathord{\left/ {\vphantom {3 7}} \right. \kern-\nulldelimiterspace} 7}}-approximation quasi-linear time algorithm for edit distance, improving the previous best factor of n^{{3 \mathord{\left/ {\vphantom {3 4}} \right. \kern-\nulldelimiterspace} 4}} [5]; if the input strings are assumed to be non-repetitive, then the approximation factor can be strengthened to n^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}.

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