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45th Annual IEEE Symposium on Foundations of Computer Science (2004)
Rome, Italy
Oct. 17, 2004 to Oct. 19, 2004
ISSN: 0272-5428
ISBN: 0-7695-2228-9
pp: 550-559
Ziv Bar-Yossef , Technion
T. S. Jayram , IBM Almaden Research Center
Robert Krauthgamer , IBM Almaden Research Center
Ravi Kumar , IBM Almaden Research Center
<p>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.</p> <p>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.</p> <p>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(k²) — still with a constant sketch — but works only for strings that are mildly "non-repetitive".</p> <p>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}}.</p>

T. S. Jayram, R. Kumar, Z. Bar-Yossef and R. Krauthgamer, "Approximating Edit Distance Efficiently," 45th Annual IEEE Symposium on Foundations of Computer Science(FOCS), Rome, Italy, 2004, pp. 550-559.
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