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Approximating the Geometric Edit Distance

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Abstract

Edit distance is a measurement of similarity between two sequences such as strings, point sequences, or polygonal curves. Many matching problems from a variety of areas, such as signal analysis, bioinformatics, etc., need to be solved in a geometric space. Therefore, the geometric edit distance (GED) has been studied. In this paper, we describe the first strictly sublinear approximate near-linear time algorithm for computing the GED of two point sequences in constant dimensional Euclidean space. Specifically, we present a randomized \(O(n\log ^2n)\) time \(O(\sqrt{n})\)-approximation algorithm. Then, we generalize our result to give a randomized \(\alpha \)-approximation algorithm for any \(\alpha \in [\sqrt{\log n}, \sqrt{n/\log n}]\), running in time \(O(n^2/\alpha ^2 \log n)\). Both algorithms are Monte Carlo and return approximately optimal solutions with high probability.

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Notes

  1. The (discrete) Fréchet and DTW distances are defined similarly to GED; however, they use one-to-many correspondences instead of one-to-one matchings, and they disallow the use of gap points. As in GED, DTW aims to minimize the sum of distances between corresponding points, while discrete Fréchet distance aims to minimize the maximum distance over corresponding points.

  2. We say an event occurs with high probability if it occurs with probability at least \(1-\frac{1}{n^c}\) for some constant \(c>0\).

  3. This variant of the string edit distance is really closer to the shortest common supersequence length of the strings rather than the traditional Levenshtein distance, but we stick with “edit distance” for simplicity.

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Acknowledgements

The authors would like to thank Anne Driemel and Benjamin Raichel for helpful discussions.

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Authors and Affiliations

  1. The University of Texas at Dallas, Richardson, TX, USA

    Kyle Fox

  2. The University of Utah, Salt Lake City, UT, USA

    Xinyi Li

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  1. Kyle Fox
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  2. Xinyi Li
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Correspondence to Kyle Fox.

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A preliminary version of this work appeared in the Proceedings of the 30th International Symposium on Algorithms and Computation [13]. Most of the work was done while the second author was a student at the University of Texas at Dallas.

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Fox, K., Li, X. Approximating the Geometric Edit Distance. Algorithmica 84, 2395–2413 (2022). https://doi.org/10.1007/s00453-022-00966-4

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