# Approximating Edit Distance within Constant Factor in Truly Sub-Quadratic Time

@article{Chakraborty2018ApproximatingED,
title={Approximating Edit Distance within Constant Factor in Truly Sub-Quadratic Time},
author={Diptarka Chakraborty and Debarati Das and Elazar Goldenberg and Michal Kouck{\'y} and Michael E. Saks},
journal={2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)},
year={2018},
pages={979-990}
}
• Published 1 October 2018
• Computer Science
• 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
Edit distance is a measure of similarity of two strings based on the minimum number of character insertions, deletions, and substitutions required to transform one string into the other. The edit distance can be computed exactly using a dynamic programming algorithm that runs in quadratic time. Andoni, Krauthgamer and Onak (2010) gave a nearly linear time algorithm that approximates edit distance within approximation factor poly(log n). In this paper, we provide an algorithm with running time…

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The main result is a very simple algorithm for this benchmark that settles the open problem of obtaining a truly sublinear time for the entire range of relevant $k$ and obtains a $k-vs-$k^2$algorithm for the one-sided preprocessing model. ## References SHOWING 1-10 OF 32 REFERENCES Edit Distance in Near-Linear Time: it's a Constant Factor • Computer Science, Mathematics 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS) • 2020 The result completes a research direction set forth in the recent breakthrough paper, which showed the first constant-factor approximation algorithm with a (strongly) sub-quadratic running time. Approximating Edit Distance in Truly Subquadratic Time: Quantum and MapReduce • Computer Science SODA • 2018 This work provides a quantum constant approximation algorithm for computing the edit distance in truly subquadratic time, and provides a MapReduce algorithm to approximate edit distance within a factor of$3, with sublinearly many machines and sublinear memory.
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Edit Distance Cannot Be Computed in Strongly Subquadratic Time (Unless SETH is False)
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SIAM J. Comput.
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Evidence is provided that the near-quadratic running time bounds known for the problem of computing edit distance might be tight, and it is shown that if the edit distance can be computed in time $O(n^{2-\delta})$ for some constant $\delta>0$, then the satisfiability of conjunctive normal form formulas with $N$ variables and $M$ clauses can be solved in time.
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On any input with edit(x,y) ≥ n 1−ζ the algorithm outputs a constant factor approximation with high probability and this result has been proven independently by Brakensiek and Rubinstein.
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