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}
}
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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A Simple Sublinear Algorithm for Gap Edit Distance
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References

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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.
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TLDR
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.
Constant-factor approximation of near-linear edit distance in near-linear time
We show that the edit distance between two strings of length n can be computed via a randomized algorithm within a factor of f(є) in n 1+є time as long as the edit distance is at least n 1−δ for some
Simpler Constant-Factor Approximation to Edit Distance Problems
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An alternative implementation of the core idea from [DGKS18] is provided, with the main goal of developing a simpler constant-factor approximation algorithm with sub-quadratic run-time, while achieving the best possible bounds.
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Constant factor approximations to edit distance on far input pairs in nearly linear 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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TLDR
Algorithms are developed 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 /spl lscr/, decide which of the two holds and develop an n/sup 3/7/-approximation quasilinear time algorithm.
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This paper presents two streaming algorithms for computing edit distance, one which runs in time $O(n+k^2)$ and the other which is known to be optimal under the Strong Exponential Time Hypothesis.
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TLDR
This is the first subpolynomial approximation algorithm for this problem that runs in near-linear time, improving on the state-of-the-art $n^{1/3+o(1)}$ approximation.
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