Edit Distance in Near-Linear Time: it's a Constant Factor

Andoni, Alexandr; Nosatzki, Negev Shekel

doi:10.1109/focs46700.2020.00096

Cited by 28 publications

(24 citation statements)

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“…There exists a randomized algorithm that solves any instance of the (0, α)-Gap Edit Distance problem in O( n 1+α ) time with success probability at least 2 3 . Theorem 2.5 (Andoni and Nosatzki [AN20]). There exist decreasing functions f AN , g AN : R + → R ≥1 and a randomized algorithm A that, given X, Y ∈ Σ n and ε ∈ R + , runs in O(g AN (ε)n 1+ε ) time and returns a value A(X, Y, ε…”

Section: Preliminariesmentioning

confidence: 99%

“…This has led to a quest for faster algorithms that find an approximate solution. A long line of research towards that goal [BEK + 03, BJKK04, BES06, AO12, AKO10, BEG + 21, CDG + 20] recently culminated with an almost-linear-time approximation algorithm by Andoni and Nosatzki [AN20] that, for any desired ε > 0, runs in O(n 1+ε ) time and its approximation factor depends only on ε, and thus for fixed ε > 0 it achieves O(1)-approximation.…”

Section: Introductionmentioning

confidence: 99%

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Gap Edit Distance via Non-Adaptive Queries: Simple and Optimal

Goldenberg¹,

Kociumaka²,

Krauthgamer³

et al. 2021

Preprint

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We study the problem of approximating edit distance in sublinear time. This is formalized as a promise problem (k, k c )-Gap Edit Distance, where the input is a pair of strings X, Y and parameters k, c > 1, and the goal is to returnRecent years have witnessed significant interest in designing sublinear-time algorithms for Gap Edit Distance.We resolve the non-adaptive query complexity of Gap Edit Distance, improving over several previous results. Specifically, we design a non-adaptive algorithm with query complexity Õ( n k c−0.5 ), and further prove that this bound is optimal up to polylogarithmic factors. Our algorithm also achieves optimal time complexity Õ( n k c−0.5 ) whenever c ≥ 1.5. For 1 < c < 1.5, the running time of our algorithm is Õ( n k 2c−1 ). For the restricted case of k c = Ω(n), this matches a known result [Batu, Ergün, Kilian, Magen, Raskhodnikova, Rubinfeld, and Sami, STOC 2003], and in all other (nontrivial) cases, our running time is strictly better than all previous algorithms, including the adaptive ones.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Gap Edit Distance via Non-Adaptive Queries: Simple and Optimal

Goldenberg¹,

Kociumaka²,

Krauthgamer³

et al. 2021

Preprint

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“…It took another eight years to obtain the first constant-factor approximation of edit distance in subquadratic time [CDG + 20] (see [BEG + 21] for a quantum analog). Finally, Andoni and Nosatzki improved the running time to near-linear while maintaining a constant factor approximation [AN20]. Using the best result in string edit distance approximation [AN20], it is possible to improve the approximation factor of [Sah14] to O(log n).…”

Section: Introductionmentioning

confidence: 99%

“…Finally, Andoni and Nosatzki improved the running time to near-linear while maintaining a constant factor approximation [AN20]. Using the best result in string edit distance approximation [AN20], it is possible to improve the approximation factor of [Sah14] to O(log n). However, designing a constant factor approximation for Dyck edit distance in subquadratic time remains wide open.…”

Section: Introductionmentioning

confidence: 99%

Improved Approximation Algorithms for Dyck Edit Distance and RNA Folding

Das¹,

Kociumaka²,

Saha³

2021

Preprint

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The Dyck language, which consists of well-balanced sequences of parentheses, is one of the most fundamental context-free languages. The Dyck edit distance quantifies the number of edits (character insertions, deletions, and substitutions) required to make a given length-n parenthesis sequence well-balanced. RNA Folding involves a similar problem, where a closing parenthesis can match an opening parenthesis of the same type irrespective of their ordering. For example, in RNA Folding, both () and )( are valid matches, whereas the Dyck language only allows () as a match. Both of these problems have been studied extensively in the literature. Using fast matrix multiplication, it is possible to compute their exact solutions in time O(n 2.824 ) (Bringmann, Grandoni, Saha, V.-Williams, FOCS'16), and a (1 + )-multiplicative approximation is known with a running time of Ω(n 2.372 ).The impracticality of fast matrix multiplication often makes combinatorial algorithms much more desirable. Unfortunately, it is known that the problems of (exactly) computing Dyck edit distance and folding distance are at least as hard as Boolean matrix multiplication. Thereby, they are unlikely to admit truly subcubic-time combinatorial algorithms. In terms of fast approximation algorithms that are combinatorial in nature, the state of the art for Dyck edit distance is a O(log n)-factor approximation algorithm that runs in near-linear time (Saha, FOCS'14), whereas for RNA Folding only an n-additive approximation in Õ( n 2 ) time (Saha, FOCS'17) is known.In this paper, we make substantial improvements to the state of the art for Dyck edit distance (with any number of parenthesis types). We design a constant-factor approximation algorithm that runs in Õ(n 1.971 ) time (the first constant-factor approximation in subquadratic time). Moreover, we develop a (1 + )-factor approximation algorithm running in Õ( n 2 ) time, which improves upon the earlier additive approximation. Finally, we design a (3 + )-approximation that takes Õ( nd ) time, where d ≥ 1 is an upper bound on the sought distance.As for RNA folding, for any s ≥ 1, we design a multiplicative s-approximation algorithm that runs in O(n + n s 3 ) time. To the best of our knowledge, this is the first nontrivial approximation algorithm for RNA Folding that can go below the n 2 barrier. All our algorithms are combinatorial in nature.

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“…These strong hardness results naturally bring up the question whether LCS or edit distance can be efficiently approximated (namely, whether an algorithm with truly subquadratic time Opn 2´ε q for any constant ε ą 0, can produce a good approximation in the worst-case). In the last two decades, significant progress has been made towards designing efficient approximation algorithms for edit distance [14,13,15,9,7,21,23,29,17]; the latest achievement is a constant-factor approximation in almost-linear 1 time [8].…”

Section: Introductionmentioning

confidence: 99%

A Linear-Time $n^{0.4}$-Approximation for Longest Common Subsequence

Bringmann¹,

Cohen-Addad²,

Das³

2021

Preprint

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We consider the classic problem of computing the Longest Common Subsequence (LCS) of two strings of length n. While a simple quadratic algorithm has been known for the problem for more than 40 years, no faster algorithm has been found despite an extensive effort. The lack of progress on the problem has recently been explained by Abboud, Backurs, and Vassilevska Williams [FOCS'15] and Bringmann and Künnemann [FOCS'15] who proved that there is no subquadratic algorithm unless the Strong Exponential Time Hypothesis fails. This major roadblock for getting faster exact algorithms has led the community to look for subquadratic approximation algorithms for the problem.Yet, unlike the edit distance problem for which a constant-factor approximation in almost-linear time is known, very little progress has been made on LCS, making it a notoriously difficult problem also in the realm of approximation. For the general setting (where we make no assumption on the length of the optimum solution or the alphabet size), only a naive Opn ε{2 q-approximation algorithm with running time r Opn 2´ε q has been known, for any constant 0 ă ε ď 1. Recently, a breakthrough result by Hajiaghayi, Seddighin, Seddighin, and Sun [SODA'19] provided a linear-time algorithm that yields a Opn 0.497956 qapproximation in expectation; improving upon the naive Op ? nq-approximation for the first time. In this paper, we provide an algorithm that in time Opn 2´ε q computes an r Opn 2ε{5 q-approximation with high probability, for any 0 ă ε ď 1. Our result (1) gives an r Opn 0.4 q-approximation in linear time, improving upon the bound of Hajiaghayi, Seddighin, Seddighin, and Sun, (2) provides an algorithm whose approximation scales with any subquadratic running time Opn 2´ε q, improving upon the naive bound of Opn ε{2 q for any ε, and (3) instead of only in expectation, succeeds with high probability.

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Edit Distance in Near-Linear Time: it's a Constant Factor

Cited by 28 publications

References 28 publications

Gap Edit Distance via Non-Adaptive Queries: Simple and Optimal

Gap Edit Distance via Non-Adaptive Queries: Simple and Optimal

Improved Approximation Algorithms for Dyck Edit Distance and RNA Folding

A Linear-Time $n^{0.4}$-Approximation for Longest Common Subsequence

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