Approximate Matching for Run-Length Encoded Strings Is 3sum-Hard

Chen, Kuan‐Yu; Hsu, Ping-Hui; Chao, Kun-Mao

doi:10.1007/978-3-642-02441-2_15

Cited by 6 publications

(2 citation statements)

References 21 publications

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“…It is often conjectured that 3SUM requires Ωpn 2 q time and that Integer3SUM requires Ωpn 2´op1q q time [32,3]. These conjectures have been shown to imply strong lower bounds on numerous problems in computational geometry [25,4,9,35] dynamic graph algorithms [32,3], and pattern matching [1,6,14,17]. For example, the 3SUM conjecture implies that the following problems require at least Ωpn 2 q time.…”

Section: Implications Of the 3sum Conjecturesmentioning

confidence: 99%

Threesomes, Degenerates, and Love Triangles

Grønlund

Pettie

2018

J. ACM

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The 3SUM problem is to decide, given a set of n real numbers, whether any three sum to zero. It is widely conjectured that a trivial Opn 2 q-time algorithm is optimal and over the years the consequences of this conjecture have been revealed. This 3SUM conjecture implies Ωpn 2 q lower bounds on numerous problems in computational geometry and a variant of the conjecture implies strong lower bounds on triangle enumeration, dynamic graph algorithms, and string matching data structures.In this paper we refute the 3SUM conjecture. We prove that the decision tree complexity of 3SUM is Opn 3{2 ?log nq and give two subquadratic 3SUM algorithms, a deterministic one running in Opn 2 {plog n{ log log nq 2{3 q time and a randomized one running in Opn 2 plog log nq 2 { log nq time with high probability. Our results lead directly to improved bounds for k-variate linear degeneracy testing for all odd k ě 3. The problem is to decide, given a linear function f px 1 , . . . , x k q " α 0`ř 1ďiďk α i x i and a set A Ă R, whether 0 P f pA k q. We show the decision tree complexity of this problem is Opn k{2 ? log nq. Finally, we give a subcubic algorithm for a generalization of the pmin,`q-product over realvalued matrices and apply it to the problem of finding zero-weight triangles in weighted graphs. We give a depth-Opn 5{2 ?log nq decision tree for this problem, as well as an algorithm running in time Opn 3 plog log nq 2 { log nq.

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Section: Implications Of the 3sum Conjecturesmentioning

confidence: 99%

Threesomes, Degenerates, and Love Triangles

Grønlund

Pettie

2018

J. ACM

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“…More recently, the 3-SUM Conjecture has been used in surprising ways to show polynomial lower bounds for purely combinatorial problems in dynamic algorithms [40,2] and Graph algorithms [40,32,48]. The only previous work relating 3-SUM to a Stringology problem, to our knowledge, is the result of Chen et al [12] showing that under the 3-SUM Conjecture, when the input strings are encodings of much longer strings, using Run-Length-Encoding, then the string matching with don't cares problem requires time that is quadratic in the lengths of the compressions. This string problem, however, is strongly related to geometric problems and is less "combinatorial" than the problems we consider here (e.g.…”

Section: -Sum Hardnessmentioning

confidence: 99%

Consequences of Faster Alignment of Sequences

Abboud

Williams

Weimann

2014

Lecture Notes in Computer Science

121

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Abstract. The Local Alignment problem is a classical problem with applications in biology. Given two input strings and a scoring function on pairs of letters, one is asked to find the substrings of the two input strings that are most similar under the scoring function. The best algorithms for Local Alignment run in time that is roughly quadratic in the string length. It is a big open problem whether substantially subquadratic algorithms exist. In this paper we show that for all ε > 0, an O(n 2−ε ) time algorithm for Local Alignment on strings of length n would imply breakthroughs on three longstanding open problems: it would imply that for some δ > 0, 3SUM on n numbers is in O(n 2−δ ) time, CNF-SAT on n variables is in O((2 − δ) n ) time, and Max Weight 4-Clique is in O(n 4−δ ) time. Our result for CNF-SAT also applies to the easier problem of finding the longest common substring of binary strings with don't cares. We also give strong conditional lower bounds for the more general Multiple Local Alignment problem on k strings, under both k-wise and SP scoring, and for other string similarity problems such as Global Alignment with gap penalties and normalized Longest Common Subsequence.

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Recent Advances in Text-to-Pattern Distance Algorithms

Uznański

2020

Lecture Notes in Computer Science

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Approximate Matching for Run-Length Encoded Strings Is 3sum-Hard

Cited by 6 publications

References 21 publications

Threesomes, Degenerates, and Love Triangles

Threesomes, Degenerates, and Love Triangles

Consequences of Faster Alignment of Sequences

Recent Advances in Text-to-Pattern Distance Algorithms

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