More Logarithmic-Factor Speedups for 3SUM, (median,+)-Convolution, and Some Geometric 3SUM-Hard Problems

Chan, Timothy M.

doi:10.1137/1.9781611975031.57

Cited by 10 publications

(9 citation statements)

References 21 publications

(35 reference statements)

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“…If the input numbers are real numbers instead of integers (now in the Real-RAM model of computation), Grønlund and Pettie [104] gave an O(n 2 (log log n) 2/3 / log 2/3 n) time algorithm. This runtime was recently improved by Chan [56] to n 2 (log log n) O(1) / log 2 n, almost matching the known running time for integer inputs.…”

Section: -Sum Hypothesismentioning

confidence: 75%

On Some Fine-Grained Questions in Algorithms and Complexity

Williams

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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In recent years, a new "fine-grained" theory of computational hardness has been developed, based on "fine-grained reductions" that focus on exact running times for problems. Mimicking NP-hardness, the approach is to (1) select a key problem X that for some function t, is conjectured to not be solvable by any O(t(n) 1−ε ) time algorithm for ε > 0, and (2) reduce X in a fine-grained way to many important problems, thus giving tight conditional time lower bounds for them. This approach has led to the discovery of many meaningful relationships between problems, and to equivalence classes.The main key problems used to base hardness on have been: the 3-SUM problem, the CNF-SAT problem (based on the Strong Exponential Time Hypothesis (SETH)) and the All Pairs Shortest Paths Problem. Research on SETH-based lower bounds has flourished in particular in recent years showing that the classical algorithms are optimal for problems such as Approximate Diameter, Edit Distance, Frechet Distance and Longest Common Subsequence.This paper surveys the current progress in this area, and highlights some exciting new developments. IntroductionArguably the main goal of the theory of algorithms is to study the worst case time complexity of fundamental computational problems. When considering a problem P , we fix a computational model, such as a Random Access Machine (RAM) or a Turing machine (TM). Then we strive to develop an efficient algorithm that solves P and to prove that for a (hopefully slow growing) function t(n), the algorithm solves P on instances of size n in O(t(n)) time in that computational model. The gold standard for the running time t(n) is linear time, O(n); to solve most problems, one needs to at least read the input, and so linear time is necessary. The theory of algorithms has developed a wide variety of techniques. These have yielded near-linear time algorithms for many diverse problems. For instance, it is known since the 1960s and 70s (e.g. [143,144,145,99]) that Depth-First Search (DFS) and Breadth-First Search (BFS) run in linear time in graphs, and that using these techniques one can obtain linear time algorithms (on a RAM) for many interesting graph problems: Single-Source Shortest paths, Topological Sort of a Directed Acyclic Graph, Strongly Connected Components, Testing Graph Planarity etc. More recent work has shown that even more complex problems such as Approximate Max Flow, Maximum Bipartite Matching, Linear Systems on Structured Matrices, and many others, admit close to linear time algorithms, by combining combinatorial and linear algebraic techniques (see e.g. [140,64,141,116,117,67,68,65,66,113]).Nevertheless, for most problems of interest, the fastest known algorithms run much slower than linear time. This is perhaps not too surprising. Time hierarchy theorems show that for most computational models, for any computable function t(n) ≥ n, there exist problems that are solvable in O(t(n)) time but are NOT solvable in O(t(n) 1−ε ) time for ε > 0 (this was first proven for TMs [95], see [124] for more).Ti...

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Section: -Sum Hypothesismentioning

confidence: 75%

On Some Fine-Grained Questions in Algorithms and Complexity

Williams

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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“…Combining the reduction provided by Theorem 1 with the real RAM algorithm for 3-SUM from Chan [14], we obtain the following.…”

Section: Problem 3 (Triangle)mentioning

confidence: 94%

“…This contrasts with our current knowledge on the related 3-SUM-hard problem of finding three collinear points, also known as GENERAL POSITION TESTING. Despite recent attempts [10,14], it is still an open problem to find a subquadratic algorithm for GENERAL POSITION TESTING.…”

Section: Corollarymentioning

confidence: 99%

“…Our results are motivated by the nontrivial improved upper bounds on the complexity of 3-SUM and k-SUM proven in the recent years. While it has long been conjectured that no subquadratic algorithm for 3-SUM existed, it is now known to be solvable in time O((n 2 / log n)(log log n) 2 ) in the real RAM model, and in time O((n 2 / log 2 n)(log log n) O (1) ) if we allow bitwise operations on fixed-length words [25,20,22,14]. The existence of an O(n 2−δ ) algorithm for some δ > 0 remains an open problem.…”

Section: Introductionmentioning

confidence: 99%

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Geometric Pattern Matching Reduces to k -SUM

Aronov

Cardinal

2021

Discrete Comput Geom

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We prove that some exact geometric pattern matching problems reduce in linear time to k-SUM when the pattern has a fixed size k. This holds in the real RAM model for searching for a similar copy of a set of k ≥ 3 points within a set of n points in the plane, and for searching for an affine image of a set of k ≥ d + 2 points within a set of n points in d-space.As corollaries, we obtain improved real RAM algorithms and decision trees for the two problems. In particular, they can be solved by algebraic decision trees of near-linear height. ACM Subject ClassificationTheory of computation → Design and analysis of algorithms; Theory of computation → Computational geometry Keywords and phrases Geometric pattern matching, k-SUM problem, Linear decision trees

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“…In 3SUM one is given n integers and is asked whether three of them sum to 0. The problem is easy to solve in O n 2 time, and slightly subquadratic time algorithms exist [4,11]. 3SUM is a central problem in fine-grained complexity [44].…”

Section: Our Contributionsmentioning

confidence: 99%

Monochromatic Triangles, Intermediate Matrix Products, and Convolutions

Lincoln,

Polak,

Williams

2020

Preprint

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The most studied linear algebraic operation, matrix multiplication, has surprisingly fast O(n ω ) time algorithms for ω < 2.373. On the other hand, the (min, +) matrix product which is at the heart of many fundamental graph problems such as All-Pairs Shortest Paths, has received only minor n o(1) improvements over its brute-force cubic running time and is widely conjectured to require n 3−o(1) time. There is a plethora of matrix products and graph problems whose complexity seems to lie in the middle of these two problems. For instance, the Min-Max matrix product, the Minimum Witness matrix product, All-Pairs Shortest Paths in directed unweighted graphs and determining whether an edge-colored graph contains a monochromatic triangle, can all be solved in O(n (3+ω)/2 ) time. While slight improvements are sometimes possible using rectangular matrix multiplication, if ω = 2, the best runtimes for these "intermediate" problems are all O(n 2.5 ).A similar phenomenon occurs for convolution problems. Here, using the FFT, the usual (+, ×)-convolution of two n-length sequences can be solved in O(n log n) time, while the (min, +)convolution is conjectured to require n 2−o(1) time, the brute force running time for convolution problems. There are analogous intermediate problems that can be solved in O(n 1.5 ) time, but seemingly not much faster: Min-Max convolution, Minimum Witness convolution, etc.Can one improve upon the running times for these intermediate problems, in either the matrix product or the convolution world? Or, alternatively, can one relate these problems to each other and to other key problems in a meaningful way?This paper makes progress on these questions by providing a network of fine-grained reductions. We show for instance that APSP in directed unweighted graphs and Minimum Witness product can be reduced to both the Min-Max product and a variant of the monochromatic triangle problem, so that a significant improvement over n (3+ω)/2 time for any of the latter problems would result in a similar improvement for both of the former problems. We also show that a natural convolution variant of monochromatic triangle is fine-grained equivalent to the famous 3SUM problem. As this variant is solvable in O(n 1.5 ) time and 3SUM is in O(n 2 ) time (and is conjectured to require n 2−o(1) time), our result gives the first fine-grained equivalence between natural problems of different running times. We also relate 3SUM to monochromatic triangle, and a coin change problem to monochromatic convolution, and thus to 3SUM.

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More Logarithmic-Factor Speedups for 3SUM, (median,+)-Convolution, and Some Geometric 3SUM-Hard Problems

Cited by 10 publications

References 21 publications

On Some Fine-Grained Questions in Algorithms and Complexity

On Some Fine-Grained Questions in Algorithms and Complexity

Geometric Pattern Matching Reduces to k -SUM

Monochromatic Triangles, Intermediate Matrix Products, and Convolutions

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