Improved Rectangular Matrix Multiplication using Powers of the Coppersmith-Winograd Tensor

Gall, François Le; Urrutia, Florent

doi:10.1137/1.9781611975031.67

Cited by 87 publications

(38 citation statements)

References 23 publications

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“…This algorithm is designed to reduce the number of multiplication operations, which is not a direct concern for distributed algorithms, in which the main cost is due to communication. Le Gall [14] improved this result for some range of sparsity by improving general rectangular matrix multiplication, for which a further improvement was recently given by Le Gall and Urrutia [17]. Kaplan et al [20] give an algorithm for multiplying sparse rectangular matrices, and Amossen and Pagh [3] give a fast algorithm for the case of sparse square matrices for which the product is also sparse.…”

Section: Related Workmentioning

confidence: 97%

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Sparse matrix multiplication and triangle listing in the Congested Clique model

Censor-Hillel

Leitersdorf

Turner

2020

Theoretical Computer Science

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We show how to multiply two n × n matrices S and T over semirings in the Congested Clique model, where n nodes communicate in a fully connected synchronous network using O(log n)-bit messages, within O(nz(S) 1/3 nz(T ) 1/3 /n+1) rounds of communication, where nz(S) and nz(T ) denote the number of non-zero elements in S and T , respectively. By leveraging the sparsity of the input matrices, our algorithm greatly reduces communication costs compared with general multiplication algorithms [Censor-Hillel et al., PODC 2015], and thus improves upon the state-of-the-art for matrices with o(n 2 ) non-zero elements. Moreover, our algorithm exhibits the additional strength of surpassing previous solutions also in the case where only one of the two matrices is such. Particularly, this allows to efficiently raise a sparse matrix to a power greater than 2. As applications, we show how to speed up the computation on non-dense graphs of 4-cycle counting and all-pairs-shortest-paths.Our algorithmic contribution is a new deterministic method of restructuring the input matrices in a sparsity-aware manner, which assigns each node with element-wise multiplication tasks that are not necessarily consecutive but guarantee a balanced element distribution, providing for communication-efficient multiplication.Moreover, this new deterministic method for restructuring matrices may be used to restructure the adjacency matrix of input graphs, enabling faster deterministic solutions for graph related problems. As an example, we present a new sparsity aware, deterministic algorithm which solves the triangle listing problem in O(m/n 5/3 + 1) rounds, a complexity that was previously obtained by a randomized algorithm [Pandurangan et al., SPAA 2018], and that matches the known lower bound ofΩ(n 1/3 ) when m = n 2 of [Izumi and Le Gall, PODC 2017, Pandurangan et al., SPAA 2018]. Naturally, our triangle listing algorithm also implies triangle counting within the same complexity of O(m/n 5/3 + 1) rounds, which is (possibly more than) a cubic improvement over the previously known deterministic O(m 2 /n 3 )-round algorithm [Dolev et al., DISC 2012]. * A preliminary version of this paper appeared in OPODIS 2018.An important case of Theorem 1, especially when squaring the adjacency matrix of a graph in order to solve graph problems, is when the sparsities of the input matrices are roughly the same. In such a case, Theorem 1 gives the following.Corollary 1. Given two n × n matrices S and T , where O(nz(S)) = O(nz(T )) = m, Algorithm SMM deterministically computes the product P = S · T over a semiring in the Congested Clique model, within O(m 2/3 /n + 1) rounds.Notice that for m = O(n 2 ), Corollary 1 gives the same complexity of O(n 1/3 ) rounds as given by the semiring multiplication of [9].We apply Algorithm SMM to 4-cycle counting, obtaining the following.Theorem 2. There is a deterministic algorithm that computes the number of 4-cycles in an n-node graph G in O(m 2/3 /n + 1) rounds in the Congested Clique model, where m is the number of edges of G.No...

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Section: Related Workmentioning

confidence: 97%

“…In many cases, multiplication is required to be carried out for sparse matrices, and this need has been generating much effort in designing algorithms that are faster given sparse inputs, both in sequential (e.g., [3,14,17,20,32]) and parallel (e.g., [4-8, 22, 23, 27]) settings.…”

Section: Introductionmentioning

confidence: 99%

Sparse matrix multiplication and triangle listing in the Congested Clique model

Censor-Hillel

Leitersdorf

Turner

2020

Theoretical Computer Science

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“…Case 2: the input graph is directed. The total preprocessing time is We substitute ω(0.7) ≤ 2.154399 and ω(0.75) ≤ 2.187543 [11], and obtain: Therefore, given a directed graph G = (V, E), there is a DSO with Õ n max{2a+ω(1−a),3−a} M 2 = Õ(n 2.723277 M 2 ).…”

Section: :7mentioning

confidence: 99%

Improved distance sensitivity oracles with subcubic preprocessing time

Ren

2022

Journal of Computer and System Sciences

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We consider the problem of building Distance Sensitivity Oracles (DSOs). Given a directed graph G = (V, E) with edge weights in {1, 2, . . . , M }, we need to preprocess it into a data structure, and answer the following queries: given vertices u, v, x ∈ V , output the length of the shortest path from u to v that does not go through x. Our main result is a simple DSO with Õ(n 2.7233 M 2 ) preprocessing time and O(1) query time. Moreover, if the input graph is undirected, the preprocessing time can be improved to Õ(n 2.6865 M 2 ). Our algorithms are randomized with correct probability ≥ 1 − 1/n c , for a constant c that can be made arbitrarily large. Previously, there is a DSO with Õ(n 2.8729 M ) preprocessing time and polylog(n) query time [Chechik and Cohen, STOC'20].At the core of our DSO is the following observation from [Bernstein and Karger, STOC'09]: if there is a DSO with preprocessing time P and query time Q, then we can construct a DSO with preprocessing time P + Õ(M n 2 ) • Q and query time O( 1). (Here Õ(•) hides polylog(n) factors.) 2012 ACM Subject Classification Theory of computation → Shortest paths; Theory of computation → Data structures design and analysis Keywords and phrases Graph theory, Failure-prone structures Digital Object Identifier 10.4230/LIPIcs.ESA.2020.79

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“…We could sharpen our results using the so-called exponent ω 2 of rectangular matrix multiplication in size (s, s) × (s, s 2 ). We can of course take ω 2 ≤ ω + 1 ≤ 3.373, but the better result ω 2 ≤ 3.252 is known [27]. We will not use these refinments in this paper.…”

Section: Polynomial and Matrix Arithmeticmentioning

confidence: 99%

Computing the Characteristic Polynomial of a Finite Rank Two Drinfeld Module

Musleh

Schost

2019

Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation

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Motivated by finding analogues of elliptic curve point counting techniques, we introduce one deterministic and two new Monte Carlo randomized algorithms to compute the characteristic polynomial of a finite rank-two Drinfeld module. We compare their asymptotic complexity to that of previous algorithms given by Gekeler, Narayanan and Garai-Papikian and discuss their practical behavior. In particular, we find that all three approaches represent either an improvement in complexity or an expansion of the parameter space over which the algorithm may be applied. Some experimental results are also presented.

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Improved Rectangular Matrix Multiplication using Powers of the Coppersmith-Winograd Tensor

Cited by 87 publications

References 23 publications

Sparse matrix multiplication and triangle listing in the Congested Clique model

Sparse matrix multiplication and triangle listing in the Congested Clique model

Improved distance sensitivity oracles with subcubic preprocessing time

Computing the Characteristic Polynomial of a Finite Rank Two Drinfeld Module

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