Algebraic Methods in the Congested Clique

Censor-Hillel, Keren; Kaski, Petteri; Korhonen, Janne H.; Lenzen, Christoph; Paz, Ami; Suomela, Jukka

doi:10.1145/2767386.2767414

Cited by 64 publications

(32 citation statements)

References 76 publications

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“…The best currently known upper bound is ω < 2.3728639 [15], implying O(n 0.372 ) rounds for the above. Later, Censor-Hillel et al [9] gave a deterministic algorithm for (general) matrix multiplication over semirings, completing in O(n 1/3 ) rounds, and a deterministic algorithm for (general) matrix multiplication over rings, completing in O(n 1−2/ω ) rounds, which by the current known upper bound on ω is O(n 0.158 ). The latter is a Strassen-like algorithm, exploiting known schemes for computing the product of two matrices over a ring without directly computing all n 3 element-wise multiplications.…”

Section: Related Workmentioning

confidence: 99%

“…Algorithm 5: Compute-Receiving: Code for node v ∈ [n], which is also denoted v i,j,k . 9 Let A i,j,1 , . .…”

Section: Compute-receivingmentioning

confidence: 99%

“…The work of Censor-Hillel et al [9] recently showed that known matrix multiplication algorithms for the parallel setting can be adapted to the distributed Congested Clique model, which consists of n nodes in a fully connected synchronous network, limited by a bandwidth of O(log n) bits per message. Subsequently, this significantly improved the state-of-the-art for a variety of tasks, including triangle and 4-cycle counting, girth computations, and (un)weighted/(un)directed allpairs-shortest-paths (APSP).…”

Section: Introductionmentioning

confidence: 99%

“…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.Notice that for m = O(n 3/2 ) this establishes 4-cycle counting in a constant number of rounds.…”

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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: 99%

“…Algorithm 5: Compute-Receiving: Code for node v ∈ [n], which is also denoted v i,j,k . 9 Let A i,j,1 , . .…”

Section: Compute-receivingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Sparse matrix multiplication and triangle listing in the Congested Clique model

Censor-Hillel

Leitersdorf

Turner

2020

Theoretical Computer Science

Self Cite

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“…The main algorithmic issue lies then in dealing with the potential congestion caused by the bandwidth restrictions. Indeed, there has been a lot of recent work in studying various fundamental problems in the Congested Clique model, including facility location [12,6], minimum spanning tree (MST) [22,14], shortest paths and distances [7,15,25], triangle finding [10,9], subgraph detection [10], ruling sets [6,14], sorting [28,21], and routing [21]. The modelling assumption in solving these problems is that the input graph G = (V, E) is "embedded" in the Congested Clique, that is, each node of G is uniquely mapped to a machine and the edges of G are naturally mapped to the links between the corresponding machines (cf.…”

Section: Introductionmentioning

confidence: 99%

Toward Optimal Bounds in the Congested Clique

Hegeman

Pandurangan

Pemmaraju

et al. 2015

Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing

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We study two fundamental graph problems, Graph Connectivity (GC) and Minimum Spanning Tree (MST), in the well-studied Congested Clique model, and present several new bounds on the time and message complexities of randomized algorithms for these problems. No non-trivial (i.e., super-constant) time lower bounds are known for either of the aforementioned problems; in particular, an important open question is whether or not constant-round algorithms exist for these problems. We make progress toward answering this question by presenting randomized Monte Carlo algorithms for both problems that run in O(log log log n) rounds (where n is the size of the clique). Our results improve by an exponential factor on the long-standing (deterministic) time bound of O(log log n) rounds for these problems due to Lotker et al. (SICOMP 2005). Our algorithms make use of several algorithmic tools including graph sketching, random sampling, and fast sorting.The second contribution of this paper is to present several almosttight bounds on the message complexity of these problems. Specifically, we show that Ω(n 2 ) messages are needed by any algorithm (including randomized Monte Carlo algorithms, and regardless of the number of rounds) that solves the GC (and hence also the MST) problem if each machine in the Congested Clique has initial knowledge only of itself (the so-called KT0 model). In contrast, if the machines have initial knowledge of their neighbors' IDs (the so-called KT1 model), we present a randomized Monte Carlo algorithm for MST that uses O(n polylog n) messages and runs in O(polylog n) rounds. To complement this, we also present a lower bound in the KT1 model that shows that Ω(n) messages are required by any al- * gorithm that solves GC, regardless of the number of rounds used. Our results are a step toward understanding the power of randomization in the Congested Clique with respect to both time and message complexity.

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Robust reconstruction of Barabási‐Albert networks in the broadcast congested clique model

et al. 2015

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In the broadcast version of the congested clique model, n nodes communicate in synchronous rounds by writing O(log n)-bit messages on a whiteboard, which is visible to all of them. The joint input to the nodes is an undirected n-node graph G, with node i receiving the list of its neighbors in G. Our goal is to design a protocol at the end of which the information contained in the whiteboard is enough for reconstructing G. It has already been shown that there is a one-round protocol for reconstructing graphs with bounded degeneracy. The main drawback of that protocol is that the degeneracy m of the input graph G must be known a priori by the nodes. Moreover, the protocol fails when applied to graphs with degeneracy larger than m. In this paper we address this issue by looking for robust reconstruction protocols, that is, protocols which always give the correct answer and work efficiently when the input is restricted to a certain class. We introduce a very simple, two-round protocol that we call Robust-Reconstruction. We prove that this protocol is robust for reconstructing the class of Barabási-Albert trees with (expected) message size O(log n). Moreover, we present computational evidence suggesting that Robust-Reconstruction also generates logarithmic size messages for arbitrary Barabási-Albert networks. Finally, we stress the importance of the preferential attachment mechanism (used in the construction of Barabási-Albert networks) by proving that RobustReconstruction does not generate short messages for random recursive 1 trees.

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Algebraic Methods in the Congested Clique

Cited by 64 publications

References 76 publications

Sparse matrix multiplication and triangle listing in the Congested Clique model

Sparse matrix multiplication and triangle listing in the Congested Clique model

Toward Optimal Bounds in the Congested Clique

Robust reconstruction of Barabási‐Albert networks in the broadcast congested clique model

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