Scaling betweenness centrality using communication-efficient sparse matrix multiplication

Solomonik, Edgar; Besta, Maciej; Vella, Flavio; Hoefler, Torsten

doi:10.1145/3126908.3126971

Cited by 73 publications

(49 citation statements)

References 44 publications

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“…Additional study appears in Solomonik et al [27] and in Azad et al [4]. The latter proposes algorithms of various types of dimensionality and also employ permutations on the input matrices.…”

Section: Related Workmentioning

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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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%

Sparse matrix multiplication and triangle listing in the Congested Clique model

Censor-Hillel

Leitersdorf

Turner

2020

Theoretical Computer Science

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“…In addition, Solomonik et al (2017) have proposed a parallel BC algorithm for weighted graphs based on novel sparse matrix multiplication routines that has achieved impressive performance, which may provide further inspiration for accelerating our algorithm. We may also consider implementing a GPU algorithm for processing dynamic networks.…”

Section: Resultsmentioning

confidence: 99%

A GPU-based solution for fast calculation of the betweenness centrality in large weighted networks

Fan

Zhao

2017

PeerJ Computer Science

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Betweenness, a widely employed centrality measure in network science, is a decent proxy for investigating network loads and rankings. However, its extremely high computational cost greatly hinders its applicability in large networks. Although several parallel algorithms have been presented to reduce its calculation cost for unweighted networks, a fast solution for weighted networks, which are commonly encountered in many realistic applications, is still lacking. In this study, we develop an efficient parallel GPU-based approach to boost the calculation of the betweenness centrality (BC) for large weighted networks. We parallelize the traditional Dijkstra algorithm by selecting more than one frontier vertex each time and then inspecting the frontier vertices simultaneously. By combining the parallel SSSP algorithm with the parallel BC framework, our GPU-based betweenness algorithm achieves much better performance than its CPU counterparts. Moreover, to further improve performance, we integrate the work-efficient strategy, and to address the loadimbalance problem, we introduce a warp-centric technique, which assigns many threads rather than one to a single frontier vertex. Experiments on both realistic and synthetic networks demonstrate the efficiency of our solution, which achieves 2.9Â to 8.44Â speedups over the parallel CPU implementation. Our algorithm is open-source and free to the community; it is publicly available through https://dx.doi.org/10.6084/m9.figshare.4542405. Considering the pervasive deployment and declining price of GPUs in personal computers and servers, our solution will offer unprecedented opportunities for exploring betweenness-related problems and will motivate follow-up efforts in network science.

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“…We strongly believe that SlimSell can be used to accelerate other graph algorithms, for example schemes for solving Betweenness Centrality [35]. Potential speedups can be much higher because BFS is a data-driven scheme with different memory access patterns across iterations while many algorithms (e.g., Pagerank) have identical communication patterns in each superstep.…”

Section: Discussionmentioning

confidence: 99%

SlimSell: A Vectorizable Graph Representation for Breadth-First Search

Besta

Marending

Solomonik

et al. 2017

2017 IEEE International Parallel and Distributed Processing Symposium (IPDPS)

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Vectorization and GPUs will profoundly change graph processing. Traditional graph algorithms tuned for 32-or 64-bit based memory accesses will be inefficient on architectures with 512-bit wide (or larger) instruction units that are already present in the Intel Knights Landing (KNL) manycore CPU. Anticipating this shift, we propose SlimSell: a vectorizable graph representation to accelerate Breadth-First Search (BFS) based on sparse-matrix dense-vector (SpMV) products. SlimSell extends and combines the state-of-the-art SIMD-friendly Sell-C-σ matrix storage format with tropical, real, boolean, and sel-max semiring operations. The resulting design reduces the necessary storage (by up to 50%) and thus pressure on the memory subsystem. We augment SlimSell with the SlimWork and SlimChunk schemes that reduce the amount of work and improve load balance, further accelerating BFS. We evaluate all the schemes on Intel Haswell multicore CPUs, the state-of-the-art Intel Xeon Phi KNL manycore CPUs, and NVIDIA Tesla GPUs. Our experiments indicate which semiring offers highest speedups for BFS and illustrate that SlimSell accelerates a tuned Graph500 BFS code by up to 33%. This work shows that vectorization can secure high-performance in BFS based on SpMV products; the proposed principles and designs can be extended to other graph algorithms.

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Scaling betweenness centrality using communication-efficient sparse matrix multiplication

Cited by 73 publications

References 44 publications

Sparse matrix multiplication and triangle listing in the Congested Clique model

Sparse matrix multiplication and triangle listing in the Congested Clique model

A GPU-based solution for fast calculation of the betweenness centrality in large weighted networks

SlimSell: A Vectorizable Graph Representation for Breadth-First Search

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