Randomized Composable Coresets for Matching and Vertex Cover

Assadi, Sepehr; Khanna, Sanjeev

doi:10.1145/3087556.3087581

Cited by 36 publications

(87 citation statements)

References 71 publications

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“…The main algorithmic question of this area is then: for which problems can we design MPC algorithms that are faster than the best PRAM algorithms? Indeed, this question has been the focus of several recent papers, see, e.g., [KSV10,LMSV11,EIM11,ANOY14,AG18,AK17,IMS17,CLM + 18]. Graph problems have been particularly well studied and one fundamental problem is connectivity in a graph.…”

Section: Introductionmentioning

confidence: 99%

Parallel Graph Connectivity in Log Diameter Rounds

Andoni

Song

Stein

et al. 2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

112

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Many modern parallel systems, such as MapReduce, Hadoop and Spark, can be modeled well by the MPC model. The MPC model captures well coarse-grained computation on large data -data is distributed to processors, each of which has a sublinear (in the input data) amount of memory and we alternate between rounds of computation and rounds of communication, where each machine can communicate an amount of data as large as the size of its memory. This model is stronger than the classical PRAM model, and it is an intriguing question to design algorithms whose running time is smaller than in the PRAM model.One fundamental graph problem is connectivity. On an undirected graph with n nodes and m edges, O(log n) round connectivity algorithms have been known for over 35 years. However, no algorithms with better complexity bounds were known. In this work, we give fully scalable, faster algorithms for the connectivity problem, by parameterizing the time complexity as a function of the diameter of the graph. Our main result is a O(log D log log m/n n) time connectivity algorithm for diameter-D graphs, using Θ(m) total memory. If our algorithm can use more memory, it can terminate in fewer rounds, and there is no lower bound on the memory per processor.We extend our results to related graph problems such as spanning forest, finding a DFS sequence, exact/approximate minimum spanning forest, and bottleneck spanning forest. We also show that achieving similar bounds for reachability in directed graphs would imply faster boolean matrix multiplication algorithms.We introduce several new algorithmic ideas. We describe a general technique called double exponential speed problem size reduction which roughly means that if we can use total memory N to reduce a problem from size n to n/k, for k = (N/n) Θ(1) in one phase, then we can solve the problem in O(log log N/n n) phases. In order to achieve this fast reduction for graph connectivity, we use a multistep algorithm. One key step is a carefully constructed truncated broadcasting scheme where each node broadcasts neighbor sets to its neighbors in a way that limits the size of the resulting neighbor sets. Another key step is random leader contraction, where we choose a smaller set of leaders than many previous works do.

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Section: Introductionmentioning

confidence: 99%

Parallel Graph Connectivity in Log Diameter Rounds

Andoni

Song

Stein

et al. 2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

112

View full text Add to dashboard Cite

show abstract

“…This is in contrast to the Ω(log n) round needed in the standard PRAM model for these problems (see, e.g., [18,25,30,35,47,49,57]). Since then, numerous algorithms have been designed for various graph problems that achieve O(1) round-complexity with local memory n 1+Ω(1) on each machine (see, e.g., [2,9,39,40] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Massively Parallel Algorithms for Finding Well-Connected Components in Sparse Graphs

Assadi

Sun

Weinstein

2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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A fundamental question that shrouds the emergence of massively parallel computing (MPC) platforms is how can the additional power of the MPC paradigm (more local storage and computational power) be leveraged to achieve faster algorithms compared to classical parallel models such as PRAM?Previous research has identified the sparse graph connectivity problem as a major obstacle to such improvement: While classical logarithmic-round PRAM algorithms for finding connected components in any n-vertex graph have been known for more than three decades, no o(log n)round MPC algorithms are known for this task with truly sublinear in n memory per machine. This problem arises when processing massive yet sparse graphs with O(n) edges, for which the interesting setting of parameters is n 1−Ω(1) memory per machine. It is conjectured that achieving an o(log n)-round algorithm for connectivity on general sparse graphs with n 1−Ω(1) per-machine memory may not be possible, and this conjecture also forms the basis for multiple conditional hardness results on the round complexity of other problems in the MPC model.In this paper, we take an opportunistic approach towards the sparse graph connectivity problem, by designing an algorithm with improved performance guarantees in terms of the connectivity structure of the input graph. Formally, we design an MPC algorithm that finds all connected components with spectral gap at least λ in a graph in O(log log n + log (1/λ)) MPC rounds and n Ω(1) memory per machine. While this algorithm still requires Ω(log n) rounds in the worst-case when components are "weakly" connected (i.e., λ ≈ 1/n), it achieves an exponential round reduction on sparse "well-connected" components (i.e., λ ≥ 1/polylog(n)) using only n Ω(1) memory per machine and O(n) total memory, and still operates in o(log n) rounds even when λ = 1/n o(1) . To best of our knowledge, this is the first non-trivial (and indeed exponential) improvement in the round complexity over PRAM algorithms, for a natural class of sparse connectivity instances.

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“…The idea is to partition the data, compute on each part a representative subset called a core-set, then solve the problem on the union of the core-sets. Relevant to this is the work of Assadi et al [3] (extending work by Assadi and Khanna [4]), which designs 3/2 + ε (resp. 3 + ε) approximate core-sets for unweighted matching (resp.…”

Section: Related Workmentioning

confidence: 89%

Greedy and Local Ratio Algorithms in the MapReduce Model

Harvey

Liaw

Liu

2018

Proceedings of the 30th on Symposium on Parallelism in Algorithms and Architectures

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MapReduce has become the de facto standard model for designing distributed algorithms to process big data on a cluster. There has been considerable research on designing efficient MapReduce algorithms for clustering, graph optimization, and submodular optimization problems. We develop new techniques for designing greedy and local ratio algorithms in this setting. Our randomized local ratio technique gives 2-approximations for weighted vertex cover and weighted matching, and an f -approximation for weighted set cover, all in a constant number of MapReduce rounds. Our randomized greedy technique gives algorithms for maximal independent set, maximal clique, and a (1+ε) ln ∆-approximation for weighted set cover. We also give greedy algorithms for vertex colouring with (1 + o(1))∆ colours and edge colouring with (1 + o(1))∆ colours.

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Randomized Composable Coresets for Matching and Vertex Cover

Cited by 36 publications

References 71 publications

Parallel Graph Connectivity in Log Diameter Rounds

Parallel Graph Connectivity in Log Diameter Rounds

Massively Parallel Algorithms for Finding Well-Connected Components in Sparse Graphs

Greedy and Local Ratio Algorithms in the MapReduce Model

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