Parallel Graph Connectivity in Log Diameter Rounds

Andoni, Alexandr; Song, Zhao; Stein, Clifford; Wang, Zhengyu; Zhong, Peilin

doi:10.1109/focs.2018.00070

Cited by 68 publications

(117 citation statements)

References 28 publications

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“…Independently and concurrently to our work, Andoni et al [6] have also studied MPC algorithms for the sparse connectivity problem with the goal of achieving improved performance on graphs with "better connectivity" structure by parametrizing based on the diameter of each connected component (as opposed to spectral gap in our paper). They develop an algorithm with n Ω(1) memory per machine and O(log D · log log N/n (n)) rounds, where D is the largest diameter of any connected component and N = Ω(m) is the total memory.…”

Section: Recent Developmentmentioning

confidence: 99%

“…Our algorithm uses a random walk data structure (new to our paper) to transform each connected component of the input graph to a random graph, and after that applies a novel leader-election algorithm to find components in O(log log n) rounds. On the other hand, [6] design a leader-election algorithm that runs in O(log log n) phases that are interleaved with an O(log D)-round procedure that increases the degree of vertices in the remaining graph (by partially computing the transitive closure of the graph) to prepare for the next phase. We note that the combination of our random walk primitive and our leader-election algorithm for random graphs is the main reason we achieve the improved round complexity compared to [6], albeit by depending on spectral gap instead of diameter (the result of [6] implies Ω((log log n) 2 ) rounds even on our final random graph instances as diameter of these graphs is Ω(log n)).…”

Section: Recent Developmentmentioning

confidence: 99%

See 1 more Smart Citation

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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Section: Recent Developmentmentioning

confidence: 99%

Section: Recent Developmentmentioning

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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“…Connectivity O(log log m/n n) O(log D · log log m/n n) [ O(log D · log log m/n n) [2] Figure 1: Round complexities of our algorithms in the AMPC model compared to the state-of-theart MPC algorithms. D denotes the diameter of the input graph.…”

Section: Ampc Mpcmentioning

confidence: 99%

“…In this section we provide an implementation of the recent undirected connectivity algorithm due to Andoni et al [2] in the AMPC model in O(log log T /n n) rounds. This improves upon the original result by a factor of log D, where D is the diameter of the input graph.…”

Section: Undirected Graph Connectivitymentioning

confidence: 99%

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Massively Parallel Computation via Remote Memory Access

Behnezhad

Dhulipala

Esfandiari

et al. 2019

The 31st ACM Symposium on Parallelism in Algorithms and Architectures

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We introduce the Adaptive Massively Parallel Computation (AMPC) model, which is an extension of the Massively Parallel Computation (MPC) model. At a high level, the AMPC model strengthens the MPC model by storing all messages sent within a round in a distributed data store. In the following round, all machines are provided with random read access to the data store, subject to the same constraints on the total amount of communication as in the MPC model. Our model is inspired by the previous empirical studies of distributed graph algorithms [28,9] using MapReduce and a distributed hash table service [17].This extension allows us to give new graph algorithms with much lower round complexities compared to the best known solutions in the MPC model. In particular, in the AMPC model we show how to solve maximal independent set in O(1) rounds and connectivity/minimum spanning tree in O(log log m/n n) rounds both using O(n δ ) space per machine for constant δ < 1. In the same memory regime for MPC, the best known algorithms for these problems require poly log n rounds. Our results imply that the 2-Cycle conjecture, which is widely believed to hold in the MPC model, does not hold in the AMPC model.

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Breaking the Linear-Memory Barrier in $$\mathsf {MPC}$$: Fast $$\mathsf {MIS}$$ on Trees with Strongly Sublinear Memory

Brandt

Fischer

Uitto

2019

Structural Information and Communication Complexity

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Recently, studying fundamental graph problems in the Massively Parallel Computation (MPC) framework, inspired by the MapReduce paradigm, has gained a lot of attention. An assumption common to a vast majority of approaches is to allow Ω(n) memory per machine, where n is the number of nodes in the graph and Ω hides polylogarithmic factors. However, as pointed out by Karloff et al. [SODA'10] and Czumaj et al. [STOC'18], it might be unrealistic for a single machine to have linear or only slightly sublinear memory.In this paper, we thus study a more practical variant of the MPC model which only requires substantially sublinear or even subpolynomial memory per machine. In contrast to the linear-memory MPC model and also to streaming algorithms, in this low-memory MPC setting, a single machine will only see a small number of nodes in the graph. We introduce a new and strikingly simple technique to cope with this imposed locality.In particular, we show that the Maximal Independent Set (MIS) problem can be solved efficiently, that is, in O(log 3 log n) rounds, when the input graph is a tree. This constitutes an almost exponential speed-up over the low-memory MPC algorithm in O( √ log n)-algorithm in a concurrent work by Ghaffari and Uitto [SODA'19] and substantially reduces the local memory from Ω(n) required by the recent O(log log n)-round MIS algorithm of Ghaffari et al. [PODC'18] to n ε for any ε > 0, without incurring a significant loss in the round complexity. Moreover, it demonstrates how to make use of the all-to-all communication in the MPC model to almost exponentially improve on the corresponding bound in the LOCAL and PRAM models by Lenzen and Wattenhofer [PODC'11].

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Parallel Graph Connectivity in Log Diameter Rounds

Cited by 68 publications

References 28 publications

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

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

Massively Parallel Computation via Remote Memory Access

Breaking the Linear-Memory Barrier in $$\mathsf {MPC}$$: Fast $$\mathsf {MIS}$$ on Trees with Strongly Sublinear Memory

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