MST in <i>O</i>(1) Rounds of Congested Clique

Jurdziński, Tomasz; Nowicki, Krzysztof

doi:10.1137/1.9781611975031.167

Cited by 66 publications

(57 citation statements)

References 18 publications

(52 reference statements)

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“…The next set of improvements reduced the memory per machine to O(n) (possibly at the cost of a slight increase in the number of rounds). For example, an O(1) round algorithm for MST and connectivity using only O(n) memory per machine has been proposed in [33] building on previous work in [27,31,43] (see also [3,13,42] for related results). A series of very recent papers [7,8,19,26,38], initiated by a breakthrough result of [19], have also achieved an O(log log n)-round algorithms for different graph problems such as matching, vertex cover, and MIS in the MPC model, when the memory per machine is O(n) or even O(n/polylog(n)).…”

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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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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“…The paper[JN18] gives an O(1) algorithm for the MST problem in the Congested Clique model, which can be simulated in the MPC model with a O(n) memory limit of a single machine. Furthermore, small changes in the analysis provided in[JN18] give O(m + n log 2 n) bound on global memory of the Connected Components algorithm.…”

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confidence: 99%

Faster Algorithms for Edge Connectivity via Random 2-Out Contractions

Ghaffari

Nowicki

Thorup

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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We provide a simple new randomized contraction approach to the global minimum cut problem for simple undirected graphs. The contractions exploit 2-out edge sampling from each vertex rather than the standard uniform edge sampling. We demonstrate the power of our new approach by obtaining better algorithms for sequential, distributed, and parallel models of computation. Our end results include the following randomized algorithms for computing edge connectivity, with high probability 1 :• Two sequential algorithms with complexities O(m log n) and O(m+n log 3 n). These improve on a long line of developments including a celebrated O(m log 3 n) algorithm of Karger [STOC'96] and the state of the art O(m log 2 n(log log n) 2 ) algorithm of Henzinger et al. [SODA'17]. Moreover, our O(m + n log 3 n) algorithm is optimal when m = Ω(n log 3 n).• AnÕ(n 0.8 D 0.2 + n 0.9 ) round distributed algorithm, where D denotes the graph diameter. This improves substantially on a recent breakthrough of Daga et al. [STOC'19], which achieved a round complexity ofÕ(n 1−1/353 D 1/353 + n 1−1/706 ), hence providing the first sublinear distributed algorithm for exactly computing the edge connectivity.• The first O(1) round algorithm for the massively parallel computation setting with linear memory per machine.

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“…Distributed MST is one of the most fundamental problems in CONGEST, with a wide range of works, a very short subset of which include [14,26,15,8,9,22,17,23,10]. In particular, Ghaffari et al [15] gave a simple MST algorithm using a framework called low-congestion shortcuts.…”

Section: Related Workmentioning

confidence: 99%

A distributed algorithm for directed minimum-weight spanning tree

Fischer

Oshman

2021

Distrib. Comput.

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In the directed minimum spanning tree problem (DMST, also called minimum weight arborescence), the network is given a root node r, and needs to construct a minimum-weight directed spanning tree, rooted at r and oriented outwards. In this paper we present the first sub-quadratic DMST algorithms in the distributed CONGEST network model, where the messages exchanged between the network nodes are bounded in size. We consider three versions: a model where the communication links are bidirectional but can have different weights in the two directions; a model where communication is unidirectional; and the Congested Clique model, where all nodes can communicate directly with each other.Our algorithm is based on a variant of Lovász' DMST algorithm for the PRAM model, and uses a distributed single-source shortest-path (SSSP) algorithm for directed graphs as a black box. In the bidirectional CONGEST model, our algorithm has roughly the same running time as the SSSP algorithm; using the state-of-the-art SSSP algorithm, we obtain a running time of O(min( √ nD, √ nD 1/4 + n 3/5 + D)) rounds for the bidirectional communication case. For the unidirectional communication model we give an O(n) algorithm, and show that it is nearly optimal. And finally, for the Congested Clique, our algorithm again matches the best known SSSP algorithm: it runs in O(n 1/3 ) rounds.On the negative side, we adapt an observation of Chechik in the sequential setting to show that in all three models, the DMST problem is at least as hard as the (s, t)-shortest path problem. Thus, in terms of round complexity, distributed DMST lies between single-source shortest path and (s, t)-shortest path.

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MST in O(1) Rounds of Congested Clique

Cited by 66 publications

References 18 publications

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

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

Faster Algorithms for Edge Connectivity via Random 2-Out Contractions

A distributed algorithm for directed minimum-weight spanning tree

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