Minimum-Weight Spanning Tree Construction in <i>O</i>(log log <i>n</i>) Communication Rounds

Lotker, Zvi; Patt-Shamir, Boaz; Pavlov, Elan; Peleg, David

doi:10.1137/s0097539704441848

Cited by 121 publications

(114 citation statements)

References 13 publications

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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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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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“…The task of n × n matrix multiplication over the min-plus semiring can be reduced to APSP with a constant blowup [3, pp.202-205], hence the above bound applies also to any APSP algorithm that only uses minimum and addition operations. This means that current techniques for similar problems, like the one used in the fast MST algorithm of Lotker et al [51] cannot be extended to solve APSP.…”

Section: Lower Boundsmentioning

confidence: 99%

“…As such, it has been recently gaining increasing attention [24,25,36,37,46,49,51,57,59,63], in an attempt to understand the relative computational power of distributed computing models.…”

Section: Introductionmentioning

confidence: 99%

Algebraic Methods in the Congested Clique

Censor-Hillel

Kaski

Korhonen

et al. 2015

Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing

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In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an O(n 1−2/ω ) round matrix multiplication algorithm, where ω < 2.3728639 is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.

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“…This algorithm takes only O(n log n) messages to construct an O(log n)-approximate MST as opposed to the Ω(n 2 ) lower bound (shown by Korach et al [14]) on the number of messages needed by any distributed MST algorithm in this model. If the time complexity needs to be optimized, then the NNT scheme can easily be implemented in O(1) time by using O(n 2 ) messages, as opposed to the best known time bound of O(log log n) for the (exact) MST [17]. These results suggest that the NNT scheme can yield faster and more communication-efficient algorithms than the algorithms that compute the exact MST.…”

Section: Nearest Neighbor Tree Schemementioning

confidence: 99%

A fast distributed approximation algorithm for minimum spanning trees

Khan

Pandurangan

2007

Distrib. Comput.

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We present a distributed algorithm that constructs an O(log n)-approximate minimum spanning tree (MST) in any arbitrary network. This algorithm runs in timeÕ(D(G)+ L(G, w)) where L(G, w) is a parameter called the local shortest path diameter and D(G) is the (unweighted) diameter of the graph. Our algorithm is existentially optimal (up to polylogarithmic factors), i.e., there exist graphs which need Ω(D(G) + L(G, w)) time to compute an H -approximation to the MST for any H ∈ [1, Θ(log n)]. Our result also shows that there can be a significant time gap between exact and approximate MST computation: there exists graphs in which the running time of our approximation algorithm is exponentially faster than the time-optimal distributed algorithm that computes the MST. Finally, we show that our algorithm can be used to find an approximate MST in wireless networks and in random weighted networks in almost optimalÕ(D(G)) time.

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Minimum-Weight Spanning Tree Construction in O(log log n) Communication Rounds

Cited by 121 publications

References 13 publications

Toward Optimal Bounds in the Congested Clique

Toward Optimal Bounds in the Congested Clique

Algebraic Methods in the Congested Clique

A fast distributed approximation algorithm for minimum spanning trees

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