A time- and message-optimal distributed algorithm for minimum spanning trees

This paper presents a randomized (Las Vegas) distributed algorithm that constructs a minimum spanning tree (MST) in weighted networks with optimal (up to polylogarithmic factors) time and message complexity. This algorithm runs in Õ(D + √n) time and exchanges Õ(m) messages (both with high probability), where n is the number of nodes of the network, D is the diameter, and m is the number of edges. This is the first distributed MST algorithm that matches simultaneously the time lower bound of Ω(D + √n) [Elkin, SIAM J. Comput. 2006] and the message lower bound of Ω(m) [Kutten et al., J. ACM 2015], which both apply to randomized Monte Carlo algorithms. The prior time and message lower bounds are derived using two completely different graph constructions; the existing lower bound construction that shows one lower bound does not work for the other. To complement our algorithm, we present a new lower bound graph construction for which any distributed MST algorithm requires both Ω(D + √n) rounds and Ω(m) messages.

[1]  David Peleg,et al.  Message Lower Bounds via Efficient Network Synchronization , 2016, SIROCCO.

[2]  Eli Gafni,et al.  Improvements in the time complexity of two message-optimal election algorithms , 1985, PODC '85.

[3]  Nancy A. Lynch,et al.  Distributed Algorithms , 1992, Lecture Notes in Computer Science.

[4]  D. Peleg,et al.  On the complexity of universal leader election , 2013, PODC '13.

[5]  Robert E. Tarjan,et al.  Data structures and network algorithms , 1983, CBMS-NSF regional conference series in applied mathematics.

[6]  Bernhard Haeupler,et al.  Distributed Algorithms for Planar Networks II: Low-Congestion Shortcuts, MST, and Min-Cut , 2016, SODA.

[7]  Michael Elkin.  An Unconditional Lower Bound on the Time-Approximation Trade-off for the Distributed Minimum Spanning Tree Problem , 2006, SIAM J. Comput..

[8]  David Peleg,et al.  Tight Bounds for Distributed Minimum-Weight Spanning Tree Verification , 2013, Theory of Computing Systems.

[9]  Bernhard Haeupler,et al.  Low-Congestion shortcuts without embedding , 2016, Distributed Computing.

[10]  Alexander A. Razborov,et al.  On the Distributional Complexity of Disjointness , 1992, Theor. Comput. Sci..

[11]  David Peleg,et al.  Distributed Matroid Basis Completion via Elimination Upcast and Distributed Correction of Minimum-Weight Spanning Trees , 1998, ICALP.

[12]  Baruch Awerbuch,et al.  Sparse partitions , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[13]  Shay Kutten,et al.  Fast Distributed Construction of Small k-Dominating Sets and Applications , 1998, J. Algorithms.

[14]  Ali Mashreghi,et al.  Time-communication trade-offs for minimum spanning tree construction , 2017, ICDCN.

[15]  Pierre A. Humblet,et al.  A Distributed Algorithm for Minimum-Weight Spanning Trees , 1983, TOPL.

[16]  David Peleg,et al.  A Near-Tight Lower Bound on the Time Complexity of Distributed Minimum-Weight Spanning Tree Construction , 2000, SIAM J. Comput..

[17]  Eyal Kushilevitz,et al.  Communication Complexity , 1997, Adv. Comput..

[18]  Bernhard Haeupler,et al.  Near-Optimal Low-Congestion Shortcuts on Bounded Parameter Graphs , 2016, DISC.

[19]  Baruch Awerbuch,et al.  Sparse Partitions (Extended Abstract , 1990, FOCS 1990.

[20]  Shay Kutten,et al.  Distributed Symmetry Breaking in Hypergraphs , 2014, DISC.

[21]  Francis Y. L. Chin,et al.  An almost linear time and O(nlogn+e) Messages distributed algorithm for minimum-weight spanning trees , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[22]  Michalis Faloutsos,et al.  A linear-time optimal-message distributed algorithm for minimum spanning trees , 2004, Distributed Computing.

[23]  Andrew Chi-Chih Yao,et al.  Probabilistic computations: Toward a unified measure of complexity , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[24]  Michael Elkin,et al.  A faster distributed protocol for constructing a minimum spanning tree , 2004, SODA '04.

[25]  Danupon Nanongkai,et al.  A tight unconditional lower bound on distributed randomwalk computation , 2011, PODC '11.

[26]  Prasad Tetali,et al.  Distributed Random Walks , 2013, JACM.

[27]  Monika Henzinger,et al.  A deterministic almost-tight distributed algorithm for approximating single-source shortest paths , 2015, STOC.

[28]  Maleq Khan,et al.  A fast distributed approximation algorithm for minimum spanning trees , 2007, Distributed Computing.

[29]  Hagit Attiya,et al.  Distributed Computing: Fundamentals, Simulations and Advanced Topics , 1998 .

[30]  Hartmut Klauck,et al.  Can quantum communication speed up distributed computation? , 2012, PODC.

[31]  David Peleg,et al.  Distributed Computing: A Locality-Sensitive Approach , 1987 .

[32]  Danupon Nanongkai,et al.  Distributed approximation algorithms for weighted shortest paths , 2014, STOC.

[33]  Hsin-Hao Su,et al.  Distributed MST and Routing in Almost Mixing Time , 2017, PODC.

[34]  David Peleg,et al.  Distributed verification and hardness of distributed approximation , 2010, STOC '11.

[35]  Sriram V. Pemmaraju,et al.  Toward Optimal Bounds in the Congested Clique: Graph Connectivity and MST , 2015, PODC.

[36]  Michael Elkin,et al.  A Simple Deterministic Distributed MST Algorithm, with Near-Optimal Time and Message Complexities , 2017, PODC.

[37]  Hartmut Klauck,et al.  Distributed Computation of Large-scale Graph Problems , 2015, SODA.

[38]  Mikkel Thorup,et al.  Construction and Impromptu Repair of an MST in a Distributed Network with o(m) Communication , 2015, PODC.

[39]  Baruch Awerbuch,et al.  A trade-off between information and communication in broadcast protocols , 1990, JACM.

[40]  Marcin Paprzycki,et al.  Distributed Computing: Fundamentals, Simulations and Advanced Topics , 2001, Scalable Comput. Pract. Exp..

[41]  Baruch Awerbuch,et al.  Optimal distributed algorithms for minimum weight spanning tree, counting, leader election, and related problems , 1987, STOC.

[42]  Valmir Carneiro Barbosa,et al.  An introduction to distributed algorithms , 1996 .