MIS in the Congested Clique Model in $O(\log \log \Delta)$ Rounds

We give a maximal independent set (MIS) algorithm that runs in O(log log∆) rounds in the congested clique model, where ∆ is the maximum degree of the input graph. This improves upon the O( log(∆)·log log∆ √ logn + log log∆) rounds algorithm of [Ghaffari, PODC ’17], where n is the number of vertices of the input graph. In the first stage of our algorithm, we simulate the first O( n poly log n ) iterations of the sequential random order Greedy algorithm for MIS in the congested clique model in O(log log∆) rounds. This thins out the input graph relatively quickly: After this stage, the maximum degree of the residual graph is poly-logarithmic. In the second stage, we run the MIS algorithm of [Ghaffari, PODC ’17] on the residual graph, which completes in O(log log∆) rounds on graphs of poly-logarithmic degree.

[1]  Sriram V. Pemmaraju,et al.  Near-Constant-Time Distributed Algorithms on a Congested Clique , 2014, DISC.

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

[3]  Jukka Suomela,et al.  Towards a Complexity Theory for the Congested Clique , 2017, DISC.

[4]  Sudipto Guha,et al.  Correlation Clustering in Data Streams , 2015, ICML.

[5]  Boaz Patt-Shamir,et al.  Minimum-Weight Spanning Tree Construction in O(log log n) Communication Rounds , 2005, SIAM J. Comput..

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

[7]  François Le Gall,et al.  Further Algebraic Algorithms in the Congested Clique Model and Applications to Graph-Theoretic Problems , 2016, DISC.

[8]  Boaz Patt-Shamir,et al.  MST construction in O(log log n) communication rounds , 2003, SPAA '03.

[9]  Michael Luby,et al.  A simple parallel algorithm for the maximal independent set problem , 1985, STOC '85.

[10]  Roger Wattenhofer,et al.  Local Computation , 2010, J. ACM.

[11]  Fabian Kuhn,et al.  On the power of the congested clique model , 2014, PODC.

[12]  Leonid Barenboim,et al.  The Locality of Distributed Symmetry Breaking , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[13]  Berthold Vöcking,et al.  Improved algorithms for latency minimization in wireless networks , 2009, Theor. Comput. Sci..

[14]  Nathan Linial,et al.  Distributive graph algorithms Global solutions from local data , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[15]  Thomas Moscibroda,et al.  What Cannot Be Computed Locally , 2004 .

[16]  Tomasz Jurdzinski,et al.  MST in O(1) Rounds of Congested Clique , 2018, SODA.

[17]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[18]  Sriram V. Pemmaraju,et al.  Lessons from the Congested Clique Applied to MapReduce , 2014, SIROCCO.

[19]  Merav Parter,et al.  MST in Log-Star Rounds of Congested Clique , 2016, PODC.

[20]  Christoph Lenzen,et al.  Optimal deterministic routing and sorting on the congested clique , 2012, PODC '13.

[21]  Mohsen Ghaffari.  Distributed MIS via All-to-All Communication , 2017, PODC.

[22]  Christoph Lenzen,et al.  Algebraic methods in the congested clique , 2015, Distributed Computing.

[23]  Mohsen Ghaffari,et al.  An Improved Distributed Algorithm for Maximal Independent Set , 2015, SODA.