A randomized distributed algorithm for the maximal independent set problem in growth-bounded graphs

The efficient distributed construction of a maximal independent set (MIS) of a graph is of fundamental importance. We study the problem in the class of Growth-Bounded Graphs, which includes for example the well-known Unit Disk Graphs. In contrast to the fastest (time-optimal) existing approach [11], we assume that no geometric information (e.g., distances in the graph's embedding) is given. Instead, nodes employ randomization for their decisions. Our algorithm computes a MIS in O(log log n • log* n) rounds with very high probability for graphs with bounded growth, where n denotes the number of nodes in the graph. In view of Linial's Ω(log* n) lower bound for computing a MIS in ring networks [12], which was extended to randomized algorithms independently by Naor [18] and Linial [13], our solution is close to optimal. In a nutshell, our algorithm shows that for computing a MIS, randomization is a viable alternative to distance information.

[1]  Harry B. Hunt,et al.  Simple heuristics for unit disk graphs , 1995, Networks.

[2]  Nathan Linial,et al.  Locality in Distributed Graph Algorithms , 1992, SIAM J. Comput..

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

[4]  Imran A. Pirwani,et al.  Energy conservation via domatic partitions , 2006, MobiHoc '06.

[5]  Peng-Jun Wan,et al.  Message-optimal connected dominating sets in mobile ad hoc networks , 2002, MobiHoc '02.

[6]  Richard Cole,et al.  Deterministic Coin Tossing with Applications to Optimal Parallel List Ranking , 2018, Inf. Control..

[7]  Stefan Schmid,et al.  Algorithmic models for sensor networks , 2006, Proceedings 20th IEEE International Parallel & Distributed Processing Symposium.

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

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

[10]  Aravind Srinivasan,et al.  Improved distributed algorithms for coloring and network decomposition problems , 1992, STOC '92.

[11]  Rajiv Gandhi,et al.  Distributed Algorithms for Coloring and Domination in Wireless Ad Hoc Networks , 2004, FSTTCS.

[12]  Baruch Awerbuch,et al.  Complexity of network synchronization , 1985, JACM.

[13]  T. Nieberg,et al.  Wireless communication graphs , 2004, Proceedings of the 2004 Intelligent Sensors, Sensor Networks and Information Processing Conference, 2004..

[14]  Srikanth V. Krishnamurthy,et al.  Topology Control to Simultaneously Achieve Near-Optimal Node Degree and Low Path Stretch in Ad hoc Networks , 2006, 2006 3rd Annual IEEE Communications Society on Sensor and Ad Hoc Communications and Networks.

[15]  Roger Wattenhofer,et al.  On the locality of bounded growth , 2005, PODC '05.

[16]  Roger Wattenhofer,et al.  Maximal independent sets in radio networks , 2005, PODC '05.

[17]  Roger Wattenhofer,et al.  Local approximation schemes for ad hoc and sensor networks , 2005, DIALM-POMC '05.

[18]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[19]  Moni Naor,et al.  A Lower Bound on Probabilistic Algorithms for Distributive Ring Coloring , 1991, SIAM J. Discret. Math..

[20]  Andrew V. Goldberg,et al.  Network decomposition and locality in distributed computation , 1989, 30th Annual Symposium on Foundations of Computer Science.

[21]  Harald Niederreiter,et al.  Probability and computing: randomized algorithms and probabilistic analysis , 2006, Math. Comput..

[22]  Andrzej Czygrinow,et al.  Distributed Approximation Algorithms in Unit-Disk Graphs , 2006, DISC.

[23]  Roger Wattenhofer,et al.  Fast Deterministic Distributed Maximal Independent Set Computation on Growth-Bounded Graphs , 2005, DISC.