An efficient distributed algorithm for constructing small dominating sets

Abstract

The dominating set problem asks for a small subset D of nodes in a graph such that every node is either in D or adjacent to a node in D. This problem arises in a number of distributed network applications, where it is important to locate a small number of centers in the network such that every node is nearby at least one center. Finding a dominating set of minimum size is NP-complete, and the best known approximation is logarithmic in the maximum degree of the graph and is provided by the same simple greedy approach that gives the well-known logarithmic approximation result for the closely related set cover problem.We describe and analyze new randomized distributed algorithms for the dominating set problem that run in polylogarithmic time, independent of the diameter of the network, and that return a dominating set of size within a logarithmic factor from optimal, with high probability. In particular, our best algorithm runs in O(log n log ∆) rounds with high probability, where n is the number of nodes, ∆ is one plus the maximum degree of any node, and each round involves a constant number of message exchanges among any two neighbors; the size of the dominating set obtained is within O(log ∆) of the optimal in expectation and within O(log n) of the optimal with high probability. We also describe generalizations to the weighted case and the case of multiple covering requirements.

ACE: An Emergent Algorithm for Highly Uniform Cluster Formation

“…While these algorithms have provable theoretical asymptotic bounds on performance on arbitrary graphs, their actual average performance on a random geometric graph is undetermined. We implemented in simulation two algorithms described by Jia et al [13] and Kuhn et al [16], however neither of them had comparable performance to even the simple Node-degree or Node-ID algorithms under our particular simulation conditions. We speculate that the relatively poor performance of these algorithms in simulation may be due to the fact that they are designed for arbitrary graphs while dedicated clustering algorithms are optimized for random geometric graphs.…”

Deterministic Dominating Set Construction in Networks with Bounded Degree

“…For instance, Jia et al [16] propose an algorithm with O(log n log Δ) running time, while Kuhn et. al.…”

Virtual Backbone Construction in MANETs Using Adjustable Transmission Ranges

“…The major drawback of a cluster formation approach is its relatively slow convergency, which takes O(n) rounds in the worst case. In DS formation approaches [13,15,18,28], the set of clusterheads may not be a MIS. The best DS formation algorithm takes O(1) rounds, but the DS size is unbounded in the worst case.…”

“…For a general graph, Jia et al [15] proposed a randomized algorithm to compute a DS, which finishes in O(log n log ∆) rounds with high probability, where ∆ is the maximal node degree, and has an expected O(log n) approximation ratio. Kuhn and Wattenhofer [18] proposed another…”