by Unknown Authors

In this chapter we present another randomized algorithm that demonstrates the power of randomization to break symmetries. We study the problem of finding a small dominating set of the network graph. As it is the case for the MIS, an efficient dominating set algorithm can be used as a basic building block to solve a number of problems in distributed computing. For example, whenever we need to partition the network into a small number of local clusters, the computation of a small dominating set usually occurs in some way. A particularly important application of dominating sets is the construction of an efficient backbone for routing. Definition 19.1 (Dominating Set). Given an undirected graph G = (V, E), a dominating set is a subset S ⊆ V of its nodes such that for all nodes v ∈ V, either v ∈ S or a neighbor u of v is in S. Remarks: • It is well-known that computing a dominating set of minimal size is NPhard. We therefore look for approximation algorithms, that is, algorithms which produce solutions which are optimal up to a certain factor. • Note that every MIS (cf. Chapter 7) is a dominating set. In general, the size of every MIS can however be larger than the size of an optimal minimum dominating set by a factor of Ω(n). As an example, connect the centers of two stars by an edge. Every MIS contains all the leaves of at least one of the two stars whereas there is a dominating set of size 2. All the dominating set algorithms that we study throughout this chapter operate in the following way. We start with S = ∅ and add nodes to S until S is a dominating set. To simplify presentation, we color nodes according to their state during the execution of an algorithm. We call nodes in S black, nodes which are covered (neighbors of nodes in S) gray, and all uncovered nodes white. By W (v), we denote the set of white nodes among the direct neighbors of v, including v itself. We call w(v) = |W (v) | the span of v.

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