The Expected Size of the Rule k Dominating Set: II

Rule k (k=1,2,3,...) is a well known family of approximation algorithms that can be used to find connected dominating sets in a graph. They were originally proposed by Dai, Li, and Wu as the basis for efficient routing methods for ad hoc wireless networks. In this paper we study the asymptotic performance of Rules 1 and 2 on random unit disk graphs formed from n random points in an `n × `n square region of the plane, and we show that: • Rule 1 does poorly in the following sense: if `n = o( √ n), then with asymptotic probability one, Rule 1 produces a connected dominating set consisting of n− o(n) vertices. • Rule 2 produces much smaller dominating sets on average: if `n = O( √ n/ log n), then Rule 2 produces a connected dominating set whose expected size is O(n/(log log n)). These results complement our results in a companion paper [18] where we consider the asymptotic performance of Rule k in the case where k ≥ 3. keywords and phrases: coverage process, dominating set, localized algorithm, performance analysis, probabilistic analysis, Rule 2, Rule k, unit disk graph