Probabilistic Analysis of Rule 2

Li and Wu proposed Rule 2, a localized approximation algorithm that attempts to find a small connected dominating set in a graph. Here we study the asymptotic performance of Rule 2 on random unit disk graphs formed from n random points in an s_n by s_n square region of the plane. If s_n is below the threshold for connectivity, then Rule 2 produces a dominating set whose expected size is O(n/(loglog n)^{3/2}). We conjecture that this bound is not optimal.