The Expected Size of the Rule k Dominating Set

Abstract

Dai, Li, and Wu proposed Rule k, a localized approximation algorithm that attempts to find a small connected dominating set in a graph. Here we consider the "average case"performance of Rule k for the model of random unit disk graphs constructed from n random points in an ℓn × ℓn square. If k ≥ 3 and ℓn = o( √ n), then the expected size of the Rule k dominating set is Θ(ℓ 2 n ) as n → ∞. If ℓn ≤ n 10 log n , then expected size of the minimum CDS is also Θ(ℓ 2 n ).

Covering random points in a unit disk

“…We end this section with an observation that is not needed in this paper, but is worth mentioning because of its relevance in applications [7]. It is implicit in the proof of Theorem 2 that, with asymptotic probability 1, the two covering points can be chosen in such a way that the distance between them is less than 1.…”

Incremental Construction of k-Dominating Sets in Wireless Sensor Networks

“…Instead of using only two connected neighbors, Dai and Wu [5] then proposed to use k connected neighbors. They called this improved version of the algorithm Rule k. Hansen and Schmutz [11] analyzed the expected size of the produced connected dominating set. Finally, the Rule k algorithm has been further generalized by Wu and Dai [26] in the form of a generic coverage condition.…”