Analysis on a Localized Pruning Method for Connected Dominating Sets

While restricted rule-k has been succeeded in generating a connected dominating set (CDS) of small size, not much theoretical analysis on the size has been done. In this paper, an analysis on the expected size of a CDS generated by such algorithm and its relation to different node density is presented. Assume N nodes are deployed uniformly and randomly in a square of size LN × LN (where N and LN → ∞); three results are obtained. (1) It is proved that the node degree distribution of such a network follows a Poisson distribution. (2) The expected size of a CDS that is derived by the restricted pruning rule-k is a decreasing function with respect to the node density ˆn. For ˆn ≥ 30, it is found that the expected size is close to N/ˆn. (3) It is proved that the lower bound � on the expected size of a �CDS 1 ˆn for a Poissonian network of node density ˆn is given by ˆn−1 − ˆn−1 exp(−(ˆn − 1)) N. The second result is of paramount importance for practitioners. It provides the information about the expected size of a CDS when the node density ˆn is between 6 and 30. The data (expected CDS size) for this range can hardly be provided by simulations