Connected dominating sets on dynamic geometric graphs

Abstract

We propose algorithms for efficiently maintaining a constant-approximate minimum connected dominating set (MCDS) of a geometric graph under node insertions and deletions, and under node mobility. Assuming that two nodes are adjacent in the graph if and only if they are within a fixed geometric distance, we show that an $O(1)$-approximate MCDS of a graph in ${\mathbb{R}}^d$ with n nodes can be maintained in $O({\log }^{2d}n)$ time per node insertion or deletion. We also show that $\Omega (n)$ time per operation is necessary to maintain exact MCDS. This lower bound holds even for $d=1$, even for randomized algorithms, and even when running time is amortized over a sequence of insertions/deletions, or over continuous motion. The crucial fact is that a single operation may affect the entire exact solution, while an approximate solution is affected only in a small neighborhood of the node that was inserted or deleted. In the approximate case, we show how to compute these local changes by a few range searching queries and a few bichromatic closest pair queries.