Thresholding random geometric graph properties motivated by ad hoc sensor networks
We study the emerging phenomenon of ad hoc, sensor-based communication networks. The communication is modeled by the random geometric graph model G(n, r, l) where n points randomly placed within [0, l](d) form the nodes, and any two nodes that correspond to points at most distance r away from each other are connected. We study fundamental properties of G(n, r, l) of interest: connectivity, coverage, and routing-stretch. We use a technique that we call bin-covering that we apply uniformly to get (asymptotically) tight thresholds for each of these properties. Typically, in the past, random geometric graph analyses involved sophisticated methods from continuum percolation theory; on contrast, our bin-covering approach is discrete and very simple, yet it gives us tight threshold bounds. The technique also yields algorithmic benefits as illustrated by a simple local routing algorithm for finding paths with low stretch. Our specific results should also prove interesting to the sensor networking community that has seen a recent increase in the study of random geometric graphs motivated by engineering ad hoc networks. (C) 2010 Elsevier Inc. All rights reserved.
thresholds random geometric graphs connectivity coverage stretch sensor network models local algorithm