Near optimal multicriteria spanner constructions in wireless ad-hoc networks

This paper studies asymmetric power assignments for which the induced communication graph is a good spanner of the Euclidean graph, induced on the wireless nodes V, while the total energy is minimized. We propose two spanner models: distance and energy. We consider a random wireless ad-hoc network with |V| = n nodes distributed uniformly and independently in a unit square. For the first model, we propose an approximation algorithm, which constructs a power assignment so that the induced network is an O ((1 + α)n-m/m log n+α)-spanner of G_V with high probability, such that the total energy is at most β· m+2 times the optimum, for any α> 1, β ≥ 1+ 2/α-1, and any positive integer m ≤ n in O (mn ²) time. For the second model, we develop a power assignment such that the resulting network is a 2-spanner with a total energy of at most O(log n) times the optimum in O(n⁴ log n) time. We also analyze a power assignment developed in [1] and show it is a low cost good k-fault resistant spanner. To the best of our knowledge, these are the first provable theoretic results for low cost spanners in wireless ad-hoc networks.