Energy Optimized Topologies for Distributed Averaging in Wireless Sensor Networks

We study the energy efficient implementation of averaging/consensus algorithms in wireless sensor networks. For static, time-invariant topologies we start from the recent result that a bidirectional spanning tree is preferable in terms of convergence time. We formulate the combinatorial optimization problem of selecting such a minimal energy tree as a mixed integer linear programming problem. Since the problem is NP-complete we devise a semi-definite relaxation and establish bounds on the optimal cost. We also develop a series of graph-based algorithms that yield energy efficient bidirectional spanning trees and establish associated bounds on the optimal cost. For dynamic, time-varying topologies we consider a recently proposed load-balancing algorithm which has preferable convergence time properties. We formulate the problem of selecting a minimal energy interconnected network over which we can run the algorithm as a sequential decision problem and cast it into a dynamic programming framework. We first consider the scenario of a large enough time horizon and show that the problem is equivalent to constructing a Minimum Spanning Tree. We then consider the scenario of a limited time horizon and employ a "rollout" heuristic that leverages the spanning tree solution and yields efficient solutions for the original problem. Some of our algorithms can be run in a distributed manner and numerical results establish that they produce near-optimal solutions.