Minimum Power Energy Spanners in Wireless Ad Hoc Networks

A power assignment is an assignment of transmission power to each of the nodes of a wireless network, so that the induced communication graph has some desired properties. The cost of a power assignment is the sum of the powers. The energy of a transmission path from node u to node v is the sum of the squares of the distances between adjacent nodes along the path. For a constant t > 1, an energy t-spanner is a graph G´, such that for any two nodes u and v, there exists a path from u to v in G´, whose energy is at most t times the energy of a minimum-energy path from a ton in the complete Euclidean graph. In this paper, we study the problem of finding a power assignment, such that (i) its induced communication graph is a ´good´ energy spanner, and (ii) its cost is ´low´. We show that for any constant t > 1, one can find a power assignment, such that its induced communication graph is an energy t-spanner, and its cost is bounded by some constant times the cost of an optimal power assignment (where the sole requirement is strong connectivity of the induced communication graph). This is a very significant improvement over the best current result due to Shpungin and Segal, presented in last year´s conference.