Minimum power energy spanners in wireless ad hoc networks

A {\em 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 {\em cost} of a power assignment is the sum of the powers. The {\em 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 {\em 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 $u$ to $v$ 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~\cite{Shpungin}, presented in last year's conference.