Energy-Efficient Paths in Radio Networks

We consider a radio network consisting of n stations represented as the complete graph on a set of n points in the Euclidean plane with edge weights ω ( p , q )=| pq | δ + C p , for some constant δ >1 and nonnegative offset costs C p . Our goal is to find paths of minimal energy cost between any pair of points that do not use more than some given number k of hops. We present an exact algorithm for the important case when δ =2, which requires time per query pair ( p , q ). For the case of an unrestricted number of hops we describe a family of algorithms with query time , where α >0 can be chosen arbitrarily. If we relax the exactness requirement, we can find an approximate (1+ ε ) solution in constant time by querying a data structure which has linear size and which can be build in time. The dependence on ε is polynomial in 1/ ε . One tool we employ might be of independent interest: For any pair of points ( p , q )∈( P × P ) we can report in constant time the cluster pair ( A , B ) representing ( p , q ) in a well-separated pair decomposition of P .