Energy-Efficient Paths in Radio Networks

Abstract

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 $\mathcal {O}(kn\log n)$ time per query pair (p,q). For the case of an unrestricted number of hops we describe a family of algorithms with query time $\mathcal {O}(n^{1+\alpha})$, where α>0 can be chosen arbitrarily. If we relax the exactness requirement, we can find an approximate (1+e) solution in constant time by querying a data structure which has linear size and which can be build in $\mathcal {O}(n\log n)$ time. The dependence on e is polynomial in 1/e. 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.