The range assignment problem in static ad-hoc networks on metric spaces

Abstract

In this paper we study the range assignment problem in static ad-hoc networks on metric spaces. We consider the h-strong connectivity and h-broadcast problems on trees, high dimensional Euclidean spaces and general finite metric spaces. Both homogeneous and non-homogeneous cases are explored. We show that the h-broadcast problem is polynomial solvable on trees and present an O(n2)-approximation algorithm for the h-strong connectivity problem on trees, where n is the number of stations. Furthermore, we propose a probabilistic O(log n log log n)-approximation algorithm for the h-broadcast problem and a probabilistic O(n2 log n log log n)-approximation algorithm for the h-strong connectivity problem on high dimensional Euclidean spaces and general metric spaces. In the case of high dimensional real normed spaces, if the distance-power gradient α ≤ 1 + O(log log log n/ log log n), an O(logα n)-approximation algorithm and an O(n2 logα n)-approximation algorithm are developed for the h-broadcast problem and the h-strong connectivity problem, respectively. They are the first algorithms for the range assignment problem in static ad-hoc networks on general metric spaces. And the approximation ratio of O(log n log log n) for the h-broadcast problem on general metric spaces is close to the known lower bound O(logn) [19]. © Springer-Verlag Berlin Heidelberg 2004.