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(n ²)-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 loglog n)-approximation algorithm for the h-broadcast problem and a probabilistic O(n ² log n loglog 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(logloglog n/loglog n), an O(log^α n)-approximation algorithm and an O(n ² log^α 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 loglog n) for the h-broadcast problem on general metric spaces is close to the known lower bound O(log n) [19].