Assign ranges in general ad-hoc networks

In this paper we study the Minimum Range Assignment problem in static ad-hoc networks with arbitrary structure, where the transmission distances can violate triangle inequality. We consider two versions of the Minimum Range Assignment problem, where the communication graph has to fulfill either the h-strong connectivity condition (Min-Range( h -SC)) or the h-broadcast condition (Min-Range( h -B)). Both homogeneous and non-homogeneous cases are studied. By approximating arbitrary edge-weighted graphs by paths, we present probabilistic O(log n)-approximation algorithms for Min-Range( h -SC) and Min-Range( h -B), which improves the previous best ratios O(log n loglog n) and O(n 2log n loglog n), respectively [21]. The result for Min-Range( h -B) matches the lower bound [20] for the case that triangle inequality for transmission distance holds (which is a special case of our model). Furthermore, we show that if the network fulfils certain property and the distance power gradient α is sufficiently small, the approximation ratio is improved to O((loglog n) α ).