In this paper we theoretically study the M INIMUM R ANGE A SSIGNMENT problem in static ad-hoc networks with arbitrary structure, where the transmission distances can violate triangle inequality. We consider two versions of the M INIMUM R ANGE A SSIGNMENT problem, where the communication graph has to fulfill either the h-strong connectivity condition (M IN-R ANGE( h -SC)) or the h-broadcast condition (M IN-R ANGE( 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 M IN-R ANGE( h -SC) and M IN-R ANGE( h -B), which improves the previous best ratios O ( log n log log n ) and O ( n 2 log n log log n ) , respectively [D. Ye, H. Zhang, The range assignment problem in static ad-hoc networks on metric spaces, Proceedings of the 11th Colloquium on Structural Information and Communication Complexity, Sirocco 2004, Lecture Notes in Computer Science, vol. 3104, pp. 291–302]. The result for M IN-R ANGE( h -B) matches the lower bound [G. Rossi, The range assignment problem in static ad-hoc wireless networks. Ph.D. Thesis, 2003] 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 ( ( log log n ) α ) . Finally we discuss the applications of our algorithms in mobile ad-hoc networks.
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