Maximal Independent Sets in Heterogeneous Wireless Ad Hoc Networks

In ad hoc wireless networks, a Connected Dominating Set (CDS) has been extensively used as a Virtual Backbone (VB) for routing. The majority of approximation algorithms for constructing a small CDS in wireless ad hoc networks follow a general two-phased approach. The first phase is to construct a Dominating Set (DS) and the second phase is to connect the nodes in it. Generally, in the first phase, a Maximum Independent Set (MIS) is used as the DS. The relation between the size of a Maximum Independent Set and a Minimum Connected Dominating Set (MCDS) plays the key role in the performance analyses of these two-phased algorithms. In homogeneous wireless ad hoc networks modeled as Unit Disk Graphs (UDG) and Unit Ball Graphs (UBG), the relation between them has been well studied. However, in heterogeneous wireless networks which generally modeled as Disk Graphs with Bidirectional links (DGB) and Ball Graphs with Bidirectional links (BGB), upper bounds for the size of MISs have seldom been studied. In this paper, we give tighter upper bounds for the size of MISs in heterogeneous wireless ad hoc networks. When the maximum and minimum transmission range are relatively close, our result is much better. In DGB, when the transmission range ratio is (1,1.152], (1.152,1.307], (1.307,1.407], (1.407,1.462], (1.462,1.515], (1.515,1.618], (1.618,1.932], we prove that the size of any Maximal Independent Set (MIS) is upper bounded by 6opt + 1, 7opt + 1, 8opt + 1, 9opt + 1, 10opt + 1, 11opt + 1, 16.7778opt + 1.2222, where opt denotes the size of an optimal solution of the CDS problem.