Minimum connected dominating sets in heterogeneous 3D wireless ad hoc networks

The Minimum Connected Dominating Set (MCDS) problem is a fundamental problem in wireless ad hoc networks. The majority of approximation algorithms for this NP-hard problem follow a two-phased approach: The first phase is to construct a Maximal Independent Set (MIS), and the second phase is to connect the nodes in it. The upper bounds of the MISs play a key role in the design of constant approximation MCDS algorithms. This paper considers this problem for 3D heterogeneous ad hoc networks, where the transmission ranges of nodes are allowed to be different. We prove upper bounds of MISs with two classical mathematical problems, the Spherical Code Problem and the Sphere Packing Problem. When the transmission range ratio (the ratio of the maximum transmission range over the minimum transmission range) is (1, 1.023], (1.023, 1.055], (1.055, 1.082], ., we reduce the MIS upper bounds from the best-known results 22|OPT|+1, 23|OPT|+1, 24|OPT|+1, ., to 12|OPT|+1, 13|OPT|+1, 14|OPT|+1, ., where OPT is an optimal CDS and |OPT| is the size of OPT. With the bounds of MISs, the approximation ratio of MCDS algorithms can be reduced from 25.02 to 16.02 in heterogeneous 3D wireless ad hoc networks.