Abstract
In a wireless ad hoc network, the size of the virtual backbone (VB) is an important factor for measuring the quality of the VB. The smaller the VB is, the less the overhead caused by the VB. Since ball graphs (BGs) have been used to model 3-dimensional wireless ad hoc networks and since a connected dominating set can be used to represent a VB undertaking routing-related tasks, the problem of finding the smallest VB is transformed into the problem of finding a minimum connected dominating set (MCDS). Many research results on the MCDS problem have been obtained for unit disk graphs and unit ball graphs, in which the transmission ranges of all nodes are identical. In some situations, the node powers can vary. One can model such a network as a graph with different transmission ranges for different nodes. In this paper, we focus on the problem of minimum strongly connected dominating and absorbing sets (MSCDASs) in a strongly connected directed ball graph with different transmission ranges, which is also NP-hard. We design an algorithm considering the construction of a strongly connected dominating and absorbing set (SCDAS), whose size does not exceed 319/15k3+116/5k2+29/5kopt+29/3k3+116/5k2+87/5k+13/15, where opt is the size of an MCDAS and k denotes the ratio of rmax to rmin in the ad hoc network with transmission range rmin,rmax. Our simulations show the feasibility of the algorithm proposed in this paper.
Highlights
Along with the rapid development of wireless radio communication technologies, embedded sensors, and VLSI, the cost of establishing a wireless ad hoc network is decreasing and the performance of wireless ad hoc networks is improving
We study the minimum strongly connected dominating and absorbing sets (MSCDASs) problem of a wireless ad hoc network with different transmission ranges in three-dimensional space
We mainly study the problem of constructing MSCDASs in a directed strongly connected ball graph with different transmission ranges for its nodes, which is NP-hard for obtaining an optimal solution
Summary
Along with the rapid development of wireless radio communication technologies, embedded sensors, and VLSI, the cost of establishing a wireless ad hoc network is decreasing and the performance of wireless ad hoc networks is improving. In some situations, such as undersea resource exploration, disaster prevention, offshore exploration, and ocean environmental monitoring (see [12, 13]), it is not suitable to use MCDS of the unit disk graph to describe the VB of the wireless ad hoc networks For this reason, people have studied the MCDS problem in three-dimensional space. According to the different requirements of the measured frequency in collisions, a node may have to change its transmission range In such situations, the MCDS problem in a unit disk graph becomes that of a minimum strongly connected dominating and absorbing set (MSCDAS) in a disk graph in reference [17], which was the first paper to study MSCDAS in a network with different transmission ranges. A strongly connected dominating and absorbing set (SCDAS) can usually be used to denote a VB of a wireless ad hoc network in which the nodes have different transmission ranges.
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