Wireless sensor networks (WSNs) are used to cover destination areas for a lot of practical applications. To enhance the performance of the WSN, the virtual backbone based on the connected dominating set is an efficient way with respect to the routing cost between sensors, lifetime of entire network, and so on. In this paper, especially for the WSN with different transmission radii among different sensors, we study the problem of constructing the minimum -range connected dominating set under the constraint -times of the minimum routing cost(MOC-CDS), where 5 and is the ratio of the maximum-to-minimum transmission radius. Our contributions are three folds. First, we propose a polynomial time approximation scheme which generates the MOC-CDS with the size of at most (1 + ) times of the optimum solution, where is the error parameter. Second, we propose a polynomial time algorithm and prove that it has two approximation ratios(6+1)2(2+1)2 and 10(2/)(ln 3/(ln(1/ cos )))(ln /(ln(2 cos(/5)))), where < arcsin(1/3). Finally,we propose the distributed version of the constant approximation ratio algorithm which has both the time complexity and message complexity O(n3), where n is the number of sensor nodes.Besides, the simulation results demonstrate the efficiency of our algorithms.
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