Efficient Distributed Algorithms for Topology Control Problem with Shortest Path Constraints

A Connected Dominating Set (CDS) can be used to construct a virtual backbone for wireless and mobile ad-hoc networks to make the system hierarchical and efficient. A virtual backbone can significantly improve network throughput, optimize broadband utilization, extend network lifetime, and reduce interference as well as packet retransmissions. Calculating a minimum backbone for a network is critical to reduce routing computation and energy consumption. This problem is a well-known NP-hard optimization problem, which has various applications in practice. In this paper, we propose a new problem based on customer fairness, which looks for a minimum CDS in a given communication model with shortest path constraints. It guarantees that any two clients can communicate with each other through this CDS with hop counts the same as the best path from the original graph, which means that routing on such a CDS will not bring additional traffic for every client. We name this problem as shortest path connected dominating set (SPCDS) and prove its NP-hardness by reduction from Hitting Set .T hen we propose a centralized greedy algorithm and an efficient distributed approximation algorithm with approximation ratio ∆ to solve SPCDS, where ∆ is the maximum vertex degree in the given topology. We also analyze the time complexity, message complexity, and evaluate the efficiency of our distributed heuristic by several numerical experiments and comparisons with previous literatures.