Page 2 Lecture Notes in Computer Science 2865 Edited by G. Goos ...

Range Assignment for High Connectivityin Wireless Ad Hoc NetworksGruia Calinescu and Peng-Jun WanDepartment of Computer Science, Illinois Institute of Technology,Chicago, IL 60616calinesc@iit.edu, wan@cs.iit.eduAbstract. Depending on whether bidirectional links or unidirectionallinks are used for communications, the network topology under a givenrange assignment is either an undirected graph referred to as the symmetrictopology, or a directed graph referred to as the asymmetric topology.The Min-Power Symmetric (resp., Asymmetric) k-Node Connectivityproblem seeks a range assignment of minimum total power subjectto the constraint the induced symmetric (resp. asymmetric) topology isk-connected. Similarly, the Min-Power Symmetric (resp., Asymmetric) k-Edge Connectivity problem seeks a range assignment of minimum totalpower subject to the constraint the induced symmetric (resp., asymmetric)topology is k-edge connected.The Min-Power Symmetric Biconnectivity problem and the Min-PowerSymmetric Edge-Biconnectivity problem has been studied by Lloyd et.al [21]. They show that range assignment based the approximation algorithmof Khuller and Raghavachari [17], which we refer to as AlgorithmKR, has an approximation ratio of at most 2(2 − 2/n)(2 + 1/n) for Min-Power Symmetric Biconnectivity, and range assignment based on theapproximation algorithm of Khuller and Vishkin [18], which we refer toas Algorithm KV, has an approximation ratio of at most 8(1 − 1/n)for Min-Power Symmetric Edge-Biconnectivity.In this paper, we first establish the NP-hardness of Min-Power Symmetric(Edge-)Biconnectivity. Then we show that Algorithm KR has anapproximation ratio of at most 4 for both Min-Power Symmetric Biconnectivityand Min-Power Asymmetric Biconnectivity, and AlgorithmKV has an approximation ratio of at most 2k for both Min-Power Symmetrick-Edge Connectivity and Min-Power Asymmetric k-Edge Connectivity.We also propose a new simple constant-approximation algorithmfor both Min-Power Symmetric Biconnectivity and Min-Power AsymmetricBiconnectivity. This new algorithm is best suited for distributedimplementation.1 IntroductionRecently, range assignment problems for wireless ad hoc networks have beenstudied extensively. In wireless ad hoc networks no wired backbone infrastructureis installed and communication sessions are achieved either through a singlehoptransmission if the communication parties are close enough, or throughS. Pierre, M. Barbeau, and E. Kranakis (Eds.): ADHOC-NOW 2003, LNCS 2865, pp. 235–246, 2003.c○ Springer-Verlag Berlin Heidelberg 2003