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

236 G. Calinescu and P.-J. Wan3.5 4.5v 1 v 1 v 2vv124v 2v 4 4v 3v 4v 3 v 4v 33 34.54.5(a)(b)(c)Fig. 1. The network topology: (a) the nodes and their transmission ranges, (b) theasymmetric topology, and (c) symmetric topology.relaying by intermediate nodes otherwise. Omnidirectional antennas are usedby all nodes to transmit and receive signals. Such antennas are attractive dueto their broadcast nature. A single transmission by a node can be received bymany nodes within its vicinity. We assume that every node can dynamicallyadjust its transmitting power based on the distance to the receiving node andthe background noise. In the most common power-attenuation model [22], thesignal power falls as 1dwhere d is the distance from the transmitter antennaκand κ is a real constant between 2 and 5 dependent on the wireless environment.We assume that all receivers have the same threshold for signal detection, andnormalize this threshold to one. With these assumptions, the power required tosupport a link between two nodes separated by a distance d is d κ .The network topology of a wireless ad hoc network, which consists of allpossible one-hop communication links among the nodes, is determined by thetransmission ranges of the nodes. Depending on whether unidirectional links orbidirectional links are used for communications, the network topology is representedby either a directed graph referred to as the asymmetric topology, oranundirected graph referred to as the symmetric topology. In the asymmetric topology,there is an arc from a node u to another node v if and only v is within thetransmission range of u. In the symmetric topology, there is an edge between twonodes u and v if and only they are within the transmission ranges of each other.An example is depicted in Figure 1. Figure 1 (a) gives the positions and thetransmission ranges of all nodes. The asymmetric topology and the symmetrictopology are given in Figure 1 (a) and (b) respectively.Connectivity is one of the most important properties of an wireless ad hocnetwork. By asymmetric k-node (resp., k-edge) connectivity we mean the asymmetrictopology is k-node (resp., k-edge) (strongly) connected, and by symmetrick-node (resp., k-edge) connectivity we mean the symmetric topology is k-node