Wireless Sensor Networks and Computational Geometry

Wireless Sensor Networks Due to its potential applications in various situations such as battle eld, emergency relief, environment monitoring, and so on, wireless sensor networks [50, 75, 118, 130] have recently emerged as a premier research topic. Sensor networks consist of a set of sensor nodes which are spread over a geographical area. These nodes are able to perform processing as well as sensing and are additionally capable of communicating with each other by means of a wireless ad hoc network. With coordination among these sensor nodes, the network together will achieve a larger sensing task both in urban environments and in inhospitable terrain. The sheer numbers of these sensors and the expected dynamics in these environments present unique challenges in the design of wireless sensor networks. Many excellent researches have been conducted to study problems in this new eld [50, 69, 75, 118, 119, 130]. In this chapter, we consider a wireless sensor network consisting of a set V of n wireless sensor nodes distributed in a two-dimensional plane. Each wireless sensor node has an omni-directional antenna. This is attractive because a single transmission of a node can be received by many nodes within its vicinity which, we assume, is a disk centered at the node. We call the radius of this disk the transmission range of this sensor node. In other words, node v can receive the signal from node u if node v is within the transmission range of the sender u. Otherwise, two nodes communicate through multi-hop wireless links by using intermediate nodes to relay the message. Consequently, each node in the sensor network also acts as a router, forwarding data packets for other nodes. By a proper scaling, we assume that all nodes have the maximum transmission range equal to one unit. These wireless sensor nodes de ne a unit disk graph UDG(V ) in which there is an edge between two nodes if and only if their Euclidean distance is at most one. In addition, we assume that each node has a low-power Global Position System (GPS) receiver, which provides the position information of the node itself. If GPS is not available, the distance between neighboring nodes can be estimated on the basis of incoming signal strengths. Relative co-ordinates of neighboring nodes can be obtained by exchanging such information between neighbors [26]. With the position information, we can apply computational geometry techniques to solve some challenging questions in sensor networks.

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