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coordinates using the iterative procedure (xu , yu ) =
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where N(u) is the set of nodes which are neighbors of u. The relaxation equations imply that nonperimeter nodes having perimeter nodes among their neighbors will tend to move toward perimeter nodes that are closest to them in terms of the number of hops. Perimeter Nodes Are Known. In this second case an algorithm is designed that does not use the fact that perimeter nodes know their exact geographic location. This is done by prefacing the previous relaxation method with a phase where perimeter nodes compute their own approximate virtual coordinates. The algorithm is in three steps. In the rst step, each perimeter node broadcasts a hello message to the entire network so as to discover its relative position (distances in hops) to all other perimeter nodes in the network: Call h(u, v) this distance measured in number of hops between perimeter nodes. In the second step, perimeter nodes broadcast their perimeter vector (i.e., vector of these distances to all other perimeter nodes) to the entire network. Finally, in the third step, every perimeter node uses a triangulation algorithm to compute the coordinates of all other perimeter nodes (including its own coordinates). Coordinates are chosen so as to minimize (h(u, v) d(u, v))2
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where d(u, v) represents the Euclidean distance between the virtual coordinates of nodes u and v. Since at the end of the third step perimeter nodes know their own virtual coordinates, nonperimeter nodes can use the previous relaxation algorithm in order to compute their own virtual coordinates. An important point raised by Ratnasamy et al. [13] is that message loss and node failure can cause perimeter nodes to have incomplete knowledge of the interperimeter distances and cause different perimeter nodes compute inconsistent coordinates. To address this problem, they use two designated bootstrapping beacons that ood the network with hello messages in order to canonicalize the computation and make all nodes performing it arrive at the same solution. Without Location Information. In this third case, the assumption that perimeter nodes know they are on the perimeter is relaxed. A preparatory stage is added to the previous algorithm where a subset of nodes identify themselves as perimeter nodes. This is achieved by leveraging one of the bootstrap beacon nodes described before. Since these bootstrap nodes broadcast hello messages to the entire network, every node discovers its distance (in number of hops) to these bootstrap nodes. To decide if they are perimeter nodes, respective nodes use the perimeter node criterion:
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A node decides that it is on the perimeter if it is the farthest away, among all its two-hop neighbors from the rst bootstrap node. Furthermore, additional coordinate projection to a circle mechanism is added, allowing maintainance of a consistent virtual coordinate space even in the face of node mobility and failures.
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2.4 (RELATIVE) LOCATION DETERMINATION Although localization of network nodes is helpful in clustering, routing, and network map building, there are instances where either sensor nodes do not have GPS capability or satellite signals are blocked by obstacles or even a GPS receiver is affected by noise. In this section, methods are presented for localizing nodes that may not have GPS receiver capability. They all rely on the fact that the position of a nonlocalized node can be determined if the positions of localized neighbor nodes can be obtained. This information is relatively easy to obtain using advertisement messages and network ooding mechanisms. Firstly, there are methods using triangulation and distance estimates to localized neighbors. These distance estimates can be obtained by examining features of received radio signals (i.e., radio-location) or a logical method. Secondly, there are methods that are not based on distance estimates, but rather try to delimit the area in which a node yet to be localized should logically be contained. A method for doing triangulation is discussed in Subsection 2.4.1. Estimation of distances using radio location is discussed in Subsection 2.4.2. Subsection 2.4.3 presents the logical distance estimation method used in the distance vector-hop algorithm. An area delimiting method, called the bounding box algorithm, is reviewed in Subsection 2.4.4. Finally, Subsection 2.4.5 explains how to build a consistent coordinate communication infrastructure using radio-location even when nodes cannot access directly GPS information. 2.4.1 Triangulation The distance from node A to node B de nes a circle around node B, and the position of A is on the circumference of this circle. In a two-dimensional model, the position of A is unambiguously determined as the intersection of three such circles. Each circle can be modeled by a quadratic equation, and the intersection point of these circles is theoretically the location of node A. The triangulation algorithm discussed in the sequel avoids quadratic equations and uses instead linear equations, which is a problem of lower dif culty that can be solved with linear algebra tools alone. This form of triangulation is used by GPS (see references 15 and 16) for their ad hoc positioning system. Figure 2.6 depicts a node u that needs to determine its position. The positions of localized neighbors are used. The triangulation is an iterative procedure, which starts with an arbitrary estimate ( u , yu ) of the position of node u. The estimate is re ned x from one iteration to the next. The iterative procedure is repeated until the change from the previous estimate is below a given threshold. Firstly, the following procedure is applied for each localized neighbor v. (Figure 2.6 depicts an example using node
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