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ALGORITHM 3. Coordinate Determination among Perimeter Nodes in NoGeo 1. Each perimeter node broadcasts a HELLO message so that each may calculate its distance to every other perimeter node. The result is stored in a perimeter vector. 2. Each perimeter node broadcasts its perimeter vector to the network so that each perimeter node knows the distance between every pair of perimeter nodes in the network. 3. Each perimeter node triangulates to nd the coordinates of every perimeter node in the network. Coordinates are chosen so as to minimize (measured dist(i, j) euclid dist(i, j))2
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i,j perimeter
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where measured dist(i, j) represents the distance between nodes i, j as measured in 1, and dist(i, j) is the Euclidean distance between the virtual coordinates of i and j.
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The solution is the reason why two beacons must exist. The two beacons share their perimeter vectors and participate in the triangulation process. During this process, each of the perimeter nodes calculates the center of gravity of the network. The combination of the two beacons with the center of gravity provides a set of axes on which all nodes can agree: The center of gravity becomes the origin, while each of the two beacons de ne the positive x, y axes. An example topology at equilibrium, following the execution of perimeter identi cation, coordinate determination, and dissemination, appears in Figure 8.10b. This is the virtual network space that corresponds to the actual network in Figure 8.10a. Using this scheme, the virtual coordinates maintain a structure similar to the actual coordinates, but with some caveats. For example, the network appears to have been rotated about the virtual center of gravity. However, the rotation is uniform and consistent, meaning that the rotation is network-wide. Note that holes in the virtual networks constructed in Figures 8.10b and 8.11c are much larger in size than in the actual network of Figure 8.10a. As discussed below, this salient feature does not re ect itself in the routing mechanism. Routing in the virtual Cartesian coordinate system takes the form of pure greedy routing as described in Section 8.3.1: Messages are forwarded to the neighbor that most reduces the distance to destination until no such neighbor exists or the destination is found. Despite the inaccuracies of location determination and the increase in hole sizes (as re ected in physical space), greedy routing over virtual coordinate systems performs better than it does over physical coordinate systems. The rate of successful delivery rate is higher when routing over virtual coordinate systems. The underlying reason for an increase in the rate of delivery is that locations are assigned coordinates relative to the locations of other nodes in the network, not relative to the space that the network occupies. Hence, a virtual coordinate system accurately re ects not network geography, but rather network connectivity. This relationship,
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when re ected in actual space, is responsible for the apparent growth in hole sizes. (Despite this fact, the physical coverage area is unaffected, a necessary property of sensor networks.) It is also responsible for the improvement in the performance of greedy routing because it re ects network paths and connections more than node location in the physical space. One drawback to approaches in this class of position-based routing is the expense incurred by the computation and transmission necessary to establish a reasonable coordinate system. Consider NoGeo, for example, that it is somewhat impractical in that initialization requires O( n) nodes to ood the network, and it is also inpractical for least this number of nodes to store O(n) state (since the distance vectors occupy at O( n n) space). We proceed with an approach that reduces this expense in exchange for a loss in topological accuracy.
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8.4.2 Hop-Based Coordinates and Routing One alternative approach to constructing virtual coordinates foregoes Euclidean approximations, altogether. Instead, designated coordinates are comprised of a vector that contains a set of hop-distances to beacons located around the sensor network [45 48]. As is the case with NoGeo, the goal of these methods is to deliver packets in an environment with no a priori knowledge of node locations, in a point-to-point manner. We demonstrate this approach to position-based routing using Beacon Vector Routing (BVR) in reference 47 and reserve discussion for the differences with other projects until later. BVR is a protocol that assigns routing coordinates and de nes a distance function used in forwarding decisions. A node s coordinate is a tuple recording the hop distance to each of a subset of available beacons, information that is disseminated using reverse path tree constructions. (A reverse path tree construction occurs when a beacon broadcasts its existence to the network and all remaining nodes record the shortest hop distance to that beacon.) A distance function is used to route greedily. In the event that greedy routing fails, a correction mechanism exists to guarantee delivery. Let r denote the total number of beacons, and let qi denote the distance in hops from a node to beacon i. The position of node q is the tuple (q1 , q2 , . . . , qr ). By this de nition, it is possible for multiple nodes to share the same coordinate so a node identi er is necessary to disambiguate between nodes with identical coordinates. The distance function must favor greedy forwarding to maintain a high level of ef ciency. When evaluating the distance function, the BVR metric aims to minimize the difference in coordinates componentwise. This metric is based on the idea that it is better to move toward a beacon close to the destination than it is to move away. Hence, the distance function is designed to move a message toward a beacon if the destination is closer to the beacon than the current node; that is, it also moves a message away from a beacon if the destination is further away. (Note that using this intuition, movement toward a beacon always reduces the distance to the destination but moving away is not: The destination may sit on the other side of the beacon from the current node.)
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