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greedy routing. Their work is a derivation from previous work intended to test graph connectivity [44], and it assumes that nodes may accurately measure the distances to their immediate neighbors. Virtual Coordinate Construction. For clarity, explanation of the NoGeo algorithm is presented in a cumulative fashion where at each additional step we remove some knowledge from the system. Hence we begin by describing the coordinate construction under the assumption that nodes on the network boundary, or perimeter, are aware of their position relative to the network as well as their exact location. The algorithm in reference 12 is based on an iterative relaxation procedure from reference 44 which is used to determine the locations of all remaining nodes in the network. The procedure is such that each link is represented by a force that pulls its adjoining neighbors together. The force in each of the x, y directions is proportional to the difference in the x, y coordinates. At any iteration a node s neighbors are held with xed position; its equilibrium position, where the sum of the forces acting on it is zero, is the average of all its neighbors x coordinates in the x direction, and similarly for the y direction. This relationship motivates the iterative procedure repeated periodically at each node i using the relaxation equations
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in each of the x and y directions, respectively. As an example, consider the wireless sensor network in Figure 8.10a. This network consists of 3200 nodes within a 200 200 unit space where each node has a
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Figure 8.10. The (a) real and (b) virtual coordinates of a 3200-node network. (a) Physical network in 200 200 space. (b) Virtual network where perimeter nodes are initially unknown.
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Figure 8.11. Intermittent virtual coordinates of 3200 nodes in a 200 200 space where nodes are known in advance to lie on the network s perimeter. (a) After 10 iterations. (b) After 100 iterations. (c) After 1000 iterations.
communication range of 8 units. As the iterative procedure described above is executed over this network, sensor coordinates will gradually shift to match their true coordinates within the network. This shaping is depicted following 10, 100, and 1000 iterations in Figures 8.11a 8.11c, respectively. In this example the initial coordinates of each node have been set to the center of the network at (100, 100). (We later return to compare the virtual and actual topologies.) This coordinate construction may be prefaced with additional steps if there is no advance knowledge of perimeter nodes, provided there are two beacons somewhere in the network. [Beacons are distinguished from the remaining network because they either (a) hold and disseminate information or (b) play some speci c coordination role required for successful setup and communications.] Either beacon may be used to identify nodes on the perimeter, after which point the perimeter nodes exchange messages to determine their locations. The rst step in accomplishing this task is to identify nodes on the perimeter. Recall the assumption that two beacon nodes exist. Either beacon is designated as the primary beacon, which broadcasts a HELLO message. Each node uses receipt of the HELLO message to determine its location from the beacon, according to the perimeter node criterion. The Perimeter Node Criterion says that if a node is farthest away from the bootstrap beacon among all nodes in its two-hop neighborhood, then this node decides it lays on the perimeter of the network. This is by no means an exact determination, but simulations show that it identi es a suf cient number of perimeter nodes. Once the nodes have identi ed their perimeter status, they must coordinate to exchange information and calculate their coordinates according to Algorithm 3. Each perimeter node, following Step 3 in Algorithm 3, has established its own virtual coordinate. One challenge remains. Each perimeter node has established its own virtual coordinate, yet the network lacks a global orientation. The reason is that any set of coordinates satisfying Eq. (8.3) may be rotated, translated, or transposed. The result is that perimeter nodes cannot be guaranteed to share the same sense of direction.