A recently developed routing algorithm for broadcasting in an *ad* *hoc* wireless communication network takes account of, and exploits, the spatial relationships among the locations of nodes, in addition to transmission power levels and distances between the nodes. In contrast, most prior algorithms for discovering routes through *ad hoc* networks rely heavily on transmission power levels and utilize limited graph-topology techniques that do not involve consideration of the aforesaid spatial relationships. The present algorithm extracts the relevant spatial-relationship information by use of a construct denoted the relative-neighborhood graph (RNG).

The RNG algorithm is best described by reference to, and comparison with, two prior algorithms: one based on a construct denoted the minimum-radius (minR) graph, the other based on a construct denoted the minimum spanning tree (MST). The minR algorithm starts from the assumption that every node must use the same transmission radius, *d*. [As used here, "transmission radius" does not signify radius as the term is commonly understood; instead, it signifies an effective radius that depends on the ratio between the transmission power and the bit rate.] An iterative binary search is performed to find the smallest *d* that guarantees connectivity of the network. On each iteration, a graph with transmission radius *d* is computed. If the graph is found to be connected, then *d* is decreased for the next iteration. The algorithm stops at the finding of the value of *d* such that the graph is connected but at the next smaller increment of *d*, the graph is partitioned into two or more sub-graphs. The number of operations needed to compute the minR graph is proportional to a number of the order of *n*^{2}log(s), where *n* is the number of nodes and s is the length of the side of a square of the same area as that containing the nodes.

The RNG of a set of nodes is the graph constructed according to the rule that any two of the nodes separated by distance *l* are connected by an edge only if there does not exist another node at a distance <*l* from either first-mentioned node. The two nodes so connected are denoted relative neighbors. In the RNG, unlike in the minR graph, a different radius can be used for each pair of nodes. The RNG is a super-graph of the MST (which is described next) and a sub-graph of the graph of the Delaunay triangulation, which is a technique for connecting pairs of grid points with straight lines in such a way as to obtain an optimum tessellation of a grid area. The number of operations needed to compute the RNG is proportional to a number of the order of *n*log(*n*).

The MST is a sub-graph of the RNG. Therefore, one can use the RNG in the computation of the MST. An edge is included in the MST if it does not create a cycle in the graph. The construction of the MST involves disjoint sets. Nodes that are connected are placed in the same set. If a tested edge connects two nodes belonging to different sets, then the edge is added to MST and the two sets are united. If a tested edge connects two nodes belonging to the same set, then this edge creates a cycle and it is rejected. The number of operations needed to compute the MST is bounded by a number proportional to *n*, given that the RNG was pre-computed.

The comparison among the three algorithms is unavoidably complex because each offers both advantages and disadvantages with respect to the others. The figure presents an example of comparison with respect to the maximum node degree, which is only one of several different measures applied to results of computational simulations. The only general conclusion that one can draw from the comparisons by this and other measures is that the graph properties of the RNG are good, relative to those of the minR graph and the MST and that the RNG is suitable for efficient dissemination of information to all the nodes of a network.

*This work was done by Clayton Okino and Esther Jennings of Caltech for ***NASA's Jet Propulsion Laboratory***. For further information, access the Technical Support Package (TSP) ***free on-line at www.techbriefs.com/tsp*** under the Electronics/Computers category.*

*This algorithm is available for commercial licensing. Please contact Don Hart of the California Institute of Technology at (818) 393-3425. Refer to NPO-30453.*

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This article first appeared in the February, 2004 issue of *NASA Tech Briefs* Magazine.

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