We consider routing problems in ad hoc wireless networks modeled as unit graphs in which nodes are points in the plane and two nodes can communicate if the distance between them is less than some fixed unit. We describe the first distributed algorithms for routing that do not require duplication of packets or memory at the nodes and yet guaranty that a packet is delivered to its destination. These algorithms can be extended to yield algorithms for broadcasting and geocasting that do not require packet duplication. A byproduct of our results is a simple distributed protocol for extracting a planar subgraph of a unit graph. We also present simulation results on the performance of our algorithms.

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We consider online routing algorithms for routing between the vertices of embedded planar straight line graphs. Our results include (1) two deterministic memoryless routing algorithms, one that works for all Delaunay triangulations and the other that works for all regular triangulations; (2) a randomized memoryless algorithm that works for all triangulations; (3) an O(1) memory algorithm that works for all convex subdivisions; (4) an O(1) memory algorithm that approximates the shortest path in Delaunay triangulations; and (5) theoretical and experimental results on the competitiveness of these algorithms.

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Abstract. Given an n-vertex outer-planar graph G and a set P of n points in the plane, we present an O(nlog s n) time and O(n) space algorithm to compute a straight-line embedding of G in P, improving upon the algorithm in [GMPP91, CU96] that requires O(n 2) time. Our algorithm is near-optimal as there is an ~(nlogn) lower bound for the problem [BMS95]. We present a simpler O(nd) time and O(n) space algorithm to compute a straight-line embedding of G in P where log n _ d < 2n is the length of the longest vertex disjoint path in the dual of G. Therefore, the time complexity of the simpler algorithm varies between O(n log r~) and O(n 2) depending on the value of d. More efficient algorithms are presented for certain restricted cases. If the dual of G is a path, then an optimal O(n log n) time algorithm is presented. If the given point set is in convex position then we show that O(n) time suffices.