A generalization of the FACE ROUTING algorithm to a class of non-planar networks

We consider the problem of routing with guaranteed delivery in ad-hoc wireless networks using the positions of the mobile hosts. Such networks can be modeled as geometric graphs. FACE ROUTING [Bose, P et al. (1999), Karp, B et al. (2000)] is a position-based routing algorithm for planar geometric graphs that guarantees delivery of messages without flooding control packets throughout the network. For general ad hoc networks, FACE ROUTING can use a planar sub-graph of the original graph; many local and distributed algorithms have been proposed to extract such a planar sub-graph. However, these planarization algorithms may fail in some situations, such as when the transmission ranges are not the same, for example, due to the presence of obstacles, which in turn may cause a routing failure. In this paper, we describe a generalization of FACE ROUTING that can guarantee delivery in planar graphs with disjoint crossing edges added. Our algorithm needs O(ℓ) memory, where ℓ is the maximum number of edges in any face in a graph obtained by removing one edge in each pair of crossing edges.