We address the problem of discovering routes in strongly connected planar geometric networks with directed links. Motivated by the necessity for establishing communication in wireless ad hoc networks in which the only information available to a vertex is its immediate neighborhood, we are considering routing algorithms that use the neighborhood information of a vertex for routing with constant memory only. We solve the problem for three types of directed planar geometric networks: Eulerian (in which every vertex has the same number of incoming and outgoing edges), Outerplanar (in which a single face contains all vertices of the network), and Strongly Face Connected, a new class of geometric networks that we define in the article, consisting of several faces, each face being a strongly connected outerplanar graph.
A black hole is a highly harmful host that disposes of visiting agents upon their arrival. It is known that it is possible for a team of mobile agents to locate a black hole in an asynchronous ring network if each node is equipped with a whiteboard of at least O(log n) dedicated bits of storage. In this paper, we consider the less powerful token model: each agent has has available a bounded number of tokens that can be carried, placed on a node or removed from it. All tokens are identical (i.e., indistinguishable) and no other form of communication or coordination is available to the agents. We first of all prove that a team of two agents is sufficient to locate the black hole in finite time even in this weaker coordination model. Furthermore, we prove that this can be accomplished using only O(nlogn) moves in total, which is optimal, the same as with whiteboards. Finally, we show that to achieve this result the agents need to use only O(1) tokens each.