Routing on the Visibility Graph

We consider the problem of routing on a network in the presence of line segment constraints (i.e., obstacles that edges in our network are not allowed to cross). Let P be a set of n points in the plane and let S be a set of non-crossing line segments whose endpoints are in P . We present two deterministic 1-local O(1)-memory routing algorithms that are guaranteed to find a path of at most linear size between any pair of vertices of the visibility graph of P with respect to a set of constraints S (i.e., the algorithms never look beyond the direct neighbours of the current location and store only a constant amount of additional information). Contrary to all existing deterministic local routing algorithms, our routing algorithms do not route on a plane subgraph of the visibility graph. Additionally, we provide lower bounds on the routing ratio of any deterministic local routing algorithm on the visibility graph.