Routing in histograms

Let P be a histogram with n vertices, i.e., an x-monotone orthogonal polygon whose upper boundary is a single edge. Two points p, q ∈ P are co-visible if and only if the (axis-parallel) bounding rectangle of p and q is in P. In the r-visibility graph of P, we connect two vertices of P with an unweighted edge if and only if they are co-visible. We consider routing with preprocessing in P. We may preprocess P to obtain a label and a routing table for each vertex of P. Then, we
must be able to route a packet between any two vertices s and t of P, where each step may use only the label of the target node t, the routing table, and the neighborhood of the current node. We present a routing scheme for histograms that sends any data packet along a shortest path. Each label needs O(log n) bits, while the routing table of each node consists of a single bit