Routing in histograms Let P be an xmonotone orthogonal polygon with n vertices. We call P a simple histogram if its upper boundary is a single edge; and a double histogram if it has a horizontal chord from the left boundary to the right boundary. Two points p and q in P are covisible if and only if the (axisparallel) rectangle spanned by p and q completely lies in P. In the rvisibility graph G(P) of P, we connect two vertices of P with an edge if and only if they are covisible. We consider routing with preprocessing in G(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, and the packet header. The routing problem has been studied extensively for general graphs, where truly compact and efficient routing schemes with polylogartihmic routing tables have turned out to be impossible. Thus, special graph classes are of interest.

Triangles and Girth in Disk Graphs and Transmission Graphs Let S subset mathbbR2 be a set of n sites, where each s in S has an associated radius r_s u003e 0. The disk graph D(S) is the undirected graph with vertex set S and an undirected edge between two sites s, t in S if and only if st leq r_s  r_t, i.e., if the disks with centers s and t and respective radii r_s and r_t intersect. Disk graphs are used to model sensor networks. Similarly, the transmission graph T(S) is the directed graph with vertex set S and a directed edge from a site s to a site t if and only if st leq r_s, i.e., if t lies in the disk with center s and radius r_s.  :123,provide algorithms for detecting (directed) triangles and, more generally, computing the length of a shortest cycle (the girth) in D(S) and in T(S). These problems are notoriously hard in general, but better solutions exist for special graph classes such as planar graphs. We obtain similarly efficient results for disk graphs and for transmission graphs. More precisely, we show that a shortest (Euclidean) triangle in D(S) and in T(S) can be found in O(n log n) expected time, and that the (weighted) girth of D(S) can be found in O(n log n) expected time. For this, we develop new tools for batched range searching that may be of independent interest.