We consider the problem of routing a data packet through the visibility graph of a polygonal domain P with n vertices and h holes. 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 data packet between any two vertices p and q of P , where each step must use only the label of the target node q and the routing table of the current node.For any fixed ε > 0, we present a routing scheme that always achieves a routing path whose length exceeds the shortest path by a factor of at most 1 + ε. The labels have O(log n) bits, and the routing tables are of size O((ε −1 + h) log n). The preprocessing time is O(n 2 log n). It can be improved to O(n 2 ) for simple polygons.
Suppose we are given a set D of n pairwise intersecting disks in the plane. A planar point set P stabs D if and only if each disk in D contains at least one point from P . We present a deterministic algorithm that takes O(n) time to find five points that stab D. Furthermore, we give a simple example of 13 pairwise intersecting disks that cannot be stabbed by three points.This provides a simple -albeit slightly weaker -algorithmic version of a classical result by Danzer that such a set D can always be stabbed by four points.
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