Routing in polygonal domains

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(n²log⁡n). It can be improved to O(n²) for simple polygons.