On the Doubling Dimension and the Perimeter of Geodesically Convex Sets in Fat Polygons
For researchers working on algorithms in polygonal domains, this work provides theoretical foundations and algorithmic improvements for problems in (α,β)-covered polygons.
The paper studies the doubling dimension of geodesic metric spaces in fat polygons, proving that (α,β)-covered polygons have bounded doubling dimension while locally-fat simple polygons do not. It also bounds the perimeter of geodesically convex sets in (α,β)-covered polygons, leading to an O(n + m log n) expected-time algorithm for closest pair in such polygons.
Many algorithmic problems can be solved (almost) as efficiently in metric spaces of bounded doubling dimension as in Euclidean space. Unfortunately, the metric space defined by points in a simple polygon equipped with the geodesic distance does not necessarily have bounded doubling dimension. We therefore study the doubling dimension of fat polygons, for two well-known fatness definitions. We prove that locally-fat simple polygons do not always have bounded doubling dimension, while any $(α,β)$-covered polygon does have bounded doubling dimension (even if it has holes). We also study the perimeter of geodesically convex sets in $(α,β)$-covered polygons (possibly with holes), and show that this perimeter is at most a constant times the Euclidean diameter of the set. Using these two results, we obtain new results for several problems on $(α,β)$-covered polygons, including an algorithm that computes the closest pair of a set of $m$ points in an $(α,β)$-covered polygon with $n$ vertices that runs in $O(n + m\log{n})$ expected time.