On the Doubling Dimension and the Perimeter of Geodesically Convex Sets in Fat Polygons

2026-04-15

AI Summary

This work investigates the boundedness of the doubling dimension under geodesic distance and upper bounds on the perimeter of geodesically convex sets in simple polygons. Focusing on two classes of fat polygons—locally-fat and (α,β)-covered—the study combines computational geometry and metric space theory to establish, for the first time, that (α,β)-covered polygons exhibit a bounded doubling dimension and that the perimeter of any geodesically convex subset is at most a constant factor times its Euclidean diameter. In contrast, locally-fat polygons do not possess these properties. Leveraging these theoretical results, the paper presents an expected O(n + m log n)-time algorithm for finding the closest pair among m points within an (α,β)-covered polygon.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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.

Problem

• doubling dimension
• geodesic distance
• fat polygons
• geodesically convex sets
• perimeter

Innovation

• doubling dimension
• geodesic distance
• fat polygons
• geodesically convex sets
• closest pair algorithm