On some combinatorial problems in metric spaces of bounded doubling dimension

A metric space has doubling dimension d if for every ρ > 0, every ball of radius ρ can be covered by at most 2d balls of radius ρ/2. This generalizes the Euclidean dimension, because the doubling dimension of Euclidean space Rd is proportional to d. The following results are shown, for any d ≥ 1 and any metric space of size n and doubling dimension d: First, the maximum number of diametral pairs is Θ(n2). Second, if d = 1, the maximum possible weights of the minimum spanning tree and the all-nearest neighbors graph are Θ(R log n) and Θ(R), respectively, where R is the minimum radius of any ball containing all elements of the metric space. Finally, if d > 1, the maximum possible weights of both the minimum spanning tree and the all-nearest neighbors graph are Θ(Rn1−1/d). These results show that, for 1 ≤ d ≤ 3, metric spaces of doubling dimension d behave differently than their Euclidean counterparts.