The Gap Property in Doubling Metrics

We introduce the weak gap property as a variant of the gap property. We show that in any metric space of bounded doubling dimension, any directed graph whose vertex set S has size n and which satisfies the weak gap property has total weight O(wt(MST (S)) log n), where wt(MST (S)) denotes the weight of a minimum spanning tree of S. We show that 2-optimal TSP tours and greedy spanners satisfy the weak gap property. Parts of these notes are based on discussions with Hubert Chan, Anupam Gupta, and Giri Narasimhan, during the workshop Geometric Networks and Metric Space Embeddings, which was held in Dagstuhl (Germany) in November 2006.