The Weak Gap Property in Metric Spaces of Bounded Doubling Dimension

We introduce the weak gap property for directed graphs whose vertex set S is a metric space of size n. We prove that, if the doubling dimension of S is a constant, any directed graph satisfying the weak gap property has O(n) edges and total weight $O( \log n ) \cdot wt({\mathord{\it MST}}(S))$, where $wt({\mathord{\it MST}}(S))$ 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.

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