Doubling space

Abstract: We consider approximate nearest neighbor searching in metric spaces of constant doubling dimension. More formally, we are given a set Sof npoints and an error bound i¾?> 0. The objective is to build a data structure so that given any query point qin the space, it is possible to efficiently determine a point of Swhose distance from qis within a factor of (1 + i¾?) of the distance between qand its nearest neighbor in S. In this paper we obtain the following space-time tradeoffs. Given a parameter i¾?i¾? [2,1/i¾?], we show how to construct a data structure of space $n \gamma^{O(\dim)} \log(1/\varepsilon)$ space that can answer queries in time $O(\log(n\gamma)) + (1/(\varepsilon \gamma))^{O(\dim)}$. This is the first result that offers space-time tradeoffs for approximate nearest neighbor queries in doubling spaces. At one extreme it nearly matches the best result currently known for doubling spaces, and at the other extreme it results in a data structure that can answer queries in time O(log(n/i¾?)), which matches the best query times in Euclidean space. Our approach involves a novel generalization of the AVD data structure from Euclidean space to doubling space.

Abstract: In this paper we prove that if (X, d, mu) is a metric doubling space with segment property, then inf r(E)/r(B) > 0 if and only if inf mu (E)/mu (B) > 0, where the infimum is taken over any collection C of balls E;B such that E subset of B subset of X. As a consequence we show that if X is a linear metric doubling space, then it must be finite dimensional.

Abstract: An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a tree-shaped branching structure in many applications such as trees, blood vessels, draining and irrigation systems. Here, we extend the study of ramified optimal transportation between probability measures from Euclidean spaces to a geodesic metric space. We investigate the existence as well as the behavior of optimal transport paths under various properties of the metric such as completeness, doubling, or curvature upper boundedness. We also introduce the transport dimension of a probability measure on a complete geodesic metric space, and show that the transport dimension of a probability measure is bounded above by the Minkowski dimension and below by the Hausdorff dimension of the measure. Moreover, we introduce a metric, called "the dimensional distance", on the space of probability measures. This metric gives a geometric meaning to the transport dimension: with respect to this metric, the transport dimension of a probability measure equals to the distance from it to any finite atomic probability measure.

Abstract: We consider approximate nearest neighbor searching in metric spaces of constant doubling dimension. More formally, we are given a set Sof npoints and an error bound i¾?> 0. The objective is to build a data structure so that given any query point qin the space, it is possible to efficiently determine a point of Swhose distance from qis within a factor of (1 + i¾?) of the distance between qand its nearest neighbor in S. In this paper we obtain the following space-time tradeoffs. Given a parameter i¾?i¾? [2,1/i¾?], we show how to construct a data structure of space $n \gamma^{O(\dim)} \log(1/\varepsilon)$ space that can answer queries in time $O(\log(n\gamma)) + (1/(\varepsilon \gamma))^{O(\dim)}$. This is the first result that offers space-time tradeoffs for approximate nearest neighbor queries in doubling spaces. At one extreme it nearly matches the best result currently known for doubling spaces, and at the other extreme it results in a data structure that can answer queries in time O(log(n/i¾?)), which matches the best query times in Euclidean space. Our approach involves a novel generalization of the AVD data structure from Euclidean space to doubling space.