Recently, Arya, da Fonseca, and Mount [STOC 2011, SODA 2012] made notable progress in improving the epsilon-dependencies in the space/query-time tradeoffs for (1+epsilon)-factor approximate nearest neighbor search in fixed-dimensional Euclidean spaces. However, epsilon-dependencies in the preprocessing time were not considered, and so their data structures cannot be used to derive faster algorithms for offline proximity problems. Known algorithms for many such problems, including approximate bichromatic closest pair (BCP) and approximate Euclidean minimum spanning trees (EMST), typically have factors near (1/epsilon)^{d/2 +/- O(1)} in the running time when the dimension d is a constant.
We describe a technique that breaks the (1/epsilon)^{d/2} barrier and yields new results for many well-known proximity problems, including:
The improvement arises from a new time bound for exact "discrete Voronoi diagrams", which were previously used in the construction of epsilon-kernels (or extent-based coresets), a well-known tool for another class of fundamental problems. This connection leads to more results, including:
We study the complexity of geometric minimum spanning trees under a stochastic model of input: Suppose we are given a master set of points {s_1,s_2,...,s_n} in d-dimensional Euclidean space, where each point s_i is active with some independent and arbitrary but known probability p_i. We want to compute the expected length of the minimum spanning tree (MST) of the active points. This particular form of stochastic problems has not been investigated before in computational geometry to our knowledge, and is motivated by uncertainty inherent in many sources of geometric data.
Given a point set in a fixed dimension, we note that a well-separated pair decomposition can be found in linear time if we assume that the ratio of the farthest pair distance to the closest pair distance is polynomially bounded. Many consequences follow; for example, we can construct spanners or solve the all-nearest-neighbors problem in linear time (under the same assumption), and we compute an approximate Euclidean minimum spanning tree in linear time (without any assumption).
Let tau_K^{(d)} be the analogous ratio in d-dimensional space. Khuller et al. showed that tau_3^{(d)} < 1.667 for any d. We observe that tau_3^{(d)} < 1.633.
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