@conference {15638,
	title = {Space-time tradeoffs for approximate spherical range counting},
	booktitle = {Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms},
	series = {SODA {\textquoteright}05},
	year = {2005},
	month = {2005///},
	pages = {535 - 544},
	publisher = {Society for Industrial and Applied Mathematics},
	organization = {Society for Industrial and Applied Mathematics},
	address = {Philadelphia, PA, USA},
	abstract = {We present space-time tradeoffs for approximate spherical range counting queries. Given a set S of n data points in Rd along with a positive approximation factor ε, the goal is to preprocess the points so that, given any Euclidean ball B, we can return the number of points of any subset of S that contains all the points within a (1 - ε)-factor contraction of B, but contains no points that lie outside a (1 + ε)-factor expansion of B.In many applications of range searching it is desirable to offer a tradeoff between space and query time. We present here the first such tradeoffs for approximate range counting queries. Given 0 < ε <= 1/2 and a parameter γ, where 2 <= γ <= 1/ε, we show how to construct a data structure of space O(nγd log (1/ε)) that allows us to answer ε-approximate spherical range counting queries in time O(log(nγ) + 1/(εγd-1). The data structure can be built in time O(nγd log (n/ε)) log (1/ε)). Here n, ε, and γ are asymptotic quantities, and the dimension d is assumed to be a fixed constant.At one extreme (low space), this yields a data structure of space O(n log (1/e)) that can answer approximate range queries in time O(logn + 1/(ed-1) which, up to a factor of O(n log (1/e) in space, matches the best known result for approximate spherical range counting queries. At the other extreme (high space), it yields a data structure of space O((n/ed) log(1/ε)) that can answer queries in time O(logn + 1/ε). This is the fastest known query time for this problem.We also show how to adapt these data structures to the problem of computing an ε-approximation to the kth nearest neighbor, where k is any integer from 1 to n given at query time. The space bounds are identical to the range searching results, and the query time is larger only by a factor of O(1/(εγ)).Our approach is broadly based on methods developed for approximate Voronoi diagrams (AVDs), but it involves a number of significant extensions from the context of nearest neighbor searching to range searching. These include generalizing AVD node-separation properties from leaves to internal nodes of the tree and constructing efficient generator sets through a radial decomposition of space. We have also developed new arguments to analyze the time and space requirements in this more general setting.},
	isbn = {0-89871-585-7},
	url = {http://dl.acm.org/citation.cfm?id=1070432.1070505},
	author = {Arya,Sunil and Malamatos,Theocharis and Mount, Dave}
}
