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dc.contributor.authorArya, Sunil
dc.contributor.authorMalamatos, Theocharis
dc.contributor.authorMount, David M.
dc.date.accessioned2007-02-19T19:11:10Z
dc.date.available2007-02-19T19:11:10Z
dc.date.issued2006-11-13
dc.identifier.urihttp://hdl.handle.net/1903/4299
dc.description.abstractWe present space-time tradeoffs for approximate spherical range counting queries. Given a set S of n data points in real d-dimensional space along with a positive approximation factor eps, 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-eps)-factor contraction of B, but contains no points that lie outside a (1+eps)-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 positive eps and a parameter g, where 2 <= g <= 1/eps, we show how to construct a data structure of space O(n g^d log(1/eps)) that allows us to answer eps-approximate spherical range counting queries in time O(log(ng) + 1/(eps g)^{d-1}). The data structure can be built in time O(n g^d log(n/eps) log(1/eps)). Here n, eps, and g 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/eps)) that can answer approximate range queries in time O(log n + (1/eps)^{d-1}) which, up to a factor of O(log 1/eps) 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/eps^d) log(1/eps)) that can answer queries in time O(log n + log 1/eps). This is the fastest known query time for this problem. 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.en
dc.format.extent251014 bytes
dc.format.mimetypeapplication/pdf
dc.language.isoen_USen
dc.relation.ispartofseriesUM Computer Science Departmenten
dc.relation.ispartofseriesCS-TR-4842en
dc.relation.ispartofseriesUMIACSen
dc.relation.ispartofseriesUMIACS-TR-2006-57en
dc.titleSpace-Time Tradeoffs for Approximate Spherical Range Countingen
dc.typeTechnical Reporten


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