Storing a Sparse Table with 0(1) Worst Case Access Time
Journal of the ACM (JACM)
Randomized algorithms
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
An optimal algorithm for approximate nearest neighbor searching fixed dimensions
Journal of the ACM (JACM)
On some geometric optimization problems in layered manufacturing
Computational Geometry: Theory and Applications
An optimal algorithm for approximate nearest neighbor searching
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Computational Geometry: Theory and Applications
Space-efficient approximate Voronoi diagrams
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Linear-size approximate voronoi diagrams
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Smooth-surface reconstruction in near-linear time
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
A Replacement for Voronoi Diagrams of Near Linear Size
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
The skip quadtree: a simple dynamic data structure for multidimensional data
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
On the importance of idempotence
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
The effect of corners on the complexity of approximate range searching
Proceedings of the twenty-second annual symposium on Computational geometry
On approximate halfspace range counting and relative epsilon-approximations
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Space-time tradeoffs for approximate nearest neighbor searching
Journal of the ACM (JACM)
Approximate range searching: The absolute model
Computational Geometry: Theory and Applications
A unified approach to approximate proximity searching
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Approximate Halfspace Range Counting
SIAM Journal on Computing
Sampling-based algorithms for optimal motion planning
International Journal of Robotics Research
approximate range searching: the absolute model
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
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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 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.