An algorithm for approximate closest-point queries
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Approximate nearest neighbors: towards removing the curse of dimensionality
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Efficient search for approximate nearest neighbor in high dimensional spaces
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
An optimal algorithm for approximate nearest neighbor searching fixed dimensions
Journal of the ACM (JACM)
Computational Geometry: Theory and Applications
Balanced aspect ratio trees: combining the advantages of k-d trees and octrees
Journal of Algorithms
Linear-size approximate voronoi diagrams
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
FST TCS '02 Proceedings of the 22nd Conference Kanpur on Foundations of Software Technology and Theoretical Computer Science
Faster core-set constructions and data stream algorithms in fixed dimensions
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Deformable spanners and applications
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
The skip quadtree: a simple dynamic data structure for multidimensional data
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Space-time tradeoffs for approximate spherical range counting
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
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
Deformable spanners and applications
Computational Geometry: Theory and Applications
Faster core-set constructions and data-stream algorithms in fixed dimensions
Computational Geometry: Theory and Applications
Journal of Computer and System Sciences
Minimum Spanning Tree with Neighborhoods
AAIM '07 Proceedings of the 3rd international conference on Algorithmic Aspects in Information and Management
Urban Data Visualization with Voronoi Diagrams
ICCSA '08 Proceeding sof the international conference on Computational Science and Its Applications, Part I
Space-Time Tradeoffs for Proximity Searching in Doubling Spaces
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Approximating nearest neighbor among triangles in convex position
Information Processing Letters
Approximate voronoi cell computation on spatial data streams
The VLDB Journal — The International Journal on Very Large Data Bases
Space-time tradeoffs for approximate nearest neighbor searching
Journal of the ACM (JACM)
Deformable spanners and applications
Computational Geometry: Theory and Applications
Faster core-set constructions and data-stream algorithms in fixed dimensions
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
The Geometric Stability of Voronoi Diagrams with Respect to Small Changes of the Sites
Proceedings of the twenty-seventh annual symposium on Computational geometry
DClusterE: A Framework for Evaluating and Understanding Document Clustering Using Visualization
ACM Transactions on Intelligent Systems and Technology (TIST)
Self-Organizing Maps for advanced learning in cognitive radio systems
Computers and Electrical Engineering
Enclosing surfaces for point clusters using 3d discrete voronoi diagrams
EuroVis'09 Proceedings of the 11th Eurographics / IEEE - VGTC conference on Visualization
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(MATH) Given a set $S$ of $n$ points in $\IR^d$, a {\em $(t,\epsilon)$-approximate Voronoi diagram (AVD)} is a partition of space into constant complexity cells, where each cell $c$ is associated with $t$ representative points of $S$, such that for any point in $c$, one of the associated representatives approximates the nearest neighbor to within a factor of $(1+\epsilon)$. Like the Voronoi diagram, this structure defines a spatial subdivision. It also has the desirable properties of being easy to construct and providing a simple and practical data structure for answering approximate nearest neighbor queries. The goal is to minimize the number and complexity of the cells in the AVD.(MATH) We assume that the dimension $d$ is fixed. Given a real parameter $\gamma$, where $2 \le \gamma \le 1/\epsilon$, we show that it is possible to construct a $(t,\epsilon)$-AVD consisting of \[O(n \epsilon^{\frac{d-1}{2}} \gamma^{\frac{3(d-1)}{2}} \log \gamma) \] cells for $t = O(1/(\epsilon \gamma)^{(d-1)/2})$. This yields a data structure of $O(n \gamma^{d-1} \log \gamma)$ space (including the space for representatives) that can answer $\epsilon$-NN queries in time $O(\log(n \gamma) + 1/(\epsilon \gamma)^{(d-1)/2})$. (Hidden constants may depend exponentially on $d$, but do not depend on $\epsilon$ or $\gamma$).(MATH) In the case $\gamma = 1/\epsilon$, we show that the additional $\log \gamma$ factor in space can be avoided, and so we have a data structure that answers $\epsilon$-approximate nearest neighbor queries in time $O(\log (n/\epsilon))$ with space $O(n/\epsilon^{d-1})$, improving upon the best known space bounds for this query time. In the case $\gamma = 2$, we have a data structure that can answer approximate nearest neighbor queries in $O(\log n + 1/\epsilon^{(d-1)/2})$ time using optimal $O(n)$ space. This dramatically improves the previous best space bound for this query time by a factor of $O(1/\epsilon^{(d-1)/2})$.(MATH) We also provide lower bounds on the worst-case number of cells assuming that cells are axis-aligned rectangles of bounded aspect ratio. In the important extreme cases $\gamma \in \{2, 1/\epsilon\}$, our lower bounds match our upper bounds asymptotically. For intermediate values of $\gamma$ we show that our upper bounds are within a factor of $O((1/\epsilon)^{(d-1)/2}\log \gamma)$ of the lower bound.