On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
Proceedings on STACS 85 2nd annual symposium on theoretical aspects of computer science
Efficient data structures for range searching on a grid
Journal of Algorithms
The complexity and construction of many faces in arrangements of lines and of segments
Discrete & Computational Geometry - Special issue on the complexity of arrangements
Applications of spatial data structures: Computer graphics, image processing, and GIS
Applications of spatial data structures: Computer graphics, image processing, and GIS
The design and analysis of spatial data structures
The design and analysis of spatial data structures
A New Region Expansion for Quadtrees
IEEE Transactions on Pattern Analysis and Machine Intelligence
BLASTING through the information theoretic barrier with FUSION TREES
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
A general approach to connected-component labeling for arbitrary image representations
Journal of the ACM (JACM)
An algorithm to compute the Minkowski sum outer-face of two simple polygons
Proceedings of the twelfth annual symposium on Computational geometry
Improvements on bottleneck matching and related problems using geometry
Proceedings of the twelfth annual symposium on Computational geometry
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
On the complexity of the union of fat objects in the plane
SCG '97 Proceedings of the thirteenth 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
Optimal bounds for the predecessor problem
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
The complexity of the union of (&agr;, &bgr;)-covered objects
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
An optimal algorithm for approximate nearest neighbor searching
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Dynamic Data Structures for Fat Objects and Their Applications
WADS '97 Proceedings of the 5th International Workshop on Algorithms and Data Structures
A cell probe lower bound for dynamic nearest-neighbor searching
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Optimal static range reporting in one dimension
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Closest-point problems simplified on the RAM
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Optimal bounds for the predecessor problem and related problems
Journal of Computer and System Sciences - STOC 1999
On coresets for k-means and k-median clustering
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
International Journal of Mobile Communications
International Journal of Wireless and Mobile Computing
Elkan's k-means algorithm for graphs
MICAI'10 Proceedings of the 9th Mexican international conference on Artificial intelligence conference on Advances in soft computing: Part II
Computer Vision and Image Understanding
The anchors hierarchy: using the triangle inequality to survive high dimensional data
UAI'00 Proceedings of the Sixteenth conference on Uncertainty in artificial intelligence
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In this paper we investigate data-structures obtained by a recursive partitioning of the input domain into regions of equal size. One of the most well known examples of such a structure is the quadtree, used here as a basis for more complex data structures; we also provide multidimensional versions of the stratified tree by van Emde Boas [24].We show that under the assumption that the input points have limited precision (i.e. are drawn from the integer grid of size u) these data structures yield efficient solutions to many important problems. In particular, they allow us to achieve O(log log u) time per operation for dynamic approximate nearest neighbor (under insertions and deletions) and exact on-line closest pair (under insertions only) in any constant dimension. They allow O(log log u) point location in a given planar shape or in its expansion (dilation by a ball of a given radius).Finally, we provide a linear time (optimal) algorithm for computing the expansion of a shape represented by a quadtree. This result shows that the spatial order imposed by this regular data structure is sufficient to optimize the dilation by a ball operation.