Applications of spatial data structures: Computer graphics, image processing, and GIS
Applications of spatial data structures: Computer graphics, image processing, and GIS
K-d trees for semidynamic point sets
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Proceedings of the eleventh annual symposium on Computational geometry
View-dependent simplification of arbitrary polygonal environments
Proceedings of the 24th annual conference on Computer graphics and interactive techniques
Data collection for the Sloan digital sky survey—a network-flow heuristic
Journal of Algorithms
Algorithms for dynamic closest pair and n-body potential fields
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
Balanced aspect ratio trees: combining the advantages of k-d trees and octrees
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
An optimal algorithm for approximate nearest neighbor searching
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Multidimensional binary search trees used for associative searching
Communications of the ACM
Balanced Aspect Ratio Trees and Their Use for Drawing Very Large Graphs
GD '98 Proceedings of the 6th International Symposium on Graph Drawing
K-D Trees Are Better when Cut on the Longest Side
ESA '00 Proceedings of the 8th Annual European Symposium on Algorithms
Near real-time shaded display of rigid objects
SIGGRAPH '83 Proceedings of the 10th annual conference on Computer graphics and interactive techniques
On visible surface generation by a priori tree structures
SIGGRAPH '80 Proceedings of the 7th annual conference on Computer graphics and interactive techniques
Balanced aspect ratio trees
Multidimensional Binary Search Trees in Database Applications
IEEE Transactions on Software Engineering
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Spatial databases support a variety of geometric queries on point data such as range searches, nearest neighbor searches, etc. Balanced Aspect Ratio (BAR) trees are hierarchical space decomposition structures that are general-purpose and space-efficient, and, in addition, enjoy a worst case performance poly-logarithmic in the number of points for approximate queries. They maintain limits on their depth, as well as on the aspect ratio (intuitively, how skinny the regions can be). BAR trees were initially developed for 2 dimensional spaces and a fixed set of partitioning planes, and then extended to d dimensional spaces and more general partitioning planes. Here we revisit 2 dimensional spaces and show that, for any given set of 3 partitioning planes, it is not only possible to construct such trees, it is also possible to derive a simple closed-form upper bound on the aspect ratio. This bound, and the resulting algorithm, are much simpler than what is known for general BAR trees. We call the resulting BAR trees Parameterized BAR trees and empirically evaluate them for different partitioning planes. Our experiments show that our theoretical bound converges to the empirically obtained values in the lower ranges, and also make a case for using evenly oriented partitioning planes.