A data structure for dynamic trees
Journal of Computer and System Sciences
Skip lists: a probabilistic alternative to balanced trees
Communications of the ACM
Approximate closest-point queries in high dimensions
Information Processing Letters
An optimal algorithm for approximate nearest neighbor searching fixed dimensions
Journal of the ACM (JACM)
Journal of Algorithms
An effective way to represent quadtrees
Communications of the ACM
Computational Geometry: Theory and Applications
Balanced aspect ratio trees: combining the advantages of k-d trees and octrees
Journal of Algorithms
Closest-point problems simplified on the RAM
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Dynamic Algorithms for Approximate Neighbor Searching
Proceedings of the 8th Canadian Conference on Computational Geometry
Balanced aspect ratio trees
Foundations of Multidimensional and Metric Data Structures (The Morgan Kaufmann Series in Computer Graphics and Geometric Modeling)
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Fast algorithms for the all nearest neighbors problem
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
Fully retroactive approximate range and nearest neighbor searching
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
A self-adjusting data structure for multidimensional point sets
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Output-sensitive well-separated pair decompositions for dynamic point sets
Proceedings of the 21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems
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In this paper, we introduce a simple, randomized dynamic data structure for storing multidimensional point sets, called a quadtreap. This data structure is a randomized, balanced variant of a quadtree data structure. In particular, it defines a hierarchical decomposition of space into cells, which are based on hyperrectangles of bounded aspect ratio, each of constant combinatorial complexity. It can be viewed as a multidimensional generalization of the treap data structure of Seidel and Aragon. When inserted, points are assigned random priorities, and the tree is restructured through rotations as if the points had been inserted in priority order. In any fixed dimension d, we show it is possible to store a set of n points in a quadtreap of space O(n). The height h of the tree is O(log n) with high probability. It supports point insertion in time O(h). It supports point deletion in worst-case time O(h2) and expected-case time O(h), averaged over the points of the tree. It can answer ε-approximate spherical range counting queries over groups and approximate nearest neighbor queries in time O(h + (1/ε)d-1).