Efficient Regular Data Structures and Algorithms for Location and Proximity Problems

Investigates 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, which is used in this paper as a basis for more complex data structures; we also provide multidimensional versions of the stratified tree of P. van Emde Boas (1997). We show that, under the assumption that the input points have limited precision (i. e. are drawn from an 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 finding the dynamic approximate nearest neighbor (under insertions and deletions) and the exact online 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.

Authors

• Arnon Amir
• Alon Efrat
• Piotr Indyk
• Hanan Samet

Keywords

• Data structures
• Shape
• Computer science
• Tree data structures
• Multidimensional systems
• Partitioning algorithms
• Nearest neighbor searches
• Educational institutions
• Contracts
• Data processing
• Data Structure
• Local Point
• Linear Time
• Input Point
• Quadtree
• Planar Shape
• Time And Space
• Universe
• Running Time
• Tree Nodes
• Matrix M
• Hash Function
• Point Distance
• Nearest Neighbor Search
• Leaf Number
• Ball Of Radius
• Binary Search
• Query Point
• Side Of The Square
• Bottom-up Fashion
• Path Tree
• Input Shape
• Convex Objective
• Constant Factor
• Uncompressed
• Binary Tree