A formula for computing the number of quadtree node fragments created by a shift
Pattern Recognition Letters
The design and analysis of spatial data structures
The design and analysis of spatial data structures
Analytical results on the quadtree decomposition of arbitrary rectangles
Pattern Recognition Letters
Multidimensional access methods
ACM Computing Surveys (CSUR)
An effective way to represent quadtrees
Communications of the ACM
Overlapping linear quadtrees and spatio-temporal query processing
The Computer Journal
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SIGMOD '01 Proceedings of the 2001 ACM SIGMOD international conference on Management of data
Introductory Combinatorics
Computer Vision
QuadTIN: quadtree based triangulated irregular networks
Proceedings of the conference on Visualization '02
R-trees: a dynamic index structure for spatial searching
SIGMOD '84 Proceedings of the 1984 ACM SIGMOD international conference on Management of data
An introduction to spatial database systems
The VLDB Journal — The International Journal on Very Large Data Bases - Spatial Database Systems
Analysis of the n-Dimensional Quadtree Decomposition for Arbitrary Hyperrectangles
IEEE Transactions on Knowledge and Data Engineering
PROBE Spatial Data Modeling and Query Processing in an Image Database Application
IEEE Transactions on Software Engineering
Efficient Evaluation of Continuous Range Queries on Moving Objects
DEXA '02 Proceedings of the 13th International Conference on Database and Expert Systems Applications
IEEE Transactions on Knowledge and Data Engineering
Indexing Continual Range Queries for Location-Aware Mobile Services
EEE '04 Proceedings of the 2004 IEEE International Conference on e-Technology, e-Commerce and e-Service (EEE'04)
Interval query indexing for efficient stream processing
Proceedings of the thirteenth ACM international conference on Information and knowledge management
Mathematics for Algorithm and Systems Analysis
Mathematics for Algorithm and Systems Analysis
A novel gray image representation using overlapping rectangular NAM and extended shading approach
Journal of Visual Communication and Image Representation
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In this paper, we solve the classic problem of computing the average number of decomposed quadtree blocks (quadrants, nodes, or pieces) in quadtree decomposition for an arbitrary hyperrectangle aligned with the axes. We derive a closed-form formula for general cases. The previously known state-of-the-art solution provided a closed-form solution for special cases and utilized these formulas to derive linearly interpolated formulas for general cases individually. However, there is no exact and uniform closed-form formula that fits all cases. Contrary to the top-down counting approach used by most prior solutions, we employ a bottom-up enumeration approach to transform the problem into one that involves the Cartesian product of d multisets of successive 2's powers. Classic combinatorial enumeration techniques are applied to obtain an exact and uniform closed-form formula. The result is of theoretical interest since it is the first exact closed-form formula for arbitrary cases. Practically, it is nice to have a uniform formula for estimating the average number since a simple program can be conveniently constructed taking side lengths as inputs.