Computing partial sums in multidimensional arrays
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
Reporting points in halfspaces
Computational Geometry: Theory and Applications
ACM Computing Surveys (CSUR)
Efficiency of a Good But Not Linear Set Union Algorithm
Journal of the ACM (JACM)
Computational Geometry: Theory and Applications
Models and issues in data stream systems
Proceedings of the twenty-first ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
Space-time tradeoff for answering range queries (Extended Abstract)
STOC '82 Proceedings of the fourteenth annual ACM symposium on Theory of computing
Faster core-set constructions and data stream algorithms in fixed dimensions
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Space-time tradeoffs for approximate spherical range counting
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
On the importance of idempotence
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
The effect of corners on the complexity of approximate range searching
Proceedings of the twenty-second annual symposium on Computational geometry
Range Counting over Multidimensional Data Streams
Discrete & Computational Geometry
Approximate input sensitive algorithms for point pattern matching
Pattern Recognition
Approximate Halfspace Range Counting
SIAM Journal on Computing
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Range searching is a well known problem in the area of geometric data structures. We consider this problem in the context of approximation, where an approximation parameter ε 0 is provided. Most prior work on this problem has focused on the case of relative errors, where each range shape R is bounded, and points within distance ε ċ diam (R) of the range's boundary may or may not be included. We consider a different approximation model, called the absolute model, in which points within distance ε of the range's boundary may or may not be included, regardless of the diameter of the range. We consider range spaces consisting of halfspaces, Euclidean balls, simplices, axis-aligned rectangles, and general convex bodies. We consider a variety of problem formulations, including range searching under general commutative semigroups, idempotent semigroups, groups, and range emptiness. We show how idempotence can be used to improve not only approximate, but also exact halfspace range searching. Our data structures are much simpler than both their exact and relative model counterparts, and so are amenable to efficient implementation.