Optimal halfspace range reporting in three dimensions

@inproceedings{Afshani2009OptimalHR,
  title={Optimal halfspace range reporting in three dimensions},
  author={P. Afshani and Timothy M. Chan},
  booktitle={SODA},
  year={2009}
}
We give the first optimal solution to a standard problem in computational geometry: three-dimensional halfspace range reporting. We show that n points in 3-d can be stored in a linear-space data structure so that all k points inside a query halfspace can be reported in O(log n + k) time. The data structure can be built in O(n log n) expected time. The previous methods with optimal query time required superlinear (O(n log log n)) space. We also mention consequences, for example, to higher… Expand
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References

SHOWING 1-10 OF 23 REFERENCES
Reporting Points in Halfspaces
  • J. Matousek
  • Computer Science, Mathematics
  • Comput. Geom.
  • 1992
TLDR
The halfspace itrange itreporting problem, given a finite set P of points in R d, can be solved substantially more efficiently that the more general simplex range searching problem. Expand
Random Sampling, Halfspace Range Reporting, and Construction of (<= k)-Levels in Three Dimensions
TLDR
It is shown how to answer halfspace range reporting queries in O(log n+k) expected time for an output size k and the first optimal randomized algorithm for the construction of the $(\le k)$-level in an arrangement of n planes in three dimensions is obtained. Expand
A dynamic data structure for 3-D convex hulls and 2-D nearest neighbor queries
TLDR
This is the first method that guarantees polylogarithmic update and query cost for arbitrary sequences of insertions and deletions, and improves the previous O(nϵ)-time method by Agarwal and Matoušek a decade ago. Expand
On two-dimensional indexability and optimal range search indexing
TLDR
The theory of indexability is applied to the problem of two-dimensional range searching and it is shown that the special case of 3-sided querying can be solved with constant redundancy and access overhead. Expand
On range reporting, ray shooting and k-level construction
  • E. Ramos
  • Computer Science, Mathematics
  • SCG '99
  • 1999
TLDR
This work considers some classical problems in computational geometry which have been " essentially " solved in the past and makes progress in reducing the already narrow gap with respect to the lower bounds (trivial or conjectured). Expand
On Dominance Reporting in 3D
TLDR
Using 3D results as base cases, the 3D dominance reporting problem in different models of computations is studied and optimal results in the pointer machine and the external memory models and a near optimal result in the RAM model are offered. Expand
Solving query-retrieval problems by compacting Voronoi diagrams
TLDR
The fundamental idea is that k'h-order Voronoi diagrams can be compacted from k°(1)n space to O(n) space and still retain all the information that is essential for solving query problems and may be useful in improving the memory space requirement or the query-time bound for other query-retrieval problems. Expand
The power of geometric duality
TLDR
A new formulation of the notion of duality that allows the unified treatment of a number of geometric problems is used, to solve two long-standing problems of computational geometry and to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane. Expand
Efficient partition trees
We prove a theorem on partitioning point sets inEd (d fixed) and give an efficient construction of partition trees based on it. This yields a simplex range searching structure with linear space,O(nExpand
Geometric Retrieval Problems
...
1
2
3
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