Computational geometry: an introduction
Computational geometry: an introduction
Halfplanar range search in linear space and O(n0.695) query time
Information Processing Letters
A general approach to d-dimensional geometric queries
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
Partitioning space for range queries
SIAM Journal on Computing
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Implicitly representing arrangements of lines or segments
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Quasi-optimal range searching in spaces of finite VC-dimension
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Ray shooting and other applications of spanning trees with low stabbing number
SIAM Journal on Computing
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
Applications of a new space-partitioning technique
Discrete & Computational Geometry
Reporting points in halfspaces
Computational Geometry: Theory and Applications
Discrete & Computational Geometry - Special issue on ACM symposium on computational geometry, North Conway
Cutting hyperplanes for divide-and-conquer
Discrete & Computational Geometry
ACM Computing Surveys (CSUR)
Simplex range reporting on a pointer machine
Computational Geometry: Theory and Applications
On range reporting, ray shooting and k-level construction
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Vertical Decomposition of Shallow Levels in 3-Dimensional Arrangements and Its Applications
SIAM Journal on Computing
Efficient searching with linear constraints
Journal of Computer and System Sciences - Special issue on the seventeenth ACM SIGACT-SIGMOD-SIGART symposium on principles of database systems
Lectures on Discrete Geometry
On Spanning Trees with Low Crossing Numbers
Data Structures and Efficient Algorithms, Final Report on the DFG Special Joint Initiative
A 3-space partition and its applications
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Optimal halfspace range reporting in three dimensions
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Tight lower bounds for halfspace range searching
Proceedings of the twenty-sixth annual symposium on Computational geometry
Approximate polytope membership queries
Proceedings of the forty-third annual ACM symposium on Theory of computing
Indexing for vector projections
DASFAA'11 Proceedings of the 16th international conference on Database systems for advanced applications: Part II
Dynamic Connectivity: Connecting to Networks and Geometry
SIAM Journal on Computing
Polytope approximation and the Mahler volume
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
Optimal private halfspace counting via discrepancy
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Improved pointer machine and I/O lower bounds for simplex range reporting and related problems
Proceedings of the twenty-eighth annual symposium on Computational geometry
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
A simple framework for the generalized nearest neighbor problem
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Covering a bichromatic point set with two disjoint monochromatic disks
Computational Geometry: Theory and Applications
Computational Geometry: Theory and Applications
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We revisit one of the most fundamental classes of data structure problems in computational geometry: range searching. Back in SoCG'92, Matousek gave a partition tree method for d-dimensional simplex range searching achieving O(n) space and O(n1-1/d) query time. Although this method is generally believed to be optimal, it is complicated and requires O(n1+ε) preprocessing time for any fixed ε0. An earlier method by Matousek (SoCG'91) requires O(n log n) preprocessing time but O(n1-1/d logO(1)n) query time. We give a new method that achieves simultaneously O(n log n) preprocessing time, O(n) space, and O(n1-1/d) query time with high probability. Our method has several advantages: It is conceptually simpler than Matousek's SoCG'92 method. Our partition trees satisfy many ideal properties (e.g., constant degree, optimal crossing number at almost all layers, and disjointness of the children's cells at each node). It leads to more efficient multilevel partition trees, which are important in many data structural applications (each level adds at most one logarithmic factor to the space and query bounds, better than in all previous methods). A similar improvement applies to a shallow version of partition trees, yielding O(n log n) time, O(n) space, and O(n1-1=⌊d=2⌋) query time for halfspace range emptiness in even dimensions d ≥ 4. Numerous consequences follow (e.g., improved results for computing spanning trees with low crossing number, ray shooting among line segments, intersection searching, exact nearest neighbor search, linear programming queries, finding extreme points, ... ).