Voronoi diagrams and arrangements
Discrete & Computational Geometry
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
IEEE Transactions on Pattern Analysis and Machine Intelligence
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
A fast planar partition algorithm, I
Journal of Symbolic Computation
An optimal algorithm for intersecting line segments in the plane
Journal of the ACM (JACM)
Reporting points in halfspaces
Computational Geometry: Theory and Applications
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
Size-estimation framework with applications to transitive closure and reachability
Journal of Computer and System Sciences
On range reporting, ray shooting and k-level construction
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Kinetic binary space partitions for intersecting segments and disjoint triangles
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Cylindrical static and kinetic binary space partitions
Computational Geometry: Theory and Applications
The discrepancy method: randomness and complexity
The discrepancy method: randomness and complexity
On approximating the depth and related problems
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
On approximate halfspace range counting and relative epsilon-approximations
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
On approximate range counting and depth
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Bottom-k sketches: better and more efficient estimation of aggregates
Proceedings of the 2007 ACM SIGMETRICS international conference on Measurement and modeling of computer systems
Summarizing data using bottom-k sketches
Proceedings of the twenty-sixth annual ACM symposium on Principles of distributed computing
Tighter estimation using bottom k sketches
Proceedings of the VLDB Endowment
A general approach for cache-oblivious range reporting and approximate range counting
Proceedings of the twenty-fifth annual symposium on Computational geometry
Leveraging discarded samples for tighter estimation of multiple-set aggregates
Proceedings of the eleventh international joint conference on Measurement and modeling of computer systems
Data Structures for Approximate Orthogonal Range Counting
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
A general approach for cache-oblivious range reporting and approximate range counting
Computational Geometry: Theory and Applications
Approximate Halfspace Range Counting
SIAM Journal on Computing
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We present new algorithms for approximate range counting, where, for a specified ε 0, we want to count the number of data points in a query range, up to relative error of ε. We first describe a general framework, adapted from Cohen [10], for this task, and then specialize it to two important instances of range counting: halfspaces in R3 and disks in the plane. The technique reduces the approximate range counting problem to that of finding the minimum rank of a data object in the range, with respect to a random permutation of the input.A major technical step in our analysis, which we believe to be of independent interest, is a bound of O(n log n) on the expected complexity of the overlay of all the Voronoi faces that are generated during a randomized incremental construction of the Voronoi diagram of n points in the plane. The same bound holds for the expected complexity of the overlay of all the faces of the minimization diagram of the lower envelope of n planes in R3, or for the expected complexity of the overlay of all the normal (or Gaussian) diagram faces of the convex hull of n points in R3, that are generated during a randomized incremental construction of the lower envelope or of the hull, respectively. All these bounds are tight in the worst case.The first bound leads to an algorithm that, for a query point x ∈ R2, efficiently retrieves the sequence of nearest neighbors of x in P, over the random insertion process. A query takes O(log n) expected time, and the expected storage size is O(n log n). Similarly, the other bounds lead to an algorithm that, for a query direction ω ∈ S2, efficiently retrieves the sequence of the convex hull vertices that are touched by the planes with outward direction ω that support the convex hull during the random insertion process. Again, a query takes O(log n) expected time, and the expected storage size is O(n log n). These algorithms are used as the main component in the approximate range counting technique that we present, for ranges that are halfspaces in R3 or disks in the plane.