Functional approach to data structures and its use in multidimensional searching
SIAM Journal on Computing
The input/output complexity of sorting and related problems
Communications of the ACM
Reporting points in halfspaces
Computational Geometry: Theory and Applications
Algorithms for three-dimensional dominance searching in linear space
Information Processing Letters
On two-dimensional indexability and optimal range search indexing
PODS '99 Proceedings of the eighteenth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
On range reporting, ray shooting and k-level construction
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Efficient splitting and merging algorithms for order decomposable problems
Information and Computation
Efficient cross-trees for external memory
External memory algorithms
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
Cache oblivious search trees via binary trees of small height
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Introduction to Algorithms
The K-D-B-tree: a search structure for large multidimensional dynamic indexes
SIGMOD '81 Proceedings of the 1981 ACM SIGMOD international conference on Management of data
Exponential Structures for Efficient Cache-Oblivious Algorithms
ICALP '02 Proceedings of the 29th International Colloquium on Automata, Languages and Programming
Cache-oblivious data structures for orthogonal range searching
Proceedings of the nineteenth annual symposium on Computational geometry
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
The Priority R-tree: a practically efficient and worst-case optimal R-tree
SIGMOD '04 Proceedings of the 2004 ACM SIGMOD international conference on Management of data
Cache-oblivious planar orthogonal range searching and counting
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Low-Dimensional Linear Programming with Violations
SIAM Journal on Computing
On approximating the depth and related problems
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
SIAM Journal on Computing
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Simple and semi-dynamic structures for cache-oblivious planar orthogonal range searching
Proceedings of the twenty-second annual symposium on Computational geometry
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Optimal halfspace range reporting in three dimensions
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Cache-oblivious range reporting with optimal queries requires superlinear space
Proceedings of the twenty-fifth annual symposium on Computational geometry
On Approximate Range Counting and Depth
Discrete & Computational Geometry - 23rd Annual Symposium on Computational Geometry
Range Minima Queries with Respect to a Random Permutation, and Approximate Range Counting
Discrete & Computational Geometry
Improved space bounds for cache-oblivious range reporting
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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We present cache-oblivious solutions to two important variants of range searching: range reporting and approximate range counting. Our main contribution is a general approach for constructing cache-oblivious data structures that provide relative (1+@e)-approximations for a general class of range counting queries. This class includes three-sided range counting in the plane, 3-d dominance counting, and 3-d halfspace range counting. The constructed data structures use linear space and answer queries in the optimal query bound of O(log"B(N/K)) block transfers in the worst case, where K is the number of points in the query range. As a corollary, we also obtain the first approximate 3-d halfspace range counting and 3-d dominance counting data structures with a worst-case query time of O(log(N/K)) in internal memory. An easy but important consequence of our main result is the existence of O(NlogN)-space cache-oblivious data structures with an optimal query bound of O(log"BN+K/B) block transfers for the reporting versions of the above problems. Using standard reductions, these data structures allow us to obtain the first cache-oblivious data structures that use almost linear space and achieve the optimal query bound for circular range reporting and K-nearest neighbour searching in the plane, as well as for orthogonal range reporting in three dimensions.