Filtering search: a new approach to query answering
SIAM Journal on Computing
Lower bounds for orthogonal range searching: I. The reporting case
Journal of the ACM (JACM)
Reporting points in halfspaces
Computational Geometry: Theory and Applications
Lower bounds for off-line range searching
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Algorithms for three-dimensional dominance searching in linear space
Information Processing Letters
New data structures for orthogonal range searching
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Orthogonal Range Reporting in Three and Higher Dimensions
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Proceedings of the twenty-sixth annual symposium on Computational geometry
Orthogonal range searching on the RAM, revisited
Proceedings of the twenty-seventh annual symposium on Computational geometry
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
Improved range searching lower bounds
Proceedings of the twenty-eighth annual symposium on Computational geometry
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
Hi-index | 0.00 |
In this paper, we consider two fundamental problems in the pointer machine model of computation, namely orthogonal range reporting and rectangle stabbing. Orthogonal range reporting is the problem of storing a set of n points in d-dimensional space in a data structure, such that the t points in an axis-aligned query rectangle can be reported efficiently. Rectangle stabbing is the "dual" problem where a set of n axis-aligned rectangles should be stored in a data structure, such that the t rectangles that contain a query point can be reported efficiently. Very recently an optimal O(log n+t) query time pointer machine data structure was developed for the three-dimensional version of the orthogonal range reporting problem. However, in four dimensions the best known query bound of O(log2n / log log n + t) has not been improved for decades. We describe an orthogonal range reporting data structure that is the first structure to achieve significantly less than O(log2n+t) query time in four dimensions. More precisely, we develop a structure that uses O(n (log n /log log n)d) space and can answer d-dimensional orthogonal range reporting queries (for d ≥ 4) in O( log n (log n /log log n)d-4+1/(d-2) + t) time. Ignoring log log n factors, this speeds up the best previous query time by a log1-1/(d-2)n factor. For the rectangle stabbing problem, we show that any data structure that uses nh space must use Ω(log n (log n / log h)d-2 + t) time to answer a query. This improves the previous results by a log h factor, and is the first lower bound that is optimal for a large range of h, namely for h ≥ logd-2+ε n where ε0 is an arbitrarily small constant. By a simple geometric transformation, our result also implies an improved query lower bound for orthogonal range reporting.