Data structures and algorithms 3: multi-dimensional searching and computational geometry
Data structures and algorithms 3: multi-dimensional searching and computational geometry
Halfplanar range search in linear space and O(n0.695) query time
Information Processing Letters
Information and Control
A general approach to d-dimensional geometric queries
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
Lower bounds for orthogonal range searching: I. The reporting case
Journal of the ACM (JACM)
Lower bounds for orthogonal range searching: part II. The arithmetic model
Journal of the ACM (JACM)
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
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
Quasi-optimal upper bounds for simplex range searching and new zone theorems
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Reporting points in halfspaces
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
A Lower Bound on the Complexity of Orthogonal Range Queries
Journal of the ACM (JACM)
Lower Bounds on the Complexity of Simplex Range Reporting on a Pointer Machine
ICALP '92 Proceedings of the 19th International Colloquium on Automata, Languages and Programming
A 3-space partition and its applications
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Ray shooting in convex polytopes
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
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We investigate the complexity of halfspace range searching: Given n points in d-space, build a data structure that allows us to determine efficiently how many points lie in a query halfspace. We establish a tradeoff between the storage m and the worst-case query time t in the Fredman/Yao arithmetic model of computation. We show that t must be at least on the order of (n/log n)1-((d-1)/(d(d+1))m1/d. To our knowledge, this is the first nontrivial lower bound for halfspace range searching. Although the bound is unlikely to be optimal, it falls reasonably close to the recent O(n(log m/n)d+1/m1/d) upper bound established by Matousˇek. We also show that it is possible to devise a sequence of n inserts and halfspace range queries that require a total time of n2-&thgr;(1/d). Our results imply nontrivial lower bounds for spherical range searching in any fixed dimension. For example they show that, with linear storage, circular range queries in the plane require &OHgr;(n1/3) time (modulo a logarithmic factor).