Halfplanar range search in linear space and O(n0.695) query time
Information Processing Letters
Partition trees for triangle counting and other range searching problems
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Good splitters for counting points in triangles
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
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
ACM Computing Surveys (CSUR)
Space-Time Tradeoffs for Emptiness Queries
SIAM Journal on Computing
A 3-space partition and its applications
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
On the importance of idempotence
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
The Effect of Corners on the Complexity of Approximate Range Searching
Discrete & Computational Geometry
Approximate range searching: The absolute model
Computational Geometry: Theory and Applications
Proceedings of the twenty-sixth annual symposium on Computational geometry
Indexing for vector projections
DASFAA'11 Proceedings of the 16th international conference on Database systems for advanced applications: Part II
| Hi-index | 0.00 |
We establish two new lower bounds for the halfspace range searching problem: Given a set of n points in ℜd, where each point is associated with a weight from a commutative semigroup, compute the semigroup sum of the weights of the points lying within any query halfspace. Letting $m$ denote the space requirements, we prove a lower bound for general semigroups of Ω(n1-1/(d+1)/m1/(d+1)) and for integral semigroups of Ω(n/m1/d). Our lower bounds are proved in the semigroup arithmetic model. Neglecting logarithmic factors, our result for integral semigroups matches the best known upper bound due to Matoušek. Our result for general semigroups improves upon the best known lower bound due to Brönnimann, Chazelle, and Pach. Moreover, Fonseca and Mount have recently shown that, given uniformly distributed points, halfspace range queries over idempotent semigroups can be answered in O(n1-1/(d+1)/m1/(d+1)) time in the semigroup arithmetic model. As our lower bounds are established for uniformly distributed point sets, it follows that they also resolve the computational complexity of halfspace range searching over idempotent semigroups in this important special case.