Dense point sets have sparse Delaunay triangulations: or "…but not too nasty"
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
On counting point-hyperplane incidences
Computational Geometry: Theory and Applications - Special issue: The European workshop on computational geometry -- CG01
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
Proceedings of the twenty-second annual symposium on Computational geometry
Extremal point queries with lines and line segments and related problems
Computational Geometry: Theory and Applications
Tight lower bounds for halfspace range searching
Proceedings of the twenty-sixth 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 |
We develop the first nontrivial lower bounds on the complexity of online hyperplane and halfspace emptiness queries. Our lower bounds apply to a general class of geometric range query data structures called partition graphs. Informally, a partition graph is a directed acyclic graph that describes a recursive decomposition of space. We show that any partition graph that supports hyperplane emptiness queries implicitly defines a halfspace range query data structure in the Fredman/Yao semigroup arithmetic model, with the same asymptotic space and time bounds. Thus, results of Brönnimann, Chazelle, and Pach imply that any partition graph of size s that supports hyperplane emptiness queries in time t satisfies the inequality $st^d = \Omega((n/\log n)^{d - (d-1)/(d+1)})$. Using different techniques, we improve previous lower bounds for Hopcroft's problem---Given a set of points and hyperplanes, does any hyperplane contain a point?---in dimensions four and higher. Using this offline result, we show that for online hyperplane emptiness queries, $\Omega(n^d/{\mbox{ polylog }} n)$ space is required to achieve polylogarithmic query time, and $\Omega(n^{(d-1)/d}/{\mbox{ polylog }} n)$ query time is required if only O(n polylog n) space is available. These two lower bounds are optimal up to polylogarithmic factors. For two-dimensional queries, we obtain an optimal continuous tradeoff $st^2=\Omega(n^2)$ between these two extremes. Finally, using a lifting argument, we show that the same lower bounds hold for both offline and online halfspace emptiness queries in ${\mathbb{R}}^{d(d+3)/2}$.