ź-nets and simplex range queries
David Haussler,Emo Welzl +1 more
TLDR
The concept of an ɛ-net of a set of points for an abstract set of ranges is introduced and sufficient conditions that a random sample is an Â-net with any desired probability are given.Abstract:
We demonstrate the existence of data structures for half-space and simplex range queries on finite point sets ind-dimensional space,dÂ?2, with linear storage andO(nÂ?) query time, $$\alpha = \frac{{d(d - 1)}}{{d(d - 1) + 1}} + \gamma for all \gamma > 0$$ .
These bounds are better than those previously published for alldÂ?2. Based on ideas due to Vapnik and Chervonenkis, we introduce the concept of an Â?-net of a set of points for an abstract set of ranges and give sufficient conditions that a random sample is an Â?-net with any desired probability. Using these results, we demonstrate how random samples can be used to build a partition-tree structure that achieves the above query time.read more
Citations
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Journal ArticleDOI
A Two-Dimensional Kinetic Triangulation with Near-Quadratic Topological Changes
TL;DR: This is the first known KDS for maintaining a triangulation that processes a near-quadratic number of topological events, and almost matches the $\Omega(n^2)$ lower bound.
Proceedings ArticleDOI
Almost tight upper bounds for lower envelopes in higher dimensions
TL;DR: The first nontrivial general upper bound for the combinatorial complexity of the lower envelope of n surfaces or surface patches in d-space is shown, and a randomized algorithm for computing the envelope in three dimensions is presented, with expected running time O(n/sup 2+/spl epsi//), and several applications of the new bounds are given.
Journal ArticleDOI
Improved combinatorial bounds and efficient techniques for certain motion planning problems with three degrees of freedom
TL;DR: The approach reduces each three-dimensional problem into a collection of problems involving two-dimensional arrangements, improving over the best previously known algorithms for these problems, whose time complexity is O(n3logn).
Journal ArticleDOI
A note about weak ε-nets for axis-parallel boxes in d-space
TL;DR: This analysis uses a non-trivial variant of the recent technique of Aronov et al. (2009) that yields (strong) @e-nets, whose size have the above asymptotic bound, for d=2,3.
Journal ArticleDOI
A Lower Bound for Weak ɛ-Nets in High Dimension
TL;DR: It is shown that there are point sets X ⊂ Rd for which every weak ε -net has at least const ⋅ d/2 points, which distinguishes the behavior of weak δ -nets with respect to convex sets from ε-nets withrespect to classes of shapes like balls or ellipsoids in Rd.
References
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