ź-nets and simplex range queries
David Haussler,Emo Welzl +1 more
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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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Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D
Pankaj K. Agarwal,Rolf Klein,Christian Knauer,Stefan Langerman,Pat Morin,Micha Sharir,Michael Soss +6 more
TL;DR: Subquadratic algorithms for computing the detour and spanning ratio for paths, cycles, and trees embedded in $\mathbb{E}^{3}$ are developed, and it is shown that computing thedetour in $E3$ is at least as hard as Hopcroft’s problem.
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The method of hypergraph containers
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Quasi-optimal upper bounds for simplex range searching and new zone theorems
TL;DR: In this article, the authors present quasi-optimal upper bounds for simplex range searching, where the problem is to preprocess a setP ofn points in ℜd so that, given any query simplexq, the points inP ∩q can be counted or reported efficiently.
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A fast Las Vegas algorithm for triangulating a simple polygon
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Computing Many Faces in Arrangements of Lines and Segments
TL;DR: This work presents randomized algorithms for computing many faces in an arrangement of lines or of segments in the plane, which are considerably simpler and slightly faster than the previously known ones.
References
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