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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Segment intersection searching problems in general settings
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On the Union of Cylinders in Three Dimensions
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Small Weak Epsilon-Nets in Three Dimensions
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Weak ε-nets have basis of size o(1/ε log (1/ε)) in any dimension
Saurabh Ray,Nabil H. Mustafa +1 more
TL;DR: It is shown that weak ε-nets in Rd can be computed from a subset of P of size O(1/ε log(1-ε) with only the constant of proportionality depending on the dimension, unlike all previous work where the size of the subset had the dimension in the exponent of 1/ε.
References
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