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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Proceedings ArticleDOI
Tight lower bounds for halfspace range searching
TL;DR: In this article, a lower bound of Ω(n 1 −1/(d+1)/m 1/(d + 1) was established for the half-space range search problem for general semigroups, where each point is associated with a weight from a semigroup.
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Random constructions and density results
András Gács,Tamás Szőnyi +1 more
TL;DR: This paper outlines a construction method which has been used for minimal blocking sets in PG(2, q) and maximal partial line spreads in PG (n, q), and lists some more elaborate random techniques used in finite geometry.
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RRR: Rank-Regret Representative
TL;DR: A geometric interpretation of items is used to bound their ranks on ranges of functions and to utilize combinatorial geometry notions for developing effective and efficient approximation algorithms for the problem of rank-regret.
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On Constants for Cuttings in the Plane
TL;DR: The expectation of the k th degree average of the number of lines intersecting a triangle is O(n/r) for any fixed k, which is the constant of proportionality in this result.
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Erdős–Hajnal Conjecture for Graphs with Bounded VC-Dimension
Jacob Fox,János Pach,Andrew Suk +2 more
TL;DR: This work shows that every n-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least e(logn)1-o(1), and extends the “ultra-strong regularity lemma” to k-uniform hypergraphs, and proves that the number of parts in the partition can be taken to be 1/ε O(d)O(d), improving the original bound.
References
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Book ChapterDOI
On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities
TL;DR: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady.
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