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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Proceedings ArticleDOI
Semantic complexity of classes of relational queries and query independent data partitioning
TL;DR: A measure of the semantic complexity of classes of selection queries in relational databases is introduced and certain probabilistic bounds hitherto provable only for individual queries are extended to an entire class of queries.
Proceedings ArticleDOI
On Range Searching with Semialgebraic Sets II
TL;DR: A linear-size data structure for answering range queries on P with constant-complexity semi algebraic sets as ranges, in time close to O(n-1-1/d) , which essentially matches the performance of similar structures for simplex range searching, and significantly improves earlier solutions.
Posted Content
The Decision Tree Complexity for $k$-SUM is at most Nearly Quadratic
Esther Ezra,Micha Sharir +1 more
TL;DR: A further improvement to the complexity of a linear decision tree for k-SUM queries is presented, resulting in O(n^3 \log^3{n}) linear queries.
Journal ArticleDOI
Two Proofs for Shallow Packings
TL;DR: The bound on the packing number, originally shown by Haussler, is refined for shallow geometric set systems and the proof of Chazelle is extended, originally presented as a simplification forHaussler’s proof.
Proceedings ArticleDOI
Fast segment insertion and incremental construction of constrained delaunay triangulations
TL;DR: A randomized algorithm for inserting a segment into a CDT in expected time linear in the number of edges the segment crosses is given, and it is demonstrated with a performance comparison that it is faster than gift-wrapping for segments that cross many edges.
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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Algorithms in Combinatorial Geometry
TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
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On the density of families of sets
TL;DR: This paper will answer the question in the affirmative by determining the exact upper bound of T if T is a family of subsets of some infinite set S then either there exists to each number n a set A ⊂ S with |A| = n such that |T ∩ A| = 2n or there exists some number N such that •A| c for each A⩾ N and some constant c.
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Central Limit Theorems for Empirical Measures
TL;DR: In this article, the convergence of a stochastic process indexed by a Gaussian process to a certain Gaussian processes indexed by the supremum norm was studied in a Donsker class.
Journal ArticleDOI
The power of geometric duality
TL;DR: A new formulation of the notion of duality that allows the unified treatment of a number of geometric problems is used, to solve two long-standing problems of computational geometry and to obtain a quadratic algorithm for computing the minimum-area triangle with vertices chosen amongn points in the plane.