Weak ∈-nets and interval chains

Noga Alon, Haim Kaplan, Gabriel Nivasch, Micha Sharir, Shakhar Smorodinsky

Research output: Chapter in Book/Report/Conference proceedingConference contribution

7 Scopus citations

Abstract

We construct weak ∈-nets of almost linear size for certain types of point sets. Specifically, for planar point sets in convex position we construct weak 1/r-nets of size 0(r α(r)), where α(r) denotes the inverse Ackermann function. For point sets along the moment curve in R d we construct weak 1/r-nets of size r. 2 poly(α(r)), where the degree of the polynomial in the exponent depends (quadratically) on d. Our constructions result from a reduction to a new problem, which we call stabbing interval chains with j-tuples. Given the range of integers N = [l,n], an interval chain of length k is a sequence of k consecutive, disjoint, nonempty intervals contained in N. A jtuple p̄ = (p i,..., p j) is said to stab an interval chain C = I 1...I k if each p i falls on a different interval of C. The problem is to construct a small-size family double-struck Z of j-tuples that stabs all k-interval chains in N. Let z (j) k (n) denote the minimum size of such a family double-struck Z. We derive almost-tight upper and lower bounds for z (j) k (n) for every fixed j; our bounds involve functions α m(n) of the inverse Ackermann hierarchy. Specifically, we show that for j - 3 we have z (3) k (n) = Θ(nα[k/2] (n)) for all k ≥ 6. For each j ≥ 4 we construct a pair of functions P 1 j(m), Q 1 j(m), almost equal asymptotically, such that z (j) P′j(m)(n) = 0(nα m(n)) and z (j) Q'j(m)(n) = Ω(nα m(n)).

Original languageEnglish (US)
Title of host publicationProceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms
Pages1194-1203
Number of pages10
StatePublished - Dec 1 2008
Externally publishedYes
Event19th Annual ACM-SIAM Symposium on Discrete Algorithms - San Francisco, CA, United States
Duration: Jan 20 2008Jan 22 2008

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Other

Other19th Annual ACM-SIAM Symposium on Discrete Algorithms
CountryUnited States
CitySan Francisco, CA
Period1/20/081/22/08

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

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  • Cite this

    Alon, N., Kaplan, H., Nivasch, G., Sharir, M., & Smorodinsky, S. (2008). Weak ∈-nets and interval chains. In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 1194-1203). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms).