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Deterministic constructions of high-dimensional sets with small dispersion

Mario Ullrich, Jan Vybíral

TL;DR

This construction is based on the apparently new insight that low-dispersion point sets can be deduced from solutions of certain k-restriction problems, which are well-known in coding theory.

Abstract

The dispersion of a point set $P\subset[0,1]^d$ is the volume of the largest box with sides parallel to the coordinate axes, which does not intersect $P$. Here, we show a construction of low-dispersion point sets, which can be deduced from solutions of certain $k$-restriction problems, which are well-known in coding theory. It was observed only recently that, for any $\varepsilon>0$, certain randomized constructions provide point sets with dispersion smaller than $\varepsilon$ and number of elements growing only logarithmically in $d$. Based on deep results from coding theory, we present explicit, deterministic algorithms to construct such point sets in time that is only polynomial in $d$. Note that, however, the running-time will be super-exponential in $\varepsilon^{-1}$.

Deterministic constructions of high-dimensional sets with small dispersion

TL;DR

This construction is based on the apparently new insight that low-dispersion point sets can be deduced from solutions of certain k-restriction problems, which are well-known in coding theory.

Abstract

The dispersion of a point set is the volume of the largest box with sides parallel to the coordinate axes, which does not intersect . Here, we show a construction of low-dispersion point sets, which can be deduced from solutions of certain -restriction problems, which are well-known in coding theory. It was observed only recently that, for any , certain randomized constructions provide point sets with dispersion smaller than and number of elements growing only logarithmically in . Based on deep results from coding theory, we present explicit, deterministic algorithms to construct such point sets in time that is only polynomial in . Note that, however, the running-time will be super-exponential in .

Paper Structure

This paper contains 14 sections, 10 theorems, 43 equations.

Table of Contents

  1. Introduction and Results
  2. Basics from Coding Theory
  3. Universal sets
  4. $k$-restriction problems
  5. Splitters
  6. Derandomization of Sosnovec's proof
  7. Sosnovec's proof - the setting
  8. Sosnovec's proof - randomized construction
  9. Derandomization using universal sets
  10. Improving the dependence in $\varepsilon^{-1}$
  11. Enhanced analysis of the random construction
  12. Connection to $k$-restriction problems
  13. A first attempt for derandomization
  14. Derandomization using splitters

Key Result

Theorem 2.2

There is a deterministic construction of an $(n,k)$-universal set of size $2^{k+O(\log^2(k))} \log(n)$, which can be listed in linear time of the length of the output.

Theorems & Definitions (21)

  • Remark 1.1
  • Definition 2.1: $(n,k)$-universal sets
  • Theorem 2.2: NSS
  • Definition 2.3: $(n,k,b)$-universal sets
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Definition 2.6: $k$-restriction problems
  • Theorem 2.7: NSS
  • ...and 11 more