Fast almost-linear-sized nets for boxes in the plane

Let B be any set of n axis-aligned boxes in R, d ≥ 1. For any point p, we define the subset Bp of B as Bp = {B ∈ B : p ∈ B}. A box B in Bp is said to be stabbed by p. A subset N ⊆ B is a (1/c)-net for B if Np 6= ∅ for any p ∈ R such that |Bp| ≤ n/c. The number of distinct subsets Bp is O((2n)), so the set system described above has so-called finite VCdimension d. This ensures that there always exists (1/c)-nets of size O(dc log(dc)), and that they can be found in time Od(n)c, using quite general machinery (see for example the books by Matoušek [3] or by Pach and Agarwal [7]). For some set systems, such as halfplanes in R and translates of a simple closed polygon, it was shown that there exist (1/c)-nets of size O(c) [4]. This was extended to halfspaces in R and pseudo-disks in R [2]. In this paper, we investigate a fast, O(n log c)-time construction of (1/c)-nets of size O(c) for any value 1 < c ≤ n and d = 2. Until right before JCDCG, I thought I could prove the following (which unfortunately remains a conjecture):