Towards Faster Linear-Sized Nets for Axis-Aligned Boxes in the Plane

Abstract

Let ${\mathcal B}$ be any set of n axis-aligned boxes in ${\mathbb R}^{d}$, d≥ 1. We call a subset ${\mathcal N} \subseteq {\mathcal B}$ a (1/c )-net for ${\mathcal B}$ if any p ∈ ${\mathbb R}^{d}$ contained in more than n/c boxes of ${\mathcal B}$ must be contained in a box of ${\mathcal N}$, or equivalently if a point not contained in any box in ${\mathcal N}$ can only stab at most n/c boxes of ${\mathcal B}$. General VC-dimension theory guarantees the existence of (1/c)-nets of size O(clog c) for any fixed d, the constant in the big-Oh depending on d, and Matoušek [8, 9] showed how to compute such a net in time O(nc ^O(1)), or even O(n log c + c ^O(1)) which is O(n log c) if c is small enough. In this paper, we conjecture that axis-aligned boxes in ${\mathbb R}^{2}$ admit (1/c)-nets of size O(c), and that we can even compute such a net in time O(n log c), for any c between 1 and n. We show this to be true for intervals on the real line, and for various special cases (quadrants and skylines, which are unbounded in two and one directions respectively). In a follow-up version, we also show this to be true with various fatness We also investigate generalizations to higher dimensions.