Ranking Unit Squares with Few Visibilities

TL;DR

This paper establishes an optimal method for ordering unit squares in the plane to minimize vertical visibilities, proving a tight bound of 3n-7 pairwise visibilities for any set of n squares.

Contribution

It introduces a simple lexicographic ranking method that guarantees the minimal possible maximum number of visibilities among squares.

Findings

Lexicographic order yields at most 3n-7 visibilities.

The bound of 3n-7 is tight and cannot be improved.

The method applies for all sets of n squares with n ≥ 4.

Abstract

Given a set of n unit squares in the plane, the goal is to rank them in space in such a way that only few squares see each other vertically. We prove that ranking the squares according to the lexicographic order of their centers results in at most 3n-7 pairwise visibilities for n at least 4. We also show that this bound is best possible, by exhibiting a set of n squares with at least 3n-7 pairwise visibilities under any ranking.