Paths on the Doubly Covered Region of a Covering of the Plane by Unit Discs

TL;DR

This paper investigates the shortest path within the doubly covered region of a plane covered by unit discs, providing a new upper bound that improves understanding of geometric covering problems.

Contribution

The paper offers a new upper bound of 2.78 times the distance plus a constant for shortest paths in doubly covered regions, advancing prior conjectures.

Findings

Established a bound of 2.78d + O(1) for the shortest path length.

Improved upon the conjectured bound of √2 d + O(1).

Contributed to geometric covering and path planning theory.

Abstract

Given a covering of the plane by closed unit discs $\mathcal F$ and two points $A$ and $B$ in the region doubly covered by $\mathcal F$, what is the length of the shortest path connecting them that stays within the doubly covered region? This is a problem of G. Fejes-T\'oth and he conjectured that if the distance between $A$ and $B$ is $d$, then the length of this path is at most $\sqrt 2 d+O(1)$. In this paper we give a bound of $2.78 d+O(1)$.