Shortest polygonal chains covering each planar square grid

TL;DR

This paper proves the existence of shortest polygonal chains covering all points in a planar grid, introduces an algorithm for even-sized grids, and establishes a tight upper bound on the total travel distance for such covering trails.

Contribution

It provides constructive proofs for shortest covering paths and cycles in grid graphs, along with a general algorithm and a tight upper bound on travel distance.

Findings

Existence of shortest covering paths and cycles with equal link length.

A general algorithm for even-sized grid coverage cycles.

A tight upper bound on total travel distance for covering all grid points.

Abstract

Given any $n \in \mathbb{Z}^{+}$, we constructively prove the existence of covering paths and circuits in the plane which are characterized by the same link length of the minimum-link covering trails for the two-dimensional grid $G_n^2 := \{0,1, \ldots, n-1\} \times \{0, 1, \ldots, n-1\}$. Furthermore, we introduce a general algorithm that returns a covering cycle of analogous link length for any even value of $n$. Finally, we provide the tight upper bound $n^2 - 3 + 5 \cdot \sqrt{2}$ units for the minimum total distance travelled to visit all the nodes of $G_n^2$ with a minimum-link trail (i.e., a trail with $2 \cdot n - 2$ edges if $n$ is above two).