Minimum-link $C$-Oriented Paths Visiting a Sequence of Regions in the Plane

TL;DR

This paper introduces an algorithm to find the shortest $C$-oriented path visiting a sequence of disjoint segments in the plane, minimizing the number of directional changes, with potential applications in geometric routing.

Contribution

It presents the first polynomial-time algorithm for computing minimum-link $C$-oriented paths visiting a sequence of segments, advancing geometric path planning methods.

Findings

Algorithm runs in $O(|C|^2 n^2)$ time.

Effectively handles $C$-oriented segments and paths.

Provides groundwork for more complex polygonal path problems.

Abstract

Let $E=\{e_1,\ldots,e_n\}$ be a set of $C$-oriented disjoint segments in the plane, where $C$ is a given finite set of orientations that spans the plane, and let $s$ and $t$ be two points. %(We also require that for each orientation in $C$, its opposite orientation is also in $C$.) We seek a minimum-link $C$-oriented tour of $E$, that is, a polygonal path $\pi$ from $s$ to $t$ that visits the segments of $E$ in order, such that, the orientations of its edges are in $C$ and their number is minimum. We present an algorithm for computing such a tour in $O(|C|^2 \cdot n^2)$ time. This problem already captures most of the difficulties occurring in the study of the more general problem, in which $E$ is a set of not-necessarily-disjoint $C$-oriented polygons.