Euclidean Shortest Paths in Simple Cube Curves at a Glance

This paper reports about the development of two provably correct approximate algorithms which calculate the Euclidean shortest path (ESP) within a given cube-curve with arbitrary accuracy, defined by $ϵ>0$, and in time complexity $\kappa (ϵ)\cdot \mathcal{O}(n)$, where $\kappa (ϵ)$ is the length difference between the path used for initialization and the minimum-length path, divided by $ϵ$. A run-time diagram also illustrates this linear-time behavior of the implemented ESP algorithm.