Euclidean Shortest Paths in Simple Cube Curves at a Glance

TL;DR

This paper introduces two provably correct approximate algorithms for computing Euclidean shortest paths in cube-curves, achieving arbitrary accuracy with linear time complexity relative to the input size.

Contribution

The paper presents novel algorithms that efficiently approximate Euclidean shortest paths in cube-curves with guaranteed correctness and controllable accuracy.

Findings

Algorithms are provably correct and approximate Euclidean shortest paths.

Time complexity is linear in the number of cubes, scaled by a factor depending on accuracy.

Experimental results confirm linear-time behavior of the algorithms.

Abstract

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 $\epsilon >0$, and in time complexity $\kappa(\epsilon) \cdot {\cal O}(n)$, where $\kappa(\epsilon)$ is the length difference between the path used for initialization and the minimum-length path, divided by $\epsilon$. A run-time diagram also illustrates this linear-time behavior of the implemented ESP algorithm.