On the 3-distortion of a path

TL;DR

This paper establishes bounds on the 3-distortion of paths embedded in Euclidean spaces, showing it grows at least as fast as the square root of the path length in 2D and at most as fast as a power depending on the dimension.

Contribution

It provides the first asymptotic bounds on the 3-distortion of paths in various Euclidean spaces, revealing how distortion scales with path length and dimension.

Findings

3-distortion in R^2 is at least Omega(n^{1/2})

3-distortion in R^d is at most O(n^{1/d-1})

Bounds demonstrate the impact of dimension on path distortion

Abstract

We prove that, when a path of length n is embedded in R^2, the 3-distortion is an Omega(n^{1/2}), and that, when embedded in R^d, the 3-distortion is an O(n^{1/d-1}).