Rock Climber Distance: Frogs versus Dogs

TL;DR

This paper introduces new distance measures for polygonal chains, the rock climber and k-station distances, which are computationally easier to approximate than the classical Fréchet distance, especially under move limitations.

Contribution

It defines novel distance measures based on alternate agent movement, analyzes their computational complexity, and provides approximation algorithms for limited moves.

Findings

Unlimited moves make the new measures equivalent to Fréchet or Hausdorff distances.

Limited moves make the distance computation NP-hard.

A 2-approximation algorithm for minimal k with threshold distance exists.

Abstract

The classical measure of similarity between two polygonal chains in Euclidean space is the Fr\'echet distance, which corresponds to the coordinated motion of two mobile agents along the chains while minimizing their maximum distance. As computing the Fr\'echet distance takes near-quadratic time under the Strong Exponential Time Hypothesis (SETH), we explore two new distance measures, called rock climber distance and $k$-station distance, in which the agents move alternately in their coordinated motion that traverses the polygonal chains. We show that the new variants are equivalent to the Fr\'echet or the Hausdorff distance if the number of moves is unlimited. When the number of moves is limited to a given parameter $k$, we show that it is NP-hard to determine the distance between two curves. We also describe a 2-approximation algorithm to find the minimum $k$ for which the distance drops below a given threshold.