On Flipping the Fr\'{e}chet distance

TL;DR

This paper introduces and studies a novel 'flipped' Fréchet distance that measures how far apart two curves or domains can be kept while traversing them, offering a new perspective on spatial separation and social distancing.

Contribution

It defines the flipped Fréchet measure, analyzes its computational complexity, and provides algorithms for polygonal curves, polygons, and graphs, expanding the understanding of curve and domain distance measures.

Findings

Conditional lower bounds and matching algorithms for polygonal curves.

Linear time algorithms for certain polygon and graph variants.

Connections established with existing computational geometry measures.

Abstract

The classical and extensively-studied Fr\'echet distance between two curves is defined as an inf max, where the infimum is over all traversals of the curves, and the maximum is over all concurrent positions of the two agents. In this article we investigate a "flipped" Fr\'echet measure defined by a sup min -- the supremum is over all traversals of the curves, and the minimum is over all concurrent positions of the two agents. This measure produces a notion of "social distance" between two curves (or general domains), where agents traverse curves while trying to stay as far apart as possible. We first study the flipped Fr\'echet measure between two polygonal curves in one and two dimensions, providing conditional lower bounds and matching algorithms. We then consider this measure on polygons, where it denotes the minimum distance that two agents can maintain while restricted to travel in or on the boundary of the same polygon. We investigate several variants of the problem in this setting, for some of which we provide linear time algorithms. Finally, we consider this measure on graphs. We draw connections between our proposed flipped Fr\'echet measure and existing related work in computational geometry, hoping that our new measure may spawn investigations akin to those performed for the Fr\'echet distance, and into further interesting problems that arise.