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On Flipping the Fréchet Distance

Authors Omrit Filtser , Mayank Goswami, Joseph S. B. Mitchell, Valentin Polishchuk

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Author Details

Omrit Filtser
  • The Open University of Israel, Ra'anana, Israel
Mayank Goswami
  • Queens College CUNY, Flushing, NY, USA
Joseph S. B. Mitchell
  • Stony Brook University, NY, USA
Valentin Polishchuk
  • Linköping University, Sweden

Funding

  • Filtser, Omrit: This work is supported by the Eric and Wendy Schmidt Fund for Strategic Innovation, by the Council for Higher Education of Israel, and by Ben-Gurion University of the Negev.
  • Goswami, Mayank: This work is supported by US National Science Foundation (NSF) awards CRII-1755791 and CCF-1910873.
  • Mitchell, Joseph S. B.: This work is partially supported by the National Science Foundation (CCF-2007275), the US-Israel Binational Science Foundation (BSF project 2016116), Sandia National Labs, and DARPA (Lagrange).
  • Polishchuk, Valentin: This work is supported by grants from the Swedish Research Council and the Swedish Transport Administration

Acknowledgements

We thank the anonymous reviewers for their many helpful comments. We thank the many participants of the Stony Brook CG Group, where discussions about geometric social distancing problems originated in Spring 2020, as the COVID-19 crisis expanded worldwide. We would also like to thank Gaurish Telang for helping to solve a system of non-linear equations.

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Omrit Filtser, Mayank Goswami, Joseph S. B. Mitchell, and Valentin Polishchuk. On Flipping the Fréchet Distance. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 51:1-51:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ITCS.2023.51

Abstract

The classical and extensively-studied Fréchet 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" Fréchet 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 Fréchet 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 Fréchet measure and existing related work in computational geometry, hoping that our new measure may spawn investigations akin to those performed for the Fréchet distance, and into further interesting problems that arise.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
Keywords
  • curves
  • polygons
  • distancing measure

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