Complexity and Algorithms for the Discrete Fr\'echet Distance Upper Bound with Imprecise Input

TL;DR

This paper investigates the computational complexity of determining the upper bound of the discrete Fréchet distance with imprecise input, proving NP-hardness in general but providing polynomial-time algorithms when shortcuts are permitted.

Contribution

It establishes NP-hardness for the problem and introduces efficient algorithms for the case with shortcuts, solving an open problem from 2010.

Findings

NP-hardness of computing the upper bound for imprecise input

Polynomial-time algorithms for the case with shortcuts

Resolution of an open problem from 2010

Abstract

We study the problem of computing the upper bound of the discrete Fr\'{e}chet distance for imprecise input, and prove that the problem is NP-hard. This solves an open problem posed in 2010 by Ahn \emph{et al}. If shortcuts are allowed, we show that the upper bound of the discrete Fr\'{e}chet distance with shortcuts for imprecise input can be computed in polynomial time and we present several efficient algorithms.