Computing the Fr\'echet Distance Between Uncertain Curves in One Dimension

TL;DR

This paper investigates the computational complexity of calculating the Fréchet distance between uncertain curves in one dimension, providing polynomial algorithms for some cases and proving NP-hardness for others, including variants involving weak and discrete weak Fréchet distances.

Contribution

It introduces the first general algorithmic framework for 1D uncertain curves, demonstrating polynomial-time solutions and NP-hardness results across different Fréchet distance variants.

Findings

Polynomial-time algorithm for 1D curves with interval uncertainty regions.

NP-hardness of optimal vertex placement in 1D for maximum Fréchet distance.

Polynomial-time solution for 1D weak Fréchet distance despite NP-hardness in higher dimensions.

Abstract

We consider the problem of computing the Fr\'echet distance between two curves for which the exact locations of the vertices are unknown. Each vertex may be placed in a given uncertainty region for that vertex, and the objective is to place vertices so as to minimise the Fr\'echet distance. This problem was recently shown to be NP-hard in 2D, and it is unclear how to compute an optimal vertex placement at all. We present the first general algorithmic framework for this problem. We prove that it results in a polynomial-time algorithm for curves in 1D with intervals as uncertainty regions. In contrast, we show that the problem is NP-hard in 1D in the case that vertices are placed to maximise the Fr\'echet distance. We also study the weak Fr\'echet distance between uncertain curves. While finding the optimal placement of vertices seems more difficult than the regular Fr\'echet distance -- and indeed we can easily prove that the problem is NP-hard in 2D -- the optimal placement of vertices in 1D can be computed in polynomial time. Finally, we investigate the discrete weak Fr\'echet distance, for which, somewhat surprisingly, the problem is NP-hard already in 1D.