Computing the Frechet distance between uncertain curves in one dimension

We consider the problem of computing the Frechet 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 Frechet 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 Frechet distance. We also study the weak Frechet distance between uncertain curves. While finding the optimal placement of vertices seems more difficult than the regular Frechet 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 Frechet distance, for which, somewhat surprisingly, the problem is NP-hard already in 1D. (C) 2022 The Authors. Published by Elsevier B.V.