Computing the Fréchet distance between uncertain curves in one dimension.

We consider the problem of computing the Fréchet 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 Fréchet 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 Fréchet distance. We also study the weak Fréchet distance between uncertain curves. While finding the optimal placement of vertices seems more difficult than the regular Fréchet 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 Fréchet distance, for which, somewhat surprisingly, the problem is NP-hard already in 1D

Fr\'echet Distance for Uncertain Curves

In this paper we study a wide range of variants for computing the (discrete and continuous) Fr\'echet distance between uncertain curves. We define an uncertain curve as a sequence of uncertainty regions, where each region is a disk, a line segment, or a set of points. A realisation of a curve is a polyline connecting one point from each region. Given an uncertain curve and a second (certain or uncertain) curve, we seek to compute the lower and upper bound Fr\'echet distance, which are the minimum and maximum Fr\'echet distance for any realisations of the curves. We prove that both the upper and lower bound problems are NP-hard for the continuous Fr\'echet distance in several uncertainty models, and that the upper bound problem remains hard for the discrete Fr\'echet distance. In contrast, the lower bound (discrete and continuous) Fr\'echet distance can be computed in polynomial time. Furthermore, we show that computing the expected discrete Fr\'echet distance is #P-hard when the uncertainty regions are modelled as point sets or line segments. The construction also extends to show #P-hardness for computing the continuous Fr\'echet distance when regions are modelled as point sets. On the positive side, we argue that in any constant dimension there is a FPTAS for the lower bound problem when $\Delta / \delta$ is polynomially bounded, where $\delta$ is the Fr\'echet distance and $\Delta$ bounds the diameter of the regions. We then argue there is a near-linear-time 3-approximation for the decision problem when the regions are convex and roughly $\delta$-separated. Finally, we also study the setting with Sakoe--Chiba time bands, where we restrict the alignment between the two curves, and give polynomial-time algorithms for upper bound and expected discrete and continuous Fr\'echet distance for uncertainty regions modelled as point sets.Comment: 48 pages, 11 figures. This is the full version of the paper to be published in ICALP 202