In this article, we study a wide range of variants for computing the (discrete and continuous) Fréchet distance between uncertain curves. An uncertain curve is 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 Fréchet distance, which are the minimum and maximum Fréchet distance for any realisations of the curves.

We prove that both problems are NP-hard for the Fréchet distance in several uncertainty models, and that the upper bound problem remains hard for the discrete Fréchet distance. In contrast, the lower bound (discrete [5] and continuous) Fréchet distance can be computed in polynomial time in some models. Furthermore, we show that computing the expected (discrete and continuous) Fréchet distance is #P-hard in some models.

On the positive side, we present an FPTAS in constant dimension for the lower bound problem when Δ/δ is polynomially bounded, where δ is the Fréchet distance and Δ bounds the diameter of the regions. We also show a near-linear-time 3-approximation for the decision problem on roughly δ-separated convex regions. Finally, we study the setting with Sakoe–Chiba time bands, where we restrict the alignment between the curves, and give polynomial-time algorithms for the upper bound and expected discrete and continuous Fréchet distance for uncertainty modelled as point sets.

在本文中，我们研究了计算不确定曲线之间的（离散和连续）Fréchet 距离的各种变体。不确定曲线是一系列不确定区域，其中每个区域是一个圆盘、一条线段或一组点。曲线的一种实现是连接每个区域的一个点的折线。给定一条不确定曲线和第二条（确定或不确定）曲线，我们寻求计算下限和上限 Fréchet 距离，这是曲线的任何实现的最小和最大 Fréchet 距离。

我们证明，在几个不确定性模型中，这两个问题对于 Fréchet 距离都是 NP 困难的，并且上限问题对于离散 Fréchet 距离仍然是困难的。相反，在某些模型中，下界（离散 [5] 和连续）Fréchet 距离可以在多项式时间内计算。此外，我们表明，在某些模型中计算预期（离散和连续）Fréchet 距离是#P-hard 的。

从积极的一面来看，当 Δ/δ 是多项式有界时，我们为下界问题提出了恒定维度的 FPTAS，其中 δ 是 Fréchet 距离，Δ 限制了区域的直径。我们还展示了在大致 δ 分离的凸区域上决策问题的近线性时间 3 近似。最后，我们研究了 Sakoe-Chiba 时间带的设置，其中我们限制了曲线之间的对齐，并给出了上限的多项式时间算法以及建模为点集的不确定性的预期离散和连续 Fréchet 距离。