不确定曲线的 Fréchet 距离
Fréchet Distance for Uncertain Curves
AI总结：
本文系统分析不确定曲线间离散与连续 Fréchet 距离的上下界及期望值，证明多种 NP/#P 困难性，并给出 FPTAS、近似算法与 Sakoe–Chiba 限制下的多项式算法。
摘要：
本文研究计算不确定曲线之间（离散和连续）Fréchet 距离的多种变体。我们将不确定曲线定义为一系列不确定区域，其中每个区域是圆盘、线段或点集。曲线的一个实现是连接每个区域中一个点的折线。给定一条不确定曲线和第二条（确定或不确定）曲线，我们寻求计算下界和上界 Fréchet 距离，即曲线任意实现之间的最小和最大 Fréchet 距离。我们证明，在若干不确定性模型中，连续 Fréchet 距离的上界和下界问题都是 NP 难的，并且对于离散 Fréchet 距离，上界问题仍然困难。相比之下，下界（离散和连续）Fréchet 距离可在多项式时间内计算。此外，我们表明，当不确定区域建模为点集或线段时，计算期望离散 Fréchet 距离是 #P 难的。该构造还可扩展，以证明当区域建模为点集时计算连续 Fréchet 距离的 #P 困难性。在积极方面，我们论证在任意常数维中，当 Δ/δ 受多项式界约束时，下界问题存在 FPTAS，其中 δ 是 Fréchet 距离，Δ 界定区域的直径。随后我们论证，当区域为凸且大致 δ-分离时，判定问题存在近线性时间 3-近似算法。最后，我们还研究了带 Sakoe–Chiba 时间带的情形，即限制两条曲线之间的对齐，并针对建模为点集的不确定区域，给出上界以及期望离散和连续 Fréchet 距离的多项式时间算法。
英文摘要：
In this paper we study a wide range of variants for computing the (discrete and continuous) Fréchet 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 Fréchet distance, which are the minimum and maximum Fréchet distance for any realisations of the curves. We prove that both the upper and lower bound problems are NP-hard for the continuous 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 and continuous) Fréchet distance can be computed in polynomial time. Furthermore, we show that computing the expected discrete Fréchet 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 Fréchet 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 $Δ/ δ$ is polynomially bounded, where $δ$ is the Fréchet distance and $Δ$ 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 $δ$-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 Fréchet distance for uncertainty regions modelled as point sets.
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