Fréchet Distance for Uncertain Curves
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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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