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.
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Computing the Caratheodory Number of a Point
Caratheodory’s theorem says that any point in the convex hull of a set $P$ in $R^d$ is in the convex hull of a subset $P'$ of $P$ such that $|P'| \le d + 1$. For some sets P, the upper bound d + 1 can be improved. The best upper bound for P is known as the Caratheodory number [2, 15, 17]. In this paper, we study a computational problem of finding the smallest set $P'$ for a given set $P$ and a point $p$. We call the size of this set $P'$, the Caratheodory number of a point p or CNP. We show that the problem of deciding the Caratheodory number
of a point is NP-hard. Furthermore, we show that the problem is k-LDT-hard. We present two algorithms for computing a smallest set $P'$, if CNP= 2,3.
Bárány [1] generalized Caratheodory’s theorem by using d+1 sets (colored sets) such that their convex hulls intersect. We introduce a Colorful Caratheodory number of a point or CCNP which can be smaller than d+1. Then we extend our results for CNP to CCNP.
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- Award ID(s):
- 1718994
- NSF-PAR ID:
- 10196676
- Date Published:
- Journal Name:
- Canadian Conference on Computational Geometry
- Page Range / eLocation ID:
- 192-198
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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