315 research outputs found
Fast Frechet Distance Between Curves With Long Edges
Computing the Fr\'echet distance between two polygonal curves takes roughly
quadratic time. In this paper, we show that for a special class of curves the
Fr\'echet distance computations become easier. Let and be two polygonal
curves in with and vertices, respectively. We prove four
results for the case when all edges of both curves are long compared to the
Fr\'echet distance between them: (1) a linear-time algorithm for deciding the
Fr\'echet distance between two curves, (2) an algorithm that computes the
Fr\'echet distance in time, (3) a linear-time
-approximation algorithm, and (4) a data structure that supports
-time decision queries, where is the number of vertices of
the query curve and the number of vertices of the preprocessed curve
A Pontryagin Maximum Principle in Wasserstein Spaces for Constrained Optimal Control Problems
In this paper, we prove a Pontryagin Maximum Principle for constrained
optimal control problems in the Wasserstein space of probability measures. The
dynamics, is described by a transport equation with non-local velocities and is
subject to end-point and running state constraints. Building on our previous
work, we combine the classical method of needle-variations from geometric
control theory and the metric differential structure of the Wasserstein spaces
to obtain a maximum principle stated in the so-called Gamkrelidze form.Comment: 35 page
Approximating the Packedness of Polygonal Curves
In 2012 Driemel et al. \cite{DBLP:journals/dcg/DriemelHW12} introduced the
concept of -packed curves as a realistic input model. In the case when
is a constant they gave a near linear time -approximation
algorithm for computing the Fr\'echet distance between two -packed polygonal
curves. Since then a number of papers have used the model.
In this paper we consider the problem of computing the smallest for which
a given polygonal curve in is -packed. We present two
approximation algorithms. The first algorithm is a -approximation algorithm
and runs in time. In the case we develop a faster
algorithm that returns a -approximation and runs in
time.
We also implemented the first algorithm and computed the approximate
packedness-value for 16 sets of real-world trajectories. The experiments
indicate that the notion of -packedness is a useful realistic input model
for many curves and trajectories.Comment: A preliminary version to appear in ISAAC 202
Distance Measures for Embedded Graphs
We introduce new distance measures for comparing straight-line embedded
graphs based on the Fr\'echet distance and the weak Fr\'echet distance. These
graph distances are defined using continuous mappings and thus take the
combinatorial structure as well as the geometric embeddings of the graphs into
account. We present a general algorithmic approach for computing these graph
distances. Although we show that deciding the distances is NP-hard for general
embedded graphs, we prove that our approach yields polynomial time algorithms
if the graphs are trees, and for the distance based on the weak Fr\'echet
distance if the graphs are planar embedded. Moreover, we prove that deciding
the distances based on the Fr\'echet distance remains NP-hard for planar
embedded graphs and show how our general algorithmic approach yields an
exponential time algorithm and a polynomial time approximation algorithm for
this case.Comment: 27 pages, 14 Figure
Fréchet Distance for Uncertain Curves
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/δis polynomially bounded, where δis the Fréchet distance and Δbounds 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.</p
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