6 research outputs found
Complexity and Algorithms for the Discrete Fr\'echet Distance Upper Bound with Imprecise Input
We study the problem of computing the upper bound of the discrete Fr\'{e}chet
distance for imprecise input, and prove that the problem is NP-hard. This
solves an open problem posed in 2010 by Ahn \emph{et al}. If shortcuts are
allowed, we show that the upper bound of the discrete Fr\'{e}chet distance with
shortcuts for imprecise input can be computed in polynomial time and we present
several efficient algorithms.Comment: 15 pages, 8 figure
Computing Minimum Spanning Trees with Uncertainty
We consider the minimum spanning tree problem in a setting where information
about the edge weights of the given graph is uncertain. Initially, for each
edge of the graph only a set , called an uncertainty area, that
contains the actual edge weight is known. The algorithm can `update'
to obtain the edge weight . The task is to output the edge set of
a minimum spanning tree after a minimum number of updates. An algorithm is
-update competitive if it makes at most times as many updates as the
optimum. We present a 2-update competitive algorithm if all areas are
open or trivial, which is the best possible among deterministic algorithms. The
condition on the areas is to exclude degenerate inputs for which no
constant update competitive algorithm can exist. Next, we consider a setting
where the vertices of the graph correspond to points in Euclidean space and the
weight of an edge is equal to the distance of its endpoints. The location of
each point is initially given as an uncertainty area, and an update reveals the
exact location of the point. We give a general relation between the edge
uncertainty and the vertex uncertainty versions of a problem and use it to
derive a 4-update competitive algorithm for the minimum spanning tree problem
in the vertex uncertainty model. Again, we show that this is best possible
among deterministic algorithms
Computing the Fréchet distance between uncertain curves in one dimension.
We consider the problem of computing the Fréchet distance between two curves for which the exact locations of the vertices are unknown. Each vertex may be placed in a given uncertainty region for that vertex, and the objective is to place vertices so as to minimise the Fréchet distance. This problem was recently shown to be NP-hard in 2D, and it is unclear how to compute an optimal vertex placement at all. We present the first general algorithmic framework for this problem. We prove that it results in a polynomial-time algorithm for curves in 1D with intervals as uncertainty regions. In contrast, we show that the problem is NP-hard in 1D in the case that vertices are placed to maximise the Fréchet distance. We also study the weak Fréchet distance between uncertain curves. While finding the optimal placement of vertices seems more difficult than the regular Fréchet distance—and indeed we can easily prove that the problem is NP-hard in 2D—the optimal placement of vertices in 1D can be computed in polynomial time. Finally, we investigate the discrete weak Fréchet distance, for which, somewhat surprisingly, the problem is NP-hard already in 1D
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
Fr\'echet Distance for Uncertain Curves
In this paper we study a wide range of variants for computing the (discrete
and continuous) Fr\'echet 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 Fr\'echet distance, which are the minimum and maximum Fr\'echet distance
for any realisations of the curves.
We prove that both the upper and lower bound problems are NP-hard for the
continuous Fr\'echet distance in several uncertainty models, and that the upper
bound problem remains hard for the discrete Fr\'echet distance. In contrast,
the lower bound (discrete and continuous) Fr\'echet distance can be computed in
polynomial time. Furthermore, we show that computing the expected discrete
Fr\'echet 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 Fr\'echet 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 is polynomially
bounded, where is the Fr\'echet distance and 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 -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 Fr\'echet distance for uncertainty regions modelled as
point sets.Comment: 48 pages, 11 figures. This is the full version of the paper to be
published in ICALP 202