333 research outputs found
Computing the Similarity Between Moving Curves
In this paper we study similarity measures for moving curves which can, for
example, model changing coastlines or retreating glacier termini. Points on a
moving curve have two parameters, namely the position along the curve as well
as time. We therefore focus on similarity measures for surfaces, specifically
the Fr\'echet distance between surfaces. While the Fr\'echet distance between
surfaces is not even known to be computable, we show for variants arising in
the context of moving curves that they are polynomial-time solvable or
NP-complete depending on the restrictions imposed on how the moving curves are
matched. We achieve the polynomial-time solutions by a novel approach for
computing a surface in the so-called free-space diagram based on max-flow
min-cut duality
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
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
Computing a Subtrajectory Cluster from c-packed Trajectories
We present a near-linear time approximation algorithm for the subtrajectory
cluster problem of -packed trajectories. The problem involves finding
subtrajectories within a given trajectory such that their Fr\'echet
distances are at most , and at least one subtrajectory must
be of length~ or longer. A trajectory is -packed if the intersection
of and any ball with radius is at most in length.
Previous results by Gudmundsson and Wong
\cite{GudmundssonWong2022Cubicupperlower} established an lower
bound unless the Strong Exponential Time Hypothesis fails, and they presented
an time algorithm. We circumvent this conditional lower bound
by studying subtrajectory cluster on -packed trajectories, resulting in an
algorithm with an time complexity
Random projections for high-dimensional curves
Modern time series analysis requires the ability to handle datasets that are
inherently high-dimensional; examples include applications in climatology,
where measurements from numerous sensors must be taken into account, or
inventory tracking of large shops, where the dimension is defined by the number
of tracked items. The standard way to mitigate computational issues arising
from the high-dimensionality of the data is by applying some dimension
reduction technique that preserves the structural properties of the ambient
space. The dissimilarity between two time series is often measured by
``discrete'' notions of distance, e.g. the dynamic time warping, or the
discrete Fr\'echet distance, or simply the Euclidean distance. Since all these
distance functions are computed directly on the points of a time series, they
are sensitive to different sampling rates or gaps. The continuous Fr\'echet
distance offers a popular alternative which aims to alleviate this by taking
into account all points on the polygonal curve obtained by linearly
interpolating between any two consecutive points in a sequence.
We study the ability of random projections \`a la Johnson and Lindenstrauss
to preserve the continuous Fr\'echet distance of polygonal curves by
effectively reducing the dimension. In particular, we show that one can reduce
the dimension to , where is the total number of
input points while preserving the continuous Fr\'echet distance between any two
determined polygonal curves within a factor of . We conclude
with applications on clustering.Comment: 22 page
A fast implementation of near neighbors queries for Fr\'echet distance (GIS Cup)
This paper describes an implementation of fast near-neighbours queries (also
known as range searching) with respect to the Fr\'echet distance. The algorithm
is designed to be efficient on practical data such as GPS trajectories. Our
approach is to use a quadtree data structure to enumerate all curves in the
database that have similar start and endpoints as the query curve. On these
curves we run positive and negative filters to narrow the set of potential
results. Only for those trajectories where these heuristics fail, we compute
the Fr\'echet distance exactly, by running a novel recursive variant of the
classic free-space diagram algorithm.
Our implementation won the ACM SIGSPATIAL GIS Cup 2017.Comment: ACM SIGSPATIAL'17 invited paper. 9 page
Faster Fr\'echet Distance Approximation through Truncated Smoothing
The Fr\'echet distance is a popular distance measure for curves. Computing
the Fr\'echet distance between two polygonal curves of vertices takes
roughly quadratic time, and conditional lower bounds suggest that even
approximating to within a factor cannot be done in strongly-subquadratic
time, even in one dimension. The current best approximation algorithms present
trade-offs between approximation quality and running time. Recently, van der
Horst (SODA, 2023) presented an
time -approximate algorithm for curves in arbitrary dimensions, for any
. Our main contribution is an approximation algorithm for
curves in one dimension, with a significantly faster running time of . Additionally, we give an
algorithm for curves in arbitrary dimensions that improves upon the
state-of-the-art running time by a logarithmic factor, to . Both of our algorithms rely on a linear-time simplification
procedure that in one dimension reduces the complexity of the reachable free
space to without making sacrifices in the asymptotic
approximation factor.Comment: 27 pages, 11 figure
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