1,076 research outputs found
Lower Bounds for the Complexity of the Voronoi Diagram of Polygonal Curves under the Discrete Frechet Distance
We give lower bounds for the combinatorial complexity of the Voronoi diagram
of polygonal curves under the discrete Frechet distance. We show that the
Voronoi diagram of n curves in R^d with k vertices each, has complexity
Omega(n^{dk}) for dimension d=1,2 and Omega(n^{d(k-1)+2}) for d>2.Comment: 6 pages, 2 figure
Locally Correct Frechet Matchings
The Frechet distance is a metric to compare two curves, which is based on
monotonous matchings between these curves. We call a matching that results in
the Frechet distance a Frechet matching. There are often many different Frechet
matchings and not all of these capture the similarity between the curves well.
We propose to restrict the set of Frechet matchings to "natural" matchings and
to this end introduce locally correct Frechet matchings. We prove that at least
one such matching exists for two polygonal curves and give an O(N^3 log N)
algorithm to compute it, where N is the total number of edges in both curves.
We also present an O(N^2) algorithm to compute a locally correct discrete
Frechet matching
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
Four Soviets Walk the Dog-Improved Bounds for Computing the Fr\'echet Distance
Given two polygonal curves in the plane, there are many ways to define a
notion of similarity between them. One popular measure is the Fr\'echet
distance. Since it was proposed by Alt and Godau in 1992, many variants and
extensions have been studied. Nonetheless, even more than 20 years later, the
original algorithm by Alt and Godau for computing the Fr\'echet
distance remains the state of the art (here, denotes the number of edges on
each curve). This has led Helmut Alt to conjecture that the associated decision
problem is 3SUM-hard.
In recent work, Agarwal et al. show how to break the quadratic barrier for
the discrete version of the Fr\'echet distance, where one considers sequences
of points instead of polygonal curves. Building on their work, we give a
randomized algorithm to compute the Fr\'echet distance between two polygonal
curves in time on a pointer machine
and in time on a word RAM. Furthermore, we show that
there exists an algebraic decision tree for the decision problem of depth
, for some . We believe that this
reveals an intriguing new aspect of this well-studied problem. Finally, we show
how to obtain the first subquadratic algorithm for computing the weak Fr\'echet
distance on a word RAM.Comment: 34 pages, 15 figures. A preliminary version appeared in SODA 201
Progressive Simplification of Polygonal Curves
Simplifying polygonal curves at different levels of detail is an important
problem with many applications. Existing geometric optimization algorithms are
only capable of minimizing the complexity of a simplified curve for a single
level of detail. We present an -time algorithm that takes a polygonal
curve of n vertices and produces a set of consistent simplifications for m
scales while minimizing the cumulative simplification complexity. This
algorithm is compatible with distance measures such as the Hausdorff, the
Fr\'echet and area-based distances, and enables simplification for continuous
scaling in time. To speed up this algorithm in practice, we present
new techniques for constructing and representing so-called shortcut graphs.
Experimental evaluation of these techniques on trajectory data reveals a
significant improvement of using shortcut graphs for progressive and
non-progressive curve simplification, both in terms of running time and memory
usage.Comment: 20 pages, 20 figure
A Spanner for the Day After
We show how to construct -spanner over a set of
points in that is resilient to a catastrophic failure of nodes.
Specifically, for prescribed parameters , the
computed spanner has edges, where . Furthermore, for any , and
any deleted set of points, the residual graph is -spanner for all the points of except for
of them. No previous constructions, beyond the trivial clique
with edges, were known such that only a tiny additional fraction
(i.e., ) lose their distance preserving connectivity.
Our construction works by first solving the exact problem in one dimension,
and then showing a surprisingly simple and elegant construction in higher
dimensions, that uses the one-dimensional construction in a black box fashion
Approximating the Distribution of the Median and other Robust Estimators on Uncertain Data
Robust estimators, like the median of a point set, are important for data
analysis in the presence of outliers. We study robust estimators for
locationally uncertain points with discrete distributions. That is, each point
in a data set has a discrete probability distribution describing its location.
The probabilistic nature of uncertain data makes it challenging to compute such
estimators, since the true value of the estimator is now described by a
distribution rather than a single point. We show how to construct and estimate
the distribution of the median of a point set. Building the approximate support
of the distribution takes near-linear time, and assigning probability to that
support takes quadratic time. We also develop a general approximation technique
for distributions of robust estimators with respect to ranges with bounded VC
dimension. This includes the geometric median for high dimensions and the
Siegel estimator for linear regression.Comment: Full version of a paper to appear at SoCG 201
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