102 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
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
Inferring movement patterns from geometric similarity
Spatial movement data nowadays is becoming ubiquitously available, including data of animals, vehicles and people. This data allows us to analyze the underlying movement. In particular, it allows us to infer movement patterns, such as recurring places and routes. Many methods to do so rely on the notion of similarity of places or routes. Here we briefly survey how research on this has developed in the past 15 years and outline challenges for future work
- …