5,209 research outputs found
Pattern Matching for sets of segments
In this paper we present algorithms for a number of problems in geometric
pattern matching where the input consist of a collections of segments in the
plane. Our work consists of two main parts. In the first, we address problems
and measures that relate to collections of orthogonal line segments in the
plane. Such collections arise naturally from problems in mapping buildings and
robot exploration.
We propose a new measure of segment similarity called a \emph{coverage
measure}, and present efficient algorithms for maximising this measure between
sets of axis-parallel segments under translations. Our algorithms run in time
O(n^3\polylog n) in the general case, and run in time O(n^2\polylog n) for
the case when all segments are horizontal. In addition, we show that when
restricted to translations that are only vertical, the Hausdorff distance
between two sets of horizontal segments can be computed in time roughly
O(n^{3/2}{\sl polylog}n). These algorithms form significant improvements over
the general algorithm of Chew et al. that takes time . In the
second part of this paper we address the problem of matching polygonal chains.
We study the well known \Frd, and present the first algorithm for computing the
\Frd under general translations. Our methods also yield algorithms for
computing a generalization of the \Fr distance, and we also present a simple
approximation algorithm for the \Frd that runs in time O(n^2\polylog n).Comment: To appear in the 12 ACM Symposium on Discrete Algorithms, Jan 200
Characterization of co-blockers for simple perfect matchings in a convex geometric graph
Consider the complete convex geometric graph on vertices, ,
i.e., the set of all boundary edges and diagonals of a planar convex -gon
. In [C. Keller and M. Perles, On the Smallest Sets Blocking Simple Perfect
Matchings in a Convex Geometric Graph], the smallest sets of edges that meet
all the simple perfect matchings (SPMs) in (called "blockers") are
characterized, and it is shown that all these sets are caterpillar graphs with
a special structure, and that their total number is . In this
paper we characterize the co-blockers for SPMs in , that is, the
smallest sets of edges that meet all the blockers. We show that the co-blockers
are exactly those perfect matchings in where all edges are of odd
order, and two edges of that emanate from two adjacent vertices of
never cross. In particular, while the number of SPMs and the number of blockers
grow exponentially with , the number of co-blockers grows
super-exponentially.Comment: 8 pages, 4 figure
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
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
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