12,485 research outputs found
Geodesic-Preserving Polygon Simplification
Polygons are a paramount data structure in computational geometry. While the
complexity of many algorithms on simple polygons or polygons with holes depends
on the size of the input polygon, the intrinsic complexity of the problems
these algorithms solve is often related to the reflex vertices of the polygon.
In this paper, we give an easy-to-describe linear-time method to replace an
input polygon by a polygon such that (1)
contains , (2) has its reflex
vertices at the same positions as , and (3) the number of vertices
of is linear in the number of reflex vertices. Since the
solutions of numerous problems on polygons (including shortest paths, geodesic
hulls, separating point sets, and Voronoi diagrams) are equivalent for both
and , our algorithm can be used as a preprocessing
step for several algorithms and makes their running time dependent on the
number of reflex vertices rather than on the size of
Query-points visibility constraint minimum link paths in simple polygons
We study the query version of constrained minimum link paths between two
points inside a simple polygon with vertices such that there is at
least one point on the path, visible from a query point. The method is based on
partitioning into a number of faces of equal link distance from a point,
called a link-based shortest path map (SPM). Initially, we solve this problem
for two given points , and a query point . Then, the proposed
solution is extended to a general case for three arbitrary query points ,
and . In the former, we propose an algorithm with preprocessing
time. Extending this approach for the latter case, we develop an algorithm with
preprocessing time. The link distance of a - path between
, as well as the path are provided in time and , respectively, for the above two cases, where is the number of links
Shortest Path in a Polygon using Sublinear Space
\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}}
\newcommand{\SetX}{\mathsf{X}} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}}
\newcommand{\Polygon}{\mathsf{P}} \newcommand{\Space}{\overline{\mathsf{m}}}
\newcommand{\pth}[2][\!]{#1\left({#2}\right)} We resolve an open problem due
to Tetsuo Asano, showing how to compute the shortest path in a polygon, given
in a read only memory, using sublinear space and subquadratic time.
Specifically, given a simple polygon \Polygon with vertices in a read
only memory, and additional working memory of size \Space, the new algorithm
computes the shortest path (in \Polygon) in O( n^2 /\, \Space ) expected
time. This requires several new tools, which we believe to be of independent
interest
Querying for the Largest Empty Geometric Object in a Desired Location
We study new types of geometric query problems defined as follows: given a
geometric set , preprocess it such that given a query point , the
location of the largest circle that does not contain any member of , but
contains can be reported efficiently. The geometric sets we consider for
are boundaries of convex and simple polygons, and point sets. While we
primarily focus on circles as the desired shape, we also briefly discuss empty
rectangles in the context of point sets.Comment: This version is a significant update of our earlier submission
arXiv:1004.0558v1. Apart from new variants studied in Sections 3 and 4, the
results have been improved in Section 5.Please note that the change in title
and abstract indicate that we have expanded the scope of the problems we
stud
Algorithms for distance problems in planar complexes of global nonpositive curvature
CAT(0) metric spaces and hyperbolic spaces play an important role in
combinatorial and geometric group theory. In this paper, we present efficient
algorithms for distance problems in CAT(0) planar complexes. First of all, we
present an algorithm for answering single-point distance queries in a CAT(0)
planar complex. Namely, we show that for a CAT(0) planar complex K with n
vertices, one can construct in O(n^2 log n) time a data structure D of size
O(n^2) so that, given a point x in K, the shortest path gamma(x,y) between x
and the query point y can be computed in linear time. Our second algorithm
computes the convex hull of a finite set of points in a CAT(0) planar complex.
This algorithm is based on Toussaint's algorithm for computing the convex hull
of a finite set of points in a simple polygon and it constructs the convex hull
of a set of k points in O(n^2 log n + nk log k) time, using a data structure of
size O(n^2 + k)
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