2,401 research outputs found
Improved Dynamic Geodesic Nearest Neighbor Searching in a Simple Polygon
We present an efficient dynamic data structure that supports geodesic nearest neighbor queries for a set S of point sites in a static simple polygon P. Our data structure allows us to insert a new site in S, delete a site from S, and ask for the site in S closest to an arbitrary query point q in P. All distances are measured using the geodesic distance, that is, the length of the shortest path that is completely contained in P. Our data structure achieves polylogarithmic update and query times, and uses O(n log^3n log m + m) space, where n is the number of sites in S and m is the number of vertices in P. The crucial ingredient in our data structure is an implicit representation of a vertical shallow cutting of the geodesic distance functions. We show that such an implicit representation exists, and that we can compute it efficiently
Approximate Euclidean shortest paths in polygonal domains
Given a set of pairwise disjoint simple polygonal obstacles
in defined with vertices, we compute a sketch of
whose size is independent of , depending only on and the
input parameter . We utilize to compute a
-approximate geodesic shortest path between the two given points
in time. Here, is a user
parameter, and is a small positive constant (resulting from the time
for triangulating the free space of using the algorithm in
\cite{journals/ijcga/Bar-YehudaC94}). Moreover, we devise a
-approximation algorithm to answer two-point Euclidean distance
queries for the case of convex polygonal obstacles.Comment: a few updates; accepted to ISAAC 201
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
Weak Visibility Queries of Line Segments in Simple Polygons
Given a simple polygon P in the plane, we present new algorithms and data
structures for computing the weak visibility polygon from any query line
segment in P. We build a data structure in O(n) time and O(n) space that can
compute the visibility polygon for any query line segment s in O(k log n) time,
where k is the size of the visibility polygon of s and n is the number of
vertices of P. Alternatively, we build a data structure in O(n^3) time and
O(n^3) space that can compute the visibility polygon for any query line segment
in O(k + log n) time.Comment: 16 pages, 9 figures. A preliminary version of this paper appeared in
ISAAC 2012 and we have improved results in this full versio
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)
- …