572 research outputs found
Computing a rectilinear shortest path amid splinegons in plane
We reduce the problem of computing a rectilinear shortest path between two
given points s and t in the splinegonal domain \calS to the problem of
computing a rectilinear shortest path between two points in the polygonal
domain. As part of this, we define a polygonal domain \calP from \calS and
transform a rectilinear shortest path computed in \calP to a path between s and
t amid splinegon obstacles in \calS. When \calS comprises of h pairwise
disjoint splinegons with a total of n vertices, excluding the time to compute a
rectilinear shortest path amid polygons in \calP, our reduction algorithm takes
O(n + h \lg{n}) time. For the special case of \calS comprising of concave-in
splinegons, we have devised another algorithm in which the reduction procedure
does not rely on the structures used in the algorithm to compute a rectilinear
shortest path in polygonal domain. As part of these, we have characterized few
of the properties of rectilinear shortest paths amid splinegons which could be
of independent interest
Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain
We study the computation of the diameter and radius under the rectilinear
link distance within a rectilinear polygonal domain of vertices and
holes. We introduce a \emph{graph of oriented distances} to encode the distance
between pairs of points of the domain. This helps us transform the problem so
that we can search through the candidates more efficiently. Our algorithm
computes both the diameter and the radius in time, where denotes the matrix
multiplication exponent and is the number of
edges of the graph of oriented distances. We also provide a faster algorithm
for computing the diameter that runs in time
How to Walk Your Dog in the Mountains with No Magic Leash
We describe a -approximation algorithm for computing the
homotopic \Frechet distance between two polygonal curves that lie on the
boundary of a triangulated topological disk. Prior to this work, algorithms
were known only for curves on the Euclidean plane with polygonal obstacles.
A key technical ingredient in our analysis is a -approximation
algorithm for computing the minimum height of a homotopy between two curves. No
algorithms were previously known for approximating this parameter.
Surprisingly, it is not even known if computing either the homotopic \Frechet
distance, or the minimum height of a homotopy, is in NP
Computational Geometry Column 42
A compendium of thirty previously published open problems in computational
geometry is presented.Comment: 7 pages; 72 reference
- …