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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
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
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