270 research outputs found
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
Routing on the Visibility Graph
We consider the problem of routing on a network in the presence of line
segment constraints (i.e., obstacles that edges in our network are not allowed
to cross). Let be a set of points in the plane and let be a set of
non-crossing line segments whose endpoints are in . We present two
deterministic 1-local -memory routing algorithms that are guaranteed to
find a path of at most linear size between any pair of vertices of the
\emph{visibility graph} of with respect to a set of constraints (i.e.,
the algorithms never look beyond the direct neighbours of the current location
and store only a constant amount of additional information). Contrary to {\em
all} existing deterministic local routing algorithms, our routing algorithms do
not route on a plane subgraph of the visibility graph. Additionally, we provide
lower bounds on the routing ratio of any deterministic local routing algorithm
on the visibility graph.Comment: An extended abstract of this paper appeared in the proceedings of the
28th International Symposium on Algorithms and Computation (ISAAC 2017).
Final version appeared in the Journal of Computational Geometr
Competitive Local Routing with Constraints
Let be a set of vertices in the plane and a set of non-crossing
line segments between vertices in , called constraints. Two vertices are
visible if the straight line segment connecting them does not properly
intersect any constraints. The constrained -graph is constructed by
partitioning the plane around each vertex into disjoint cones, each with
aperture , and adding an edge to the `closest' visible vertex
in each cone. We consider how to route on the constrained -graph. We
first show that no deterministic 1-local routing algorithm is
-competitive on all pairs of vertices of the constrained
-graph. After that, we show how to route between any two visible
vertices of the constrained -graph using only 1-local information.
Our routing algorithm guarantees that the returned path is 2-competitive.
Additionally, we provide a 1-local 18-competitive routing algorithm for visible
vertices in the constrained half--graph, a subgraph of the
constrained -graph that is equivalent to the Delaunay graph where the
empty region is an equilateral triangle. To the best of our knowledge, these
are the first local routing algorithms in the constrained setting with
guarantees on the length of the returned path
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