45 research outputs found
Constrained generalized Delaunay graphs are plane spanners
We look at generalized Delaunay graphs in the constrained setting by introducing line segments which the edges of the graph are not allowed to cross. Given an arbitrary convex shape C, a constrained Delaunay graph is constructed by adding an edge between two vertices p and q if and only if there exists a homothet of C with p and q on its boundary that does not contain any other vertices visible to p and q. We show that, regardless of the convex shape C used to construct the constrained Delaunay graph, there exists a constant t (that depends on C) such that it is a plane t-spanner of the visibility graph
Spanners of Additively Weighted Point Sets
We study the problem of computing geometric spanners for (additively)
weighted point sets. A weighted point set is a set of pairs where
is a point in the plane and is a real number. The distance between two
points and is defined as . We show
that in the case where all are positive numbers and for all (in which case the points can be seen as
non-intersecting disks in the plane), a variant of the Yao graph is a
-spanner that has a linear number of edges. We also show that the
Additively Weighted Delaunay graph (the face-dual of the Additively Weighted
Voronoi diagram) has constant spanning ratio. The straight line embedding of
the Additively Weighted Delaunay graph may not be a plane graph. We show how to
compute a plane embedding that also has a constant spanning ratio
The Tight Spanning Ratio of the Rectangle Delaunay Triangulation
Spanner construction is a well-studied problem and Delaunay triangulations
are among the most popular spanners. Tight bounds are known if the Delaunay
triangulation is constructed using an equilateral triangle, a square, or a
regular hexagon. However, all other shapes have remained elusive. In this paper
we extend the restricted class of spanners for which tight bounds are known. We
prove that Delaunay triangulations constructed using rectangles with aspect
ratio \A have spanning ratio at most \sqrt{2} \sqrt{1+\A^2 + \A \sqrt{\A^2 +
1}}, which matches the known lower bound
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
A Framework for Algorithm Stability
We say that an algorithm is stable if small changes in the input result in
small changes in the output. This kind of algorithm stability is particularly
relevant when analyzing and visualizing time-varying data. Stability in general
plays an important role in a wide variety of areas, such as numerical analysis,
machine learning, and topology, but is poorly understood in the context of
(combinatorial) algorithms. In this paper we present a framework for analyzing
the stability of algorithms. We focus in particular on the tradeoff between the
stability of an algorithm and the quality of the solution it computes. Our
framework allows for three types of stability analysis with increasing degrees
of complexity: event stability, topological stability, and Lipschitz stability.
We demonstrate the use of our stability framework by applying it to kinetic
Euclidean minimum spanning trees
Algorithmic and Combinatorial Results on Fence Patrolling, Polygon Cutting and Geometric Spanners
The purpose of this dissertation is to study problems that lie at the intersection of geometry and computer science. We have studied and obtained several results from three different areas, namely–geometric spanners, polygon cutting, and fence patrolling. Specifically, we have designed and analyzed algorithms along with various combinatorial results in these three areas. For geometric spanners, we have obtained combinatorial results regarding lower bounds on worst case dilation of plane spanners. We also have studied low degree plane lattice spanners, both square and hexagonal, of low dilation. Next, for polygon cutting, we have designed and analyzed algorithms for cutting out polygon collections drawn on a piece of planar material
using the three geometric models of saw, namely, line, ray and segment cuts. For fence patrolling, we have designed several strategies for robots patrolling both open and closed fences