45 research outputs found

    Constrained generalized Delaunay graphs are plane spanners

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    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

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    We study the problem of computing geometric spanners for (additively) weighted point sets. A weighted point set is a set of pairs (p,r)(p,r) where pp is a point in the plane and rr is a real number. The distance between two points (pi,ri)(p_i,r_i) and (pj,rj)(p_j,r_j) is defined as ∣pipj∣−ri−rj|p_ip_j|-r_i-r_j. We show that in the case where all rir_i are positive numbers and ∣pipj∣≥ri+rj|p_ip_j|\geq r_i+r_j for all i,ji,j (in which case the points can be seen as non-intersecting disks in the plane), a variant of the Yao graph is a (1+ϵ)(1+\epsilon)-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

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    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

    The Tight Spanning Ratio of the Rectangle Delaunay Triangulation

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    Competitive Local Routing with Constraints

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    Let PP be a set of nn vertices in the plane and SS a set of non-crossing line segments between vertices in PP, called constraints. Two vertices are visible if the straight line segment connecting them does not properly intersect any constraints. The constrained Θm\Theta_m-graph is constructed by partitioning the plane around each vertex into mm disjoint cones, each with aperture θ=2π/m\theta = 2 \pi/m, and adding an edge to the `closest' visible vertex in each cone. We consider how to route on the constrained Θ6\Theta_6-graph. We first show that no deterministic 1-local routing algorithm is o(n)o(\sqrt{n})-competitive on all pairs of vertices of the constrained Θ6\Theta_6-graph. After that, we show how to route between any two visible vertices of the constrained Θ6\Theta_6-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-Θ6\Theta_6-graph, a subgraph of the constrained Θ6\Theta_6-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

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    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

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    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
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