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The Exact Spanning Ratio of the Parallelogram Delaunay Graph [PDF]
Theoretical Computer ScienceFinding the exact spanning ratio of a Delaunay graph has been one of the longstanding open problems in Computational Geometry. Currently there are only four convex shapes for which the exact spanning ratio of their Delaunay graph is known: the equilateral triangle, the square, the regular hexagon and the rectangle.
Prosenjit Bose +2 more
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Constrained generalized Delaunay graphs are plane spanners [PDF]
Computational Geometry, 2016 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 ...
Prosenjit Bose +2 more
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On the stretch factor of convex Delaunay graphs
Journal of Computational Geometry, 2010 Let C be a compact and convex set in the plane that contains the origin in its interior, and let S be a finite set of points in the plane. The Delaunay graph DGC(S) of S is defined to be the dual of the Voronoi diagram of S with respect to the convex ...
Prosenjit Bose +3 more
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Routing from Pentagon to Octagon Delaunay Graphs [PDF]
The standard Delaunay triangulation is a geometric graph whose vertices are points in the plane, and two vertices share an edge if they lie on the boundary of an empty disk. If the disk is replaced with a homothet of a fixed convex shape $C$, then the resulting graph is called a $C$-Delaunay graph.
Prosenjit Bose +2 more
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Good triangulations yield good tours [PDF]
, 2008 Consider the following heuristic for planar Euclidean instances of the Traveling Salesman Problem (TSP): select a subset of the edges which induces a planar graph, and solve either the TSP or its graphical relaxation on that graph. In this paper, we give
Pearson, N, Letchford, A N
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Kinetic stable Delaunay graphs [PDF]
Proceedings of the twenty-sixth annual symposium on Computational geometry, 2010 We consider the problem of maintaining the Euclidean Delaunay triangulation $\DT$ of a set $P$ of $n$ moving points in the plane, along algebraic trajectories of constant description complexity. Since the best known upper bound on the number of topological changes in the full $\DT$ is nearly cubic, we seek to maintain a suitable portion of it that is ...
Pankaj K. Agarwal +6 more
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Delaunay graphs are almost as good as complete graphs [PDF]
Discrete & Computational Geometry, 1987 Let S be a set of N points in the plane and let DT(S) be the graph of the Delaunay triangulation of S. For all points a and b of S, let d(a,b) be the Euclidean distance from a to b and let DT(a,b) be the length of the shortest path in DT(S) from a to b.
Dobkin, D.P. +2 more
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The Widom-Rowlinson model on the Delaunay graph [PDF]
Electronic Journal of Probability, 2019 We establish phase transitions for continuum Delaunay multi-type particle systems (continuum Potts or Widom-Rowlinson models) with a repulsive interaction between particles of different types. Our interaction potential depends solely on the length of the Delaunay edges.
Adams, Stefan, Eyers, Michael
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A Finite Algorithm for the Realizabilty of a Delaunay Triangulation [PDF]
, 2022 The \emph{Delaunay graph} of a point set $P \subseteq \mathbb{R}^2$ is the plane graph with the vertex-set $P$ and the edge-set that contains $\{p,p'\}$ if there exists a disc whose intersection with $P$ is exactly $\{p,p'\}$. Accordingly, a triangulated
Agrawal, Akanksha +2 more
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Incremental construction of the delaunay triangulation and the delaunay graph in medium dimension [PDF]
Proceedings of the twenty-fifth annual symposium on Computational geometry, 2009 We describe a new implementation of the well-known incremental algorithm for constructing Delaunay triangulations in any dimension. Our implementation follows the exact computing paradigm and is fully robust. Extensive comparisons show that our implementation outperforms the best currently available codes for exact convex hulls and Delaunay ...
Jean-Daniel Boissonnat +2 more
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