8,312 research outputs found

    A QPTAS for the Base of the Number of Triangulations of a Planar Point Set

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    The number of triangulations of a planar n point set is known to be cnc^n, where the base cc lies between 2.432.43 and 30.30. The fastest known algorithm for counting triangulations of a planar n point set runs in O∗(2n)O^*(2^n) time. The fastest known arbitrarily close approximation algorithm for the base of the number of triangulations of a planar n point set runs in time subexponential in n.n. We present the first quasi-polynomial approximation scheme for the base of the number of triangulations of a planar point set

    Uniform Infinite Planar Triangulations

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    The existence of the weak limit as n --> infinity of the uniform measure on rooted triangulations of the sphere with n vertices is proved. Some properties of the limit are studied. In particular, the limit is a probability measure on random triangulations of the plane.Comment: 36 pages, 4 figures; Journal revised versio

    On the Number of Pseudo-Triangulations of Certain Point Sets

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    We pose a monotonicity conjecture on the number of pseudo-triangulations of any planar point set, and check it on two prominent families of point sets, namely the so-called double circle and double chain. The latter has asymptotically 12nnΘ(1)12^n n^{\Theta(1)} pointed pseudo-triangulations, which lies significantly above the maximum number of triangulations in a planar point set known so far.Comment: 31 pages, 11 figures, 4 tables. Not much technical changes with respect to v1, except some proofs and statements are slightly more precise and some expositions more clear. This version has been accepted in J. Combin. Th. A. The increase in number of pages from v1 is mostly due to formatting the paper with "elsart.cls" for Elsevie

    Transforming triangulations on non planar-surfaces

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    We consider whether any two triangulations of a polygon or a point set on a non-planar surface with a given metric can be transformed into each other by a sequence of edge flips. The answer is negative in general with some remarkable exceptions, such as polygons on the cylinder, and on the flat torus, and certain configurations of points on the cylinder.Comment: 19 pages, 17 figures. This version has been accepted in the SIAM Journal on Discrete Mathematics. Keywords: Graph of triangulations, triangulations on surfaces, triangulations of polygons, edge fli

    Percolation and coarse conformal uniformization

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    We formulate conjectures regarding percolation on planar triangulations suggested by assuming (quasi) invariance under coarse conformal uniformization

    Triangulating the Square and Squaring the Triangle: Quadtrees and Delaunay Triangulations are Equivalent

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    We show that Delaunay triangulations and compressed quadtrees are equivalent structures. More precisely, we give two algorithms: the first computes a compressed quadtree for a planar point set, given the Delaunay triangulation; the second finds the Delaunay triangulation, given a compressed quadtree. Both algorithms run in deterministic linear time on a pointer machine. Our work builds on and extends previous results by Krznaric and Levcopolous and Buchin and Mulzer. Our main tool for the second algorithm is the well-separated pair decomposition(WSPD), a structure that has been used previously to find Euclidean minimum spanning trees in higher dimensions (Eppstein). We show that knowing the WSPD (and a quadtree) suffices to compute a planar Euclidean minimum spanning tree (EMST) in linear time. With the EMST at hand, we can find the Delaunay triangulation in linear time. As a corollary, we obtain deterministic versions of many previous algorithms related to Delaunay triangulations, such as splitting planar Delaunay triangulations, preprocessing imprecise points for faster Delaunay computation, and transdichotomous Delaunay triangulations.Comment: 37 pages, 13 figures, full version of a paper that appeared in SODA 201

    Planar maps, circle patterns and 2d gravity

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    Via circle pattern techniques, random planar triangulations (with angle variables) are mapped onto Delaunay triangulations in the complex plane. The uniform measure on triangulations is mapped onto a conformally invariant spatial point process. We show that this measure can be expressed as: (1) a sum over 3-spanning-trees partitions of the edges of the Delaunay triangulations; (2) the volume form of a K\"ahler metric over the space of Delaunay triangulations, whose prepotential has a simple formulation in term of ideal tessellations of the 3d hyperbolic space; (3) a discretized version (involving finite difference complex derivative operators) of Polyakov's conformal Fadeev-Popov determinant in 2d gravity; (4) a combination of Chern classes, thus also establishing a link with topological 2d gravity.Comment: Misprints corrected and a couple of footnotes added. 42 pages, 17 figure
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