Optimal Coding and Sampling of Triangulations

Poulalhon, Dominique; Schaeffer, Gilles

doi:10.1007/3-540-45061-0_83

Cited by 69 publications

(139 citation statements)

References 24 publications

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“…As our preconditioner is a combination of the tree-based and diagonal approaches, we denote it by JTPCG.We run our experiments on a wide collection of graphs including triangle meshes (obtained from the AIM@SHAPE Shape repository), 2D regular grids, and complex networks (from the Stanford Large Network Dataset Collection). We also consider random planar triangulations, generated by the uniform random sampler by Poulalhon and Schaeffer [18], as well as graphs randomly generated according to the small-world and preferential attachment models.…”

Section: Resultsmentioning

confidence: 99%

Efficient and Practical Tree Preconditioning for Solving Laplacian Systems

Aleardi

Nolin

Ovsjanikov

2015

Experimental Algorithms

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Abstract. We consider the problem of designing efficient iterative methods for solving linear systems. In its full generality, this is one of the oldest problems in numerical analysis with a tremendous number of practical applications. We focus on a particular type of linear systems, associated with Laplacian matrices of undirected graphs, and study a class of iterative methods for which it is possible to speed up the convergence through combinatorial preconditioning. We consider a class of preconditioners, known as tree preconditioners, introduced by Vaidya, that have been shown to lead to asymptotic speed-up in certain cases. Rather than trying to improve the structure of the trees used in preconditioning, we propose a very simple modification to the basic tree preconditioner, which can significantly improve the performance of the iterative linear solvers in practice. We show that our modification leads to better conditioning for some special graphs, and provide extensive experimental evidence for the decrease in the complexity of the preconditioned conjugate gradient method for several graphs, including 3D meshes and complex networks.

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Section: Resultsmentioning

confidence: 99%

Efficient and Practical Tree Preconditioning for Solving Laplacian Systems

Aleardi

Nolin

Ovsjanikov

2015

Experimental Algorithms

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“…One of our main motivations for generalizing Schnyder woods to higher genus is the great number of existing works in the domain of graph encoding and mesh compression that take advantage of spanning tree decompositions [23,30,33], and in particular of the ones underlying Schnyder woods (and related extensions) for planar graphs [1,11,12,20,21,29]. The combinatorial properties of Schnyder woods and the related characterizations (canonical orderings [22]) for planar graphs yield efficient procedures for encoding tree structures based on multiple parenthesis words.…”

Section: Schnyder Trees and Graph Encodingmentioning

confidence: 99%

“…Finally we point out that the existence of minimal orientations (orientations without counter-clockwise directed cycles) recently made it possible to design the first optimal (linear time) encodings for some popular classes of planar graphs. For the case of triangulations and 3-connected plane graphs [20,29] there exist some bijective correspondences between such graphs and some special classes of plane trees, which give nice combinatorial interpretations of enumerative formulas originally found by Tutte [34]. Very few works have been proposed for dealing with higher genus embedded graphs (corresponding to manifold meshes): this is due to the strong combinatorial properties involved in the planar case.…”

Section: Schnyder Trees and Graph Encodingmentioning

confidence: 99%

“…As done in previous works considering the extension of Schnyder woods to other classes of (not triangulated) planar graphs [17,29], one main contribution is to propose a new generalized version of Schnyder woods to genus g triangulations (see Figure 2 for an example).…”

Section: Generalized Schnyder Woodsmentioning

confidence: 99%

“…Schnyder woods, and more generally α-orientations, received a great deal of attention [32,17,22,19]. From the combinatorial point of view, the set of Schnyder woods of a fixed triangulation has an interesting lattice structure [7,3,18,13,14], and the nice characterization in terms of spanning trees motivated a large number of applications in several domains such as graph drawing [32,22], graph coding and sampling [12,21,4,29,20,5,10,1]. Previous work focused mainly on the application and extension of the combinatorial properties of Schnyder woods to 3-connected plane graphs [17,22].…”

Section: Introductionmentioning

confidence: 99%

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Schnyder Woods for Higher Genus Triangulated Surfaces, with Applications to Encoding

Aleardi

Fusy

Lewiner

2009

Discrete Comput Geom

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Schnyder woods are a well known combinatorial structure for planar graphs, which yields a decomposition into 3 vertexspanning trees. Our goal is to extend definitions and algorithms for Schnyder woods designed for planar graphs (corresponding to combinatorial surfaces with the topology of the sphere, i.e., of genus 0) to the more general case of graphs embedded on surfaces of arbitrary genus. First, we define a new traversal order of the vertices of a triangulated surface of genus g together with an orientation and coloration of the edges that extends the one proposed by Schnyder for the planar case. As a by-product we show how some recent schemes for compression and compact encoding of graphs can be extended to higher genus. All the algorithms presented here have linear time complexity.

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