Results 11  20
of
25
A Faster Algorithm for Torus Embedding
, 2004
"... Although theoretically practical algorithms for torus embedding exist, they have not yet been successfully implemented and their complexity may be prohibitive to their practicality. We describe a simple exponential algorithm for embedding graphs on the torus (a surface shaped like a doughnut) and di ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Although theoretically practical algorithms for torus embedding exist, they have not yet been successfully implemented and their complexity may be prohibitive to their practicality. We describe a simple exponential algorithm for embedding graphs on the torus (a surface shaped like a doughnut) and discuss how it was inspired by the quadratic time planar embedding algorithm of Demoucron, Malgrange and Pertuiset. We show that it is faster in practice than the only fully implemented alternative (also exponential) and explain how both the algorithm itself and the knowledge gained during its development might be used to solve the wellstudied problem of finding the complete set of torus obstructions.
I/Oefficient algorithms for planar graphs I: Separators
"... We present I/Oefficient algorithms for computing optimal separator partitions of planar graphs. Our main result shows that, given a planar graph G with N vertices and an integer r> 0, a vertex separator of size O (N / √ r) that partitions G into O(N/r) subgraphs of size at most r and boundary s ..."
Abstract
 Add to MetaCart
We present I/Oefficient algorithms for computing optimal separator partitions of planar graphs. Our main result shows that, given a planar graph G with N vertices and an integer r> 0, a vertex separator of size O (N / √ r) that partitions G into O(N/r) subgraphs of size at most r and boundary size O ( √ r) can be computed in O(sort(N)) I/Os, provided that M ≥ 56r log 2 B. Together with the planar embedding algorithm presented in the companion paper [27], this result is the basis for I/Oefficient solutions to many other fundamental problems on planar graphs, including breadthfirst search and shortest paths [5, 8], depthfirst search [6, 9], strong connectivity [9], and topological sorting [8]. Our second result shows that, given I/Oefficient solutions to these problems, a general separator algorithm for graphs with costs and weights on their vertices [3] can be made I/Oefficient. Many classical separator theorems are special cases of this result. In particular, our I/Oefficient version allows the computation of a separator as produced by our first separator algorithm, but without placing any constraints on r.
Optimizing and Visualizing Planar Graphs via Rectangular Dualization
, 2009
"... Graphs are among the most studied and used mathematical tools for representing data and their relationships. The increasing number of applications exploiting their descriptive power makes graph visualization a key issue in modern graph theory and computer science. This is witnessed by the widespread ..."
Abstract
 Add to MetaCart
Graphs are among the most studied and used mathematical tools for representing data and their relationships. The increasing number of applications exploiting their descriptive power makes graph visualization a key issue in modern graph theory and computer science. This is witnessed by the widespread interest Graph Drawing has been able to excite in the research community over the past decade. Ever more often software offers a valuable support to key fields of science (such as chemistry, biology, physics and mathematics) in elaborating and interpreting huge quantities of data. Usually, however, a drawing explains the nature of a phenomenon better than hundreds of tables and sterile figures; that is why visualization is so important in modern Computer Science. Graph Drawing is an important part of Information Visualization addressing specific visualization problems on graphs. Most of the research work in Graph Drawing aims at creating representations complying with aesthetic criteria, relying on the principle that a drawing nice and pleasant to see is also effective. Difficulties arise when the size of graphs blows up; in this case appropriate optimization techniques must be used before visualization. One widely used optimization
A Note on the SelfSimilarity of Some Orthogonal Drawings ⋆ (Short Theory Paper)
"... Abstract. Large graphs are difficult to browse and to visually explore. This note adds up evidence that some graph drawing techniques, which produce readable layouts when applied to mediumsize graphs, yield selfsimilar patterns when launched on huge graphs. To prove this, we consider the problem of ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. Large graphs are difficult to browse and to visually explore. This note adds up evidence that some graph drawing techniques, which produce readable layouts when applied to mediumsize graphs, yield selfsimilar patterns when launched on huge graphs. To prove this, we consider the problem of assessing the selfsimilarity of graph drawings, and measure the boxcounting dimension of the output of three algorithms, each using a different approach for producing orthogonal grid drawings with a reduced number of bends. 1
New 3D Fourier Descriptors for Genuszero Mesh Objects
"... Abstract. The 2D Fourier Descriptor is an elegant and powerful technique for 2D shape analysis. This paper intends to extend such technique to 3D. Though conceptually natural, such an extension is not trivial in that two critical problems, the spherical parametrization and invariants construction, m ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. The 2D Fourier Descriptor is an elegant and powerful technique for 2D shape analysis. This paper intends to extend such technique to 3D. Though conceptually natural, such an extension is not trivial in that two critical problems, the spherical parametrization and invariants construction, must be solved. By using a newly developed surface parametrization method–the discrete conformal mapping (DCM)—we propose a 3D Fourier Descriptor (3DFD) for representing and recognizing arbitrarilycomplex genuszero mesh objects. A new DCM algorithm is suggested which solves the first problem efficiently. We also derive a method to construct a truly complete set of Spherical Harmonic invariants. The 3DFD descriptors have been tested on different complex mesh objects. Experiment results for shape representation are satisfactory. 1
International Journal, Pattern Recognition Conformal Spherical Representation of 3D GenusZero Meshes
"... This paper describes an approach of representing 3D shape by using a set of invariant Spherical Harmonic (SH) coefficients after conformal mapping. Specifically, a genuszero 3D mesh object is first conformally mapped onto the unit sphere by using a modified discrete conformal mapping, where the mod ..."
Abstract
 Add to MetaCart
(Show Context)
This paper describes an approach of representing 3D shape by using a set of invariant Spherical Harmonic (SH) coefficients after conformal mapping. Specifically, a genuszero 3D mesh object is first conformally mapped onto the unit sphere by using a modified discrete conformal mapping, where the modification is based on Möbius Factorization and is aimed at obtaining a canonical conformal mapping. Then a Spherical Harmonic Analysis is applied to the resulting conformal spherical meshes. The obtained SH coefficients are further made invariant to translation and rotation, while at the same time retain their completeness, so that the original shape information has been faithfully preserved.