Canonical ordering for graphs on the cylinder, with applications to periodic straight-line drawings on the flat cylinder and torus

We extend the notion of canonical ordering (initially developed for planar triangulations and 3-connected planar maps) to cylindric (essentially simple) triangulations and more generally to cylindric (essentially internally) $3$-connected maps. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case) and of Kant (in the $3$-connected case) to this setting. Precisely, for any cylindric essentially internally $3$-connected map $G$ with $n$ vertices, we can obtain in linear time a periodic (in $x$) straight-line drawing of $G$ that is crossing-free and internally (weakly) convex, on a regular grid $\mathbb{Z}/w\mathbb{Z}\times[0..h]$, with $w\leq 2n$ and $h\leq n(2d+1)$, where $d$ is the face-distance between the two boundaries. This also yields an efficient periodic drawing algorithm for graphs on the torus. Precisely, for any essentially $3$-connected map $G$ on the torus (i.e., $3$-connected in the periodic representation) with $n$ vertices, we can compute in linear time a periodic straight-line drawing of $G$ that is crossing-free and (weakly) convex, on a periodic regular grid $\mathbb{Z}/w\mathbb{Z}\times\mathbb{Z}/h\mathbb{Z}$, with $w\leq 2n$ and $h\leq 1+2n(c+1)$, where $c$ is the face-width of $G$. Since $c\leq\sqrt{2n}$, the grid area is $O(n^{5/2})$.Comment: 37 page

Schnyder woods for higher genus triangulated surfaces, with applications to encoding

Schnyder woods are a well-known combinatorial structure for plane triangulations, which yields a decomposition into 3 spanning trees. We extend here definitions and algorithms for Schnyder woods to closed orientable surfaces of arbitrary genus. In particular, we describe a method to traverse a triangulation of genus $g$ and compute a so-called $g$-Schnyder wood on the way. As an application, we give a procedure to encode a triangulation of genus $g$ and $n$ vertices in $4n+O(g \log(n))$ bits. This matches the worst-case encoding rate of Edgebreaker in positive genus. All the algorithms presented here have execution time $O((n+g)g)$, hence are linear when the genus is fixed.Comment: 27 pages, to appear in a special issue of Discrete and Computational Geometr

Fast Spherical Drawing of Triangulations: An Experimental Study of Graph Drawing Tools

We consider the problem of computing a spherical crossing-free geodesic drawing of a planar graph: this problem, as well as the closely related spherical parameterization problem, has attracted a lot of attention in the last two decades both in theory and in practice, motivated by a number of applications ranging from texture mapping to mesh remeshing and morphing. Our main concern is to design and implement a linear time algorithm for the computation of spherical drawings provided with theoretical guarantees. While not being aesthetically pleasing, our method is extremely fast and can be used as initial placer for spherical iterative methods and spring embedders. We provide experimental comparison with initial placers based on planar Tutte parameterization. Finally we explore the use of spherical drawings as initial layouts for (Euclidean) spring embedders: experimental evidence shows that this greatly helps to untangle the layout and to reach better local minima

Schnyder woods for higher genus triangulated surfaces

The final version of this extended abstract has been published in "Discrete and Computational Geometry (2009)"International audienceSchnyder woods are a well known combinatorial structure for planar graphs, which yields a decomposition into 3 vertex-spanning 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

Dynamic updates of succinct triangulations

In a recent article, we presented a succinct representation of triangulations that supports efficient navigation operations. Here this representation is improved to allow for efficient local updates of the triangulations. Precisely, we show how a succinct representation of a triangulation with $m$ triangles can be maintained under vertex insertions in $O(1)$ amortized time and under vertex deletions/edge flips in $O(lg^2 m)$ amortized time. Our structure achieves the information theory bound for the storage for the class of triangulations with a boundary, requiring asymptotically $2.17m+o(m)$ bits, and supports adjacency queries between triangles in $O(1)$ time (an extra amount of $O(g lgm)$ bits are needed for representing triangulations of genus $g$ surfaces)

Canonical Ordering for Triangulations on the Cylinder, with Applications to Periodic Straight-line Drawings

International audienceWe extend the notion of canonical orderings to cylindric triangulations. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack to this setting. Our algorithm yields in linear time a crossing-free straight-line drawing of a cylindric triangulation $G$ with $n$ vertices on a regular grid \mZ/w\mZ\times[0..h], with $w\leq 2n$ and $h\leq n(2d+1)$, where $d$ is the (graph-) distance between the two boundaries. As a by-product, we can also obtain in linear time a crossing-free straight-line drawing of a toroidal triangulation with $n$ vertices on a periodic regular grid \mZ/w\mZ\times\mZ/h\mZ, with $w\leq 2n$ and $h\leq 1+n(2c+1)$, where $c$ is the length of a shortest non-contractible cycle. Since $c\leq\sqrt{2n}$, the grid area is $O(n^{5/2})$. Our algorithms apply to any triangulation (whether on the cylinder or on the torus) that have no loops nor multiple edges in the periodic representation

2D Triangulation Representation Using Stable Catalogs

The problem of representing triangulations has been widely studied to obtain convenient encodings and space efficient data structures. In this paper we propose a new practical approach to reduce the amount of space needed to represent in main memory an arbitrary triangulation, while maintaining constant time for some basic queries. This work focuses on the connectivity information of the triangulation, rather than the geometry information (vertex coordinates), since the combinatorial data represents the main storage part of the structure. The main idea is to gather triangles into patches, to reduce the number of pointers by eliminating the internal pointers in the patches and reducing the multiple references to vertices. To accomplish this, we define stable catalogs of patches that are close under basic standard update operations such as insertion and deletion of vertices, and edge flips. We present some bounds and results concerning special catalogs, and some experimental results for the quadrilateral-triangle catalog

Compact data structures for triangulations

International audienceThe main problem consists in designing space-efficient data structures allowing to represent the connectivity of triangle meshes while supporting fast navigation and local updates

ESQ: Editable SQuad Representation for Triangle Meshes

International audienceWe consider the problem of designing space efficient solutions for representing the connectivity information of manifold triangle meshes. Most mesh data structures are quite redundant, storing a large amount of information in order to efficiently support mesh traversal operators. Several compact data structures have been proposed to reduce storage cost while supporting constant-time mesh traversal. Some recent solutions are based on a global re-ordering approach, which allows to implicitly encode a map between vertices and faces. Unfortunately, these compact representations do not support efficient updates, because local connectivity changes (such as edge-contractions, edge-flips or vertex insertions) require re-ordering the entire mesh. Our main contribution is to propose a new way of designing compact data structures which can be dynamically maintained. In our solution, we push further the limits of the re-ordering approaches: the main novelty is to allow to re-order vertex data (such as vertex coordinates), and to exploit this vertex permutation to easily maintain the connectivity under local changes. We describe a new class of data structures, called Editable SQuad (ESQ), offering the same navigational and storage performance as previous works, while supporting local editing in amortized constant time. As far as we know, our solution provides the most compact dynamic data structure for triangle meshes. We propose a linear-time and linear-space construction algorithm, and provide worst-case bounds for storage and time cost.Cet article traite de la conception de structure de données usant peu de mémoire pour représenter des surfaces manifold triangulées. La plupart des structures utilisées sont largement redondantes pour permettre un parcours efficace des adjacences entre triangles. Par ailleurs il existe des structures compactes, basées sur une renumérotation qui code de manière implicite une correspondance entre faces et sommets. Malheureusement, ces structures ne permettent pas de modifier la triangulation car des opérations telles que insertion suppression ou bascule d'arête nécessite de renuméroter toute la triangulation. Nous proposons une nouvelle méthode de conception de structures de données compactes permettant une mise à jour dynamique en adaptant l'idée de renumérotation. Nous introduisons Editab SQuad (ESQ), une nouvelle famille de structures de données qui a les mêmes performances de stockage et de temps d'accés que les précédents travaux tout en permettant des modifications locales en temps constant amorti

Compact representation of triangulations

We consider the problem of representing compact geometric data structures maintaining an efficient implementation of navigation operations. For the case of planar triangulations with $m$ faces, we propose a compact representation of the connectivity information that improves to $2.175$ bits per triangle the asymptotic amount of space required and that supports navigation between adjacent triangles in constant time. For triangulations with $m$ faces of a surface with genus $g$, our representation requires asymptotically an extra amount of 36(g-1)\lgm bits. The structure also allows constant time access to vertex specific data, like coordinates, but the paper does not address the compression of this geometric information