Optimal encoding of triangular and quadrangular meshes with fixed topology

Extending a bijection recently introduced by Poulalhon and Schaeffer [15] for triangulations of the sphere we design an efficient algorithm for encoding (topological) triangulations and bipartite quadrangulations on an orientable surface of fixed topology τ (given by the genus g and number of boundaries b). To our knowledge, our encoding procedure is the first to be asymptotically optimal (in the information theory sense) with respect to two natural parameters, the number n of inner vertices and the number k of boundary vertices.

[1]  W. G. Brown.  Enumeration of Triangulations of the Disk , 1964 .

[2]  H. de Fraysseix,et al.  On topological aspects of orientations , 2001, Discret. Math..

[3]  Colin McDiarmid,et al.  Random graphs on surfaces , 2008, J. Comb. Theory, Ser. B.

[4]  Éric Fusy.  Combinatoire des cartes planaires et applications algorithmiques , 2007 .

[5]  W. T. Tutte,et al.  A Census of Planar Triangulations , 1962, Canadian Journal of Mathematics.

[6]  W. G. Brown,et al.  Enumeration of Quadrangular Dissections of the Disk , 1965, Canadian Journal of Mathematics.

[7]  Martin Kutz,et al.  Computing shortest non-trivial cycles on orientable surfaces of bounded genus in almost linear time , 2005, SCG '06.

[8]  Dominique Poulalhon,et al.  Optimal Coding and Sampling of Triangulations , 2003, Algorithmica.

[9]  Jarek Rossignac,et al.  Efficient Edgebreaker for surfaces of arbitrary topology , 2004, Proceedings. 17th Brazilian Symposium on Computer Graphics and Image Processing.

[10]  Stefan Felsner,et al.  Lattice Structures from Planar Graphs , 2004, Electron. J. Comb..

[11]  Jarek Rossignac,et al.  Edgebreaker: Connectivity Compression for Triangle Meshes , 1999, IEEE Trans. Vis. Comput. Graph..

[12]  Olivier Devillers,et al.  Succinct Representation of Triangulations with a Boundary , 2005, WADS.

[13]  Olivier Bernardi,et al.  Bijective Counting of Tree-Rooted Maps and Shuffles of Parenthesis Systems , 2006, Electron. J. Comb..

[14]  Ulrik Brandes,et al.  A Linear Time Algorithm for the Arc Disjoint Menger Problem in Planar Directed Graphs , 2000, Algorithmica.

[15]  Maxim Krikun,et al.  Explicit Enumeration of Triangulations with Multiple Boundaries , 2007, Electron. J. Comb..

[16]  Éric Fusy,et al.  A unified bijective method for maps: application to two classes with boundaries , 2010 .

[17]  Walter Schnyder,et al.  Embedding planar graphs on the grid , 1990, SODA '90.