Algorithms for Sampling 3-Orientations of Planar Triangulations

Given a planar triangulation, a 3-orientation is an orientation of the internal edges so all internal vertices have out-degree three. Each 3-orientation gives rise to a unique edge coloring known as a Schnyder wood that has proven powerful for various computing and combinatorics applications. We consider natural Markov chains for sampling uniformly from the set of 3-orientations. First, we study a "triangle-reversing" chain on the space of 3-orientations of a fixed triangulation that reverses the orientation of the edges around a triangle in each move. It was shown previously that this chain connects the state space and we show that (i) when restricted to planar triangulations of maximum degree six, the Markov chain is rapidly mixing, and (ii) there exists a triangulation with high degree on which this Markov chain mixes slowly. Next, we consider an "edge-flipping" chain on the larger state space consisting of 3-orientations of all planar triangulations on a fixed number of vertices. It was also shown previously that this chain connects the state space and we prove that the chain is always rapidly mixing. The triangle-reversing and edge-flipping Markov chains both arise in the context of sampling other combinatorial structures, such as Eulerian orientations and triangulations of planar point sets, so our results here may shed light on the mixing rate of these related chains as well.

[1]  Nicolas Bonichon,et al.  Wagner's Theorem on Realizers , 2002, ICALP.

[2]  Dana Randall,et al.  Slow Mixing of Markov Chains Using Fault Lines and Fat Contours , 2010, Algorithmica.

[3]  Eric Vigoda,et al.  Accelerating simulated annealing for the permanent and combinatorial counting problems , 2006, SODA '06.

[4]  Dana Randall,et al.  Analyzing Glauber Dynamics by Comparison of Markov Chains , 1998, LATIN.

[5]  Stefan Felsner,et al.  Convex Drawings of Planar Graphs and the Order Dimension of 3-Polytopes , 2001, Order.

[6]  Páidí Creed.  Sampling Eulerian orientations of triangular lattice graphs , 2009, J. Discrete Algorithms.

[7]  Martin E. Dyer,et al.  A more rapidly mixing Markov chain for graph colorings , 1998, Random Struct. Algorithms.

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

[9]  Mark Jerrum,et al.  Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains , 1987, International Workshop on Graph-Theoretic Concepts in Computer Science.

[10]  Mark Jerrum,et al.  Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains , 1987, WG.

[11]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[12]  Eric Vigoda,et al.  A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries , 2001, STOC '01.

[13]  D. Wilson.  Mixing times of lozenge tiling and card shuffling Markov chains , 2001, math/0102193.

[14]  Stefan Felsner,et al.  On the Number of Planar Orientations with Prescribed Degrees , 2008, Electron. J. Comb..

[15]  Alistair Sinclair,et al.  Algorithms for Random Generation and Counting: A Markov Chain Approach , 1993, Progress in Theoretical Computer Science.

[16]  Eric Vigoda,et al.  A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries , 2004, JACM.

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

[18]  W. Schnyder.  Planar graphs and poset dimension , 1989 .

[19]  Leslie G. Valiant,et al.  Random Generation of Combinatorial Structures from a Uniform Distribution , 1986, Theor. Comput. Sci..

[20]  P. Diaconis,et al.  COMPARISON THEOREMS FOR REVERSIBLE MARKOV CHAINS , 1993 .

[21]  Prasad Tetali,et al.  On the mixing time of the triangulation walk and other Catalan structures , 1997, Randomization Methods in Algorithm Design.

[22]  Yi-Ting Chiang,et al.  Orderly spanning trees with applications to graph encoding and graph drawing , 2001, SODA '01.

[23]  Bruce A. Reed,et al.  On the mixing rate of the triangulation walk , 1997, Randomization Methods in Algorithm Design.

[24]  Nicolas Bonichon,et al.  A bijection between realizers of maximal plane graphs and pairs of non-crossing Dyck paths , 2005, Discret. Math..

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

[26]  Dana Randall,et al.  Mixing Times of Self-Organizing Lists and Biased Permutations , 2012, ArXiv.

[27]  Stefan Felsner,et al.  Geometric Graphs and Arrangements , 2004 .

[28]  Nicolas Bonichon,et al.  Watermelon uniform random generation with applications , 2003, Theor. Comput. Sci..

[29]  Peter Winkler,et al.  On the number of Eulerian orientations of a graph , 2005, Algorithmica.

[30]  Dana Randall,et al.  Markov Chain Algorithms for Planar Lattice Structures , 2001, SIAM J. Comput..