[Paper Review] Algorithms for Sampling 3-Orientations of Planar Triangulations

Sarah Miracle, Dana Randall+2 more|arXiv (Cornell University)|Feb 22, 2012

Computational Geometry and Mesh Generation27 references4 citations

TL;DR

This paper analyzes two Markov chains—triangle-reversing and edge-flipping—for sampling 3-orientations of planar triangulations. It proves that the edge-flipping chain is rapidly mixing for all planar triangulations with n internal vertices, while the triangle-reversing chain is rapidly mixing only when the maximum internal degree is at most six, but can be slow for high-degree triangulations.

ABSTRACT

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.

Motivation & Objective

• To analyze the mixing time of Markov chains for sampling 3-orientations in planar triangulations.
• To resolve an open problem posed by Felsner and Zickfeld on efficient sampling of 3-orientations.
• To determine under what conditions natural local moves (triangle-reversing and edge-flipping) yield rapidly mixing chains.
• To establish theoretical bounds on convergence speed for sampling 3-orientations using Markov chain Monte Carlo methods.
• To extend understanding of mixing rates in related combinatorial structures such as Eulerian orientations and triangulations.

Proposed method

• Proposes a triangle-reversing Markov chain that reverses edge orientations around a directed triangle, preserving vertex out-degrees.
• Introduces an edge-flipping Markov chain that operates across all planar triangulations with n vertices, allowing transitions between different triangulations.
• Uses canonical path analysis to bound the mixing time of the edge-flipping chain, establishing polynomial runtime guarantees.
• Applies comparison techniques between the triangle-reversing and cycle-reversing chains to derive mixing time bounds.
• Employs a decomposition of transition paths into stages (e.g., red/green edge swaps) to upper bound the number of canonical paths per edge.
• Leverages known bijections between 3-orientations and perfect matchings in bipartite graphs to inform sampling complexity.

Experimental results

Research questions

• RQ1 Under what conditions is the triangle-reversing Markov chain rapidly mixing for sampling 3-orientations of a fixed planar triangulation?
• RQ2 Is the edge-flipping Markov chain rapidly mixing when sampling across all planar triangulations with n internal vertices?
• RQ3 Can the mixing time of the triangle-reversing chain be bounded for triangulations with maximum internal degree Δ ≤ 6?
• RQ4 Does the mixing time of the triangle-reversing chain become exponential when the maximum degree is unbounded?
• RQ5 Can the canonical path method be applied effectively to bound the mixing time of the edge-flipping chain in the larger state space?

Key findings

• The edge-flipping Markov chain is rapidly mixing for the union of all planar triangulations with n internal vertices, with mixing time O(n^10 log(n/ε)).
• The triangle-reversing chain is rapidly mixing for triangulations with maximum internal degree at most six, with mixing time O(n^5 log(ε⁻¹)).
• For triangulations with maximum internal degree exceeding six, the triangle-reversing chain can mix slowly, requiring exponential time.
• The canonical path analysis shows that at most O(n^4) canonical paths use any given edge in the edge-flipping chain.
• The mixing time bound for the edge-flipping chain is derived using a combination of path decomposition, path length, and stationary distribution bounds.
• The results imply that the edge-flipping chain provides an efficient sampling algorithm for 3-orientations across all planar triangulations of fixed size.