平面三角剖分3-方向采样算法
Algorithms for Sampling 3-Orientations of Planar Triangulations
摘要：
给定一个平面三角剖分，3-定向是内部边的定向，因此所有内部顶点的出度为3。每个3-方向产生一个独特的边着色，称为Schnyder木材，已被证明是各种计算和组合数学应用的强大工具。我们考虑自然马尔可夫链的采样均匀从3-方向的集合。首先，我们研究了一个“三角形反转”链上的3个方向的一个固定的三角形，反转的方向周围的三角形的边缘在每一个移动的空间。之前已经证明，该链连接状态空间，并且我们证明：（i）当限制为最大度数为六的平面三角形时，马尔可夫链快速混合，以及（ii）存在一个高度数的三角形，该马尔可夫链在该三角形上混合缓慢。接下来，我们考虑一个“边翻转”链的更大的状态空间，由3个方向的所有平面三角形上的一个固定数量的顶点。之前也表明，这个链连接状态空间，我们证明了链总是快速混合。三角反转和边翻转马尔可夫链都出现在其他组合结构的采样背景下，如欧拉方向和平面点集的三角剖分，所以我们的结果可以揭示这些相关链的混合率。
英文摘要：
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.
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