The problem of finding a diagonal flip path between two triangulations has been studied for nearly a century in the combinatorial (topological) setting and for decades in the geometric setting. In the geometric setting, finding a diagonal flip path between two triangulations that minimizes the number of diagonal flips over all such paths is NP-complete. However, when restricted to lattice triangulations - i.e. triangulations of a finite subset of the integer lattice, or an affine transformation of this lattice, bounded by a simple, closed polygon with lattice points as vertices - the problem has a polynomial time algorithm. Lattice triangulations have been studied for their uses in discriminant theory, Hilbert's 16th problem, toric varieties, quantum spin systems, and material science. Our first main result shows that there is a polynomial-time computable, unique partially-ordered set of diagonal flips such that there is a bijection between valid linear-orderings of this set and minimum diagonal flip paths between two lattice triangulations. This provides an alternative proof of the previously known result, as well as new structural insights into these diagonal flip paths. Our second main result characterizes pairs of triangulations, containing sets of edges $E$ and $E'$ respectively, such that the minimum diagonal flip path between them contains the minimum number of diagonal flips over all minimum diagonal flip paths between pairs of triangulations, containing sets of edges $E$ and $E'$ respectively. Remarkably, all of our results are derived from a simple relationship between edges in triangulations and Farey sequences.

中文翻译：

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翻转格子三角剖分之间的路径

近一个世纪以来，在组合（拓扑）设置中和几何设置中研究了在两个三角剖分之间寻找对角翻转路径的问题。在几何设置中，在两个三角剖分之间找到一条对角翻转路径来最小化所有这些路径上的对角翻转次数是 NP 完全的。然而，当仅限于格三角剖分时 - 即整数格子的有限子集的三角剖分，或该格子的仿射变换，由以格点为顶点的简单闭合多边形为界 - 问题具有多项式时间算法。格三角剖分已被研究用于判别理论、希尔伯特第 16 个问题、复曲面变体、量子自旋系统和材料科学。我们的第一个主要结果表明，存在多项式时间可计算的唯一部分有序对角翻转集，因此该集的有效线性排序与两个格子三角剖分之间的最小对角翻转路径之间存在双射。这提供了先前已知结果的替代证明，以及对这些对角翻转路径的新结构见解。我们的第二个主要结果表征了三角测量对，分别包含一组边 $E$ 和 $E'$，这样它们之间的最小对角翻转路径包含三角测量对之间所有最小对角翻转路径上的最小对角翻转次数，分别包含边的集合 $E$ 和 $E'$。值得注意的是，