发表机构
University of Science and Technology of China（中国科学技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明平面Delaunay三角剖分的伸展因子上界为1.65，通过Bellman表述和样条构造改进先前结果，缩小与下界的差距。
AI 中文摘要
Delaunay三角剖分是一类基本的平面生成子图，确定其最坏情况伸展因子一直是计算几何中的一个长期问题。我们证明了1.65的上界，改进了先前1.998的界，并将与已知下界1.5932的差距缩小了七倍以上。该结果适用于所有平面Delaunay三角剖分，包括具有共线或共圆点的配置。我们的主要贡献是证明了证明所依据的圆盘链界的Bellman表述。通过将最短路径长度与沿查询线段的加性进展进行比较，我们得到一个精确的递归，其状态仅记录当前圆盘、进入的弦以及两个前缀距离之差。我们证明，该链类的一个界成立当且仅当一个势函数满足三个局部不等式，分别用于初始化、转移和终止。相应的Bellman值函数是逐点最小的可行势函数，为构造上界提供了精确目标。我们使用一个单变量函数构造了这样的势函数。几何单调性将其可行性简化为关于该函数及其导数仿射的不等式。通过精确算术和严格区间界验证的样条构造，得到了1.65的伸展界。我们还给出了一个对偶证明，表明在相同条件下，任何可行的二次型都需要一个大于1.67的验证常数。
英文摘要
Delaunay triangulations are a fundamental class of plane spanners, and determining their worst-case stretch factor has been a longstanding problem in computational geometry. We prove an upper bound of \(1.65\), improving the bound of \(1.998\) due to Xia (2011) and reducing the gap to the known lower bound of \(1.5932\) by a factor of more than seven. Our proof works with the chains of circumdisks introduced by Xia, along which a path between two sites is assembled disk by disk. Xia measures such a path against a quantity attached to the whole chain, and because that quantity is not additive, his induction has to be carried alongside a separate global estimate. Our main idea is to measure the path against the progress it makes along the segment joining the two sites. This quantity is additive, so the bound becomes a Bellman recursion that forgets all but one number about the disks already passed, and we show that the bound holds if and only if a potential on the current state satisfies three local inequalities. The smallest feasible potential is the value function of that recursion, so searching for a potential becomes the problem of fitting this value function from above. The geometry of the disks reduces the fit to a linear program over functions of one variable, in which a GPT-based multi-agent system that we developed found a feasible point, certified in exact arithmetic.
