凸点集的贪心三角剖分的跨越比
On the Spanning Ratio of the Greedy Triangulation for Convex Point Sets
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中文总结 AI 辅助
该研究针对凸点集的贪心三角剖分，将其跨越比的标准界从约11739.1降至小于17.814，证明其为18- spanner。
中文摘要 AI 辅助
有限平面点集的贪心三角剖分通过按长度非递减顺序处理所有线段，插入不与已插入线段相交的线段得到。已知其跨越比有通用常数界，但由菱形和良好多边形性质得到的标准界约为11739.1。本文针对凸位置的点集证明了小得多的界：对任意凸位置的有限点集P⊂ℝ²及任意u,v∈P，贪心三角剖分中存在u到v的路径长度不超过κ|uv|，其中κ<17.814，故凸点集的贪心三角剖分是18- spanner（18- spanner指支撑度为18的图结构）。
英文摘要
The greedy triangulation of a finite planar point set is obtained by considering all segments in nondecreasing order of length and inserting each segment that does not cross an earlier one. Its spanning ratio is known to be bounded by a universal constant, but the standard bound obtained from the diamond and good-polygon properties is about $11739.1$. We prove a substantially smaller bound for points in convex position. In particular, for every finite point set $P\subset\mathbb{R}^2$ in convex position and every pair $u,v\in P$, the greedy triangulation contains a $u$--$v$ path of length at most $κ|uv|$, where $κ<17.814$. Thus, the greedy triangulation of a convex point set is an $18$-spanner.
