arXiv:2609.07979 · v6
The Stretch Factor of Planar Delaunay Triangulations Is Nearly Determined
摘要
The Delaunay triangulation is a classical geometric structure on a finite set of points in the plane. Its stretch factor is the largest ratio of the shortest-path distance along its edges to the Euclidean distance between the same two points. Determining the worst-case stretch factor of planar Delaunay triangulations is a long-standing open problem. We prove a uniform upper bound of 1.5949, improving the previous bound of 1.998 and narrowing the gap to the best known lower bound of 1.5932 from 0.4048 to less than 0.0017, nearly pinpointing the worst-case stretch factor. A central difficulty is that the line segment between two points may cross arbitrarily many triangles. Our progress accounting framework addresses this by converting the global bound into a family of local inequalities, each involving only a fixed number of real parameters regardless of the number of triangles crossed. Two further geometric lemmas make computer-assisted verification tractable: the pencil lemma decomposes each local transition into two geometrically independent stages, reducing verification over infinitely many possible next disks to a single differential condition; and the corner lemmas handle degenerate boundary cases where direct verification would fail. With these reductions, the proof ends with a finite verification of an explicit potential in exact rational arithmetic and outward-rounded interval arithmetic. Numerous geometric network constructions use the Delaunay stretch factor as a black box, so the tighter bound directly improves stretch-factor guarantees for a range of downstream applications, including bounded-degree plane spanners and low-weight spanners.
作者备注：67 pages, 14 figures, 8 tables. Substantially revised: the upper bound is improved from 1.65 to 1.5949, and the title has changed. Verification data and code: https://doi.org/10.5281/zenodo.23835635