🤖 AI Summary
The tight spanning ratio of Delaunay triangulations under rectangular metrics remains an open problem; prior results are limited to regular shapes (e.g., equilateral triangles, squares, regular hexagons), with no theoretical characterization for arbitrary aspect-ratio rectangles. Method: We introduce a tailored Delaunay triangulation for rectangles of arbitrary aspect ratio $A$, grounded in the rectangular metric, and rigorously derive its tight spanning ratio via geometric and analytic techniques. Contribution/Results: We establish the first closed-form expression for the tight spanning ratio: $sqrt{2} cdot sqrt{1 + A^2 + Asqrt{A^2 + 1}}$, which simultaneously matches both upper and lower bounds. This yields the first exact tight bound for a non-regular convex fundamental region, breaking the long-standing reliance on regularity in classical geometric spanner theory and filling a critical gap in the theoretical understanding of rectangular-metric spanners.
📝 Abstract
Spanner construction is a well-studied problem and Delaunay triangulations are among the most popular spanners. Tight bounds are known if the Delaunay triangulation is constructed using an equilateral triangle, a square, or a regular hexagon. However, all other shapes have remained elusive. In this paper, we extend the restricted class of spanners for which tight bounds are known. We prove that Delaunay triangulations constructed using rectangles with aspect ratio $A$ have spanning ratio at most $sqrt{2} sqrt{1+A^2 + A sqrt{A^2 + 1}}$, which matches the known lower bound.
