We study the quality of weighted shortest paths when a continuous 2-dimensional space is discretized by a weighted triangular tessellation. In order to evaluate how well the tessellation approximates the 2-dimensional space, we study three types of shortest paths: a weighted shortest path $ \mathit{SP_w}(s,t) $, which is a shortest path from $ s $ to $ t $ in the space; a weighted shortest vertex path $ \mathit{SVP_w}(s,t) $, which is an any-angle shortest path; and a weighted shortest grid path $ \mathit{SGP_w}(s,t) $, which is a shortest path whose edges are edges of the tessellation. Given any arbitrary weight assignment to the faces of a triangular tessellation, thus extending recent results by Bailey et al. [Path-length analysis for grid-based path planning. Artificial Intelligence, 301:103560, 2021], we prove upper and lower bounds on the ratios $ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} $, $ \frac{\lVert \mathit{SVP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} $, $ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SVP_w}(s,t)\rVert} $, which provide estimates on the quality of the approximation. It turns out, surprisingly, that our worst-case bounds are independent of any weight assignment. Our main result is that $ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} = \frac{2}{\sqrt{3}} \approx 1.15 $ in the worst case, and this is tight. As a corollary, for the weighted any-angle path $ \mathit{SVP_w}(s,t) $ we obtain the approximation result $ \frac{\lVert \mathit{SVP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} \lessapprox 1.15 $.

翻译：本文研究了连续二维空间被加权三角网格离散化时，加权最短路径的质量。为评估网格对二维空间的逼近程度，我们研究了三种最短路径类型：加权最短路径 $ \mathit{SP_w}(s,t) $（即空间中的实际最短路径）、加权最短顶点路径 $ \mathit{SVP_w}(s,t) $（任意角度最短路径），以及加权最短网格路径 $ \mathit{SGP_w}(s,t) $（其边必须沿网格边的路径）。通过将任意权重分配至三角网格的面元，从而扩展了Bailey等人[Path-length analysis for grid-based path planning. Artificial Intelligence, 301:103560, 2021]的最新研究成果，我们证明了比值 $ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} $、$ \frac{\lVert \mathit{SVP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} $ 和 $ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SVP_w}(s,t)\rVert} $ 的上界与下界，这些界提供了逼近质量的估计。令人惊讶的是，我们的最坏情况界与任何权重分配无关。主要研究结果表明，在最坏情况下 $ \frac{\lVert \mathit{SGP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} = \frac{2}{\sqrt{3}} \approx 1.15 $，且该界是紧的。作为推论，对于加权任意角度路径 $ \mathit{SVP_w}(s,t) $，我们得到逼近结果 $ \frac{\lVert \mathit{SVP_w}(s,t)\rVert}{\lVert \mathit{SP_w}(s,t)\rVert} \lessapprox 1.15 $。