We consider the problem of computing an approximate weighted shortest path in a weighted subdivision, with weights assigned from the set $\{0, 1, \infty\}$. We present a data structure $B$, which stores a set of convex, non-overlapping regions. These include zero-cost regions (0-regions) with a weight of $0$ and obstacles with a weight of $\infty$, all embedded in a plane with a weight of $1$. The data structure $B$ can be constructed in expected time $O(N + (n/\varepsilon^3)(\log(n/\varepsilon) + \log N))$, where $n$ is the total number of regions, $N$ represents the total complexity of the regions, and $1 + \varepsilon$ is the approximation factor, for any $0 < \varepsilon < 1$. Using $B$, one can compute an approximate weighted shortest path from any point $s$ to any point $t$ in $O(N + n/\varepsilon^3 + (n/\varepsilon^2) \log(n/\varepsilon) + (\log N)/\varepsilon)$ time. In the special case where the 0-regions and obstacles are polygons (not necessarily convex), $B$ contains a $(1 + \varepsilon)$-spanner of the input vertices.

翻译：本文研究在加权剖分中计算近似加权最短路径的问题，其中权重取自集合 $\{0, 1, \infty\}$。我们提出一种数据结构 $B$，用于存储一组凸的、互不重叠的区域。这些区域包括权重为 $0$ 的零成本区域（0-区域）和权重为 $\infty$ 的障碍物，它们均嵌入在权重为 $1$ 的平面中。该数据结构 $B$ 的期望构建时间为 $O(N + (n/\varepsilon^3)(\log(n/\varepsilon) + \log N))$，其中 $n$ 为区域总数，$N$ 表示区域的总复杂度，$1 + \varepsilon$ 为近似因子（对于任意 $0 < \varepsilon < 1$）。利用 $B$，可在 $O(N + n/\varepsilon^3 + (n/\varepsilon^2) \log(n/\varepsilon) + (\log N)/\varepsilon)$ 时间内计算出从任意点 $s$ 到任意点 $t$ 的近似加权最短路径。在特殊情况下，当 0-区域和障碍物为多边形（不必为凸多边形）时，$B$ 包含输入顶点的一个 $(1 + \varepsilon)$-生成子图。