Given an $n$-vertex planar embedded digraph $G$ with non-negative edge weights and a face $f$ of $G$, Klein presented a data structure with $O(n\log n)$ space and preprocessing time which can answer any query $(u,v)$ for the shortest path distance in $G$ from $u$ to $v$ or from $v$ to $u$ in $O(\log n)$ time, provided $u$ is on $f$. This data structure is a key tool in a number of state-of-the-art algorithms and data structures for planar graphs. Klein's data structure relies on dynamic trees and the persistence technique as well as a highly non-trivial interaction between primal shortest path trees and their duals. The construction of our data structure follows a completely different and in our opinion very simple divide-and-conquer approach that solely relies on Single-Source Shortest Path computations and contractions in the primal graph. Our space and preprocessing time bound is $O(n\log |f|)$ and query time is $O(\log |f|)$ which is an improvement over Klein's data structure when $f$ has small size.

翻译：鉴于Klein提出了一个带有非负边缘重量的GG美元和面值为G$美元的数据结构,Klein展示了一个带有O(n\log n)美元空间和预处理时间的数据结构,该结构可以回答任何查询$(u,v)美元,用于最短路径距离,从美元到美元或从美元到美元或从美元到美元到美元(log n)时间的最短路径距离,条件是美元为美元。这个数据结构是规划图中一些最先进的算法和数据结构中的一个关键工具。Klein的数据结构依赖于动态树和持久性技术,以及原始最短路径树及其两侧之间的高度非三角互动。我们数据结构的构建遵循完全不同的、我们认为完全依赖单一来源短路径计算和原始图中的收缩的非常简单的分化方法。我们的空间和预处理时间绑定是$(nlog\\ f)$(nlog\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\1\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\