A Simple Algorithm for Multiple-Source Shortest Paths in Planar Digraphs.

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.