Improved Bounds for Shortest Paths in Dense Distance Graphs

We study the problem of computing shortest paths in so-called dense distance graphs. Every planar graph $G$ on $n$ vertices can be partitioned into a set of $O(n/r)$ edge-disjoint regions (called an $r$-division) with $O(r)$ vertices each, such that each region has $O(\sqrt{r})$ vertices (called boundary vertices) in common with other regions. A dense distance graph of a region is a complete graph containing all-pairs distances between its boundary nodes. A dense distance graph of an $r$-division is the union of the $O(n/r)$ dense distance graphs of the individual pieces. Since the introduction of dense distance graphs by Fakcharoenphol and Rao, computing single-source shortest paths in dense distance graphs has found numerous applications in fundamental planar graph algorithms. Fakcharoenphol and Rao proposed an algorithm (later called FR-Dijkstra) for computing single-source shortest paths in a dense distance graph in $O\left(\frac{n}{\sqrt{r}}\log{n}\log{r}\right)$ time. We show an $O\left(\frac{n}{\sqrt{r}}\left(\frac{\log^2{r}}{\log^2\log{r}}+\log{n}\log^{\epsilon}{r}\right)\right)$ time algorithm for this problem, which is the first improvement to date over FR-Dijkstra for the important case when $r$ is polynomial in $n$. In this case, our algorithm is faster by a factor of $O(\log^2{\log{n}})$ and implies improved upper bounds for such planar graph problems as multiple-source multiple-sink maximum flow, single-source all-sinks maximum flow, and (dynamic) exact distance oracles.