Improved Bounds for Shortest Paths in Dense Distance Graphs

TL;DR

This paper presents an improved algorithm for computing shortest paths in dense distance graphs of planar graphs, significantly enhancing efficiency over previous methods and impacting various planar graph algorithms.

Contribution

The authors develop a faster algorithm for shortest path computation in dense distance graphs, improving upon the FR-Dijkstra algorithm for polynomial r, with implications for multiple planar graph problems.

Findings

Faster shortest path algorithm with improved time complexity

Enhanced bounds for multiple-source maximum flow in planar graphs

Implications for dynamic distance oracles in planar graphs

Abstract

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.