Faster Shortest Paths in Dense Distance Graphs, with Applications

TL;DR

This paper improves shortest path algorithms in dense distance graphs of planar graphs by removing a logarithmic factor, leading to faster solutions with applications in maximum flow and regional computations.

Contribution

It combines two existing techniques to eliminate the log n factor in shortest path algorithms for dense distance graphs, enhancing efficiency in planar graph problems.

Findings

Shortest-path algorithm runtime reduced to O(n r^{-1/2} log^2 r)

Maximum flow in planar graphs computed in O(n log p) time

Faster regional dense graph computations for small boundary vertices

Abstract

We show how to combine two techniques for efficiently computing shortest paths in directed planar graphs. The first is the linear-time shortest-path algorithm of Henzinger, Klein, Subramanian, and Rao [STOC'94]. The second is Fakcharoenphol and Rao's algorithm [FOCS'01] for emulating Dijkstra's algorithm on the dense distance graph (DDG). A DDG is defined for a decomposition of a planar graph $G$ into regions of at most $r$ vertices each, for some parameter $r < n$. The vertex set of the DDG is the set of $\Theta(n/\sqrt r)$ vertices of $G$ that belong to more than one region (boundary vertices). The DDG has $\Theta(n)$ arcs, such that distances in the DDG are equal to the distances in $G$. Fakcharoenphol and Rao's implementation of Dijkstra's algorithm on the DDG (nicknamed FR-Dijkstra) runs in $O(n\log(n) r^{-1/2} \log r)$ time, and is a key component in many state-of-the-art planar graph algorithms for shortest paths, minimum cuts, and maximum flows. By combining these two techniques we remove the $\log n$ dependency in the running time of the shortest-path algorithm, making it $O(n r^{-1/2} \log^2r)$. This work is part of a research agenda that aims to develop new techniques that would lead to faster, possibly linear-time, algorithms for problems such as minimum-cut, maximum-flow, and shortest paths with negative arc lengths. As immediate applications, we show how to compute maximum flow in directed weighted planar graphs in $O(n \log p)$ time, where $p$ is the minimum number of edges on any path from the source to the sink. We also show how to compute any part of the DDG that corresponds to a region with $r$ vertices and $k$ boundary vertices in $O(r \log k)$ time, which is faster than has been previously known for small values of $k$.