An algorithmic framework for the single source shortest path problem with applications to disk graphs

Abstract

Shortest path problems are among the fundamental problems in graph theory. It is folklore that the unweighted single source shortest path (SSSP) problem in general graphs can be solved optimally with breadth first search (BFS) in $O(n+m)$ time. In this paper, we develop an algorithmic framework that generalizes a batched BFS approach to give efficient SSSP algorithms for several graph classes. The running time of these algorithms depends on the running time of three main ingredients. The first is a preprocessing step, to define a shortcut graph that maintains some distance information. Then during one run of the algorithm repeatably there are the steps of efficiently finding a set of candidate vertices adjacent in the shortcut graph to a given set of vertices and finally finding the subset of the candidate vertices that actually form an edge in the original graph.

A disk graph $D\left(S\right)$ is a graph that is defined on a set S of point sites in $R^2$, where each site $s\in S$ has an associated radius $r_s$. The vertex set of $D\left(S\right)$ is S and two sites $s,t$ are connected by an edge st in $D\left(S\right)$ if and only if the disks induced by s and t intersect. These graphs are also called the intersection graph of disks. Our results are algorithms that use the framework to efficiently solve the SSSP problem in intersection graphs. For disk graphs in the $L_2$-metric, we can show that after $O(n{\log }^{2}n)$ preprocessing time we can solve the SSSP problem in $O(n\log n)$ time. This significantly improves the previous best bound of $O(n{\log }^{4}n)$ [1], [2]. In the case of intersection graphs of axis-parallel squares, we are even able to reduce the preprocessing time to an optimal $O(n\log n)$. As intersection graphs of axis parallel squares are equivalent to disk graphs in the $L_1$- and $L_{\infty }$-metric the result carries over to these metrics.

To show further applications of our framework, we restate the classical BFS, and also the optimal SSSP algorithm for unit disk graphs by Chan and Skrepetos [3] in our framework, showing its robustness.