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 $D(S)$ is a graph that is defined on a set S of point sites in ${\mathbb{R}}^2$, where each site $s\in S$ has an associated radius $r_s$. The vertex set of $D(S)$ is S and two sites $s,t$ are connected by an edge st in $D(S)$ 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.

最短路径问题是图论中的基本问题之一。民间传说一般图中的未加权单源最短路径 ( SSSP ) 问题可以通过广度优先搜索 ( BFS ) 在$欧(n+米)$时间。在本文中，我们开发了一个算法框架，该框架概括了批处理BFS方法，为多个图类提供高效的SSSP算法。这些算法的运行时间取决于三个主要成分的运行时间。第一个是预处理步骤，定义一个保持一些距离信息的快捷图。然后，在算法的一次运行中，重复执行以下步骤：在快捷图中有效地找到一组与给定顶点集相邻的候选顶点，并最终找到在原始图中实际形成边的候选顶点的子集。

磁盘图 $丁(\mathrm{小号})$是定义在一组S点站点上的图${\mathbb{R}}^{\mathrm{2个}}$，其中每个站点$秒\in \mathrm{小号}$有关联的半径$r_{秒}$. 的顶点集$丁(\mathrm{小号})$是S和两个站点$秒,吨$由一条边st in连接$丁(\mathrm{小号})$当且仅当由s和t诱导的圆盘相交。这些图也称为圆盘的交集图。我们的结果是使用该框架有效解决相交图中SSSP问题的算法。对于磁盘图${\mathrm{大号}}_{\mathrm{2个}}$-metric，我们可以证明之后$欧(n{\mathrm{日志}}^{\mathrm{2个}}n)$预处理时间我们可以解决SSSP问题$欧(n\mathrm{日志}n)$时间。这显着提高了以前的最佳界限$欧(n{\mathrm{日志}}^{\mathrm{4个}}n)$[1]、[2]。在轴平行正方形的交叉图的情况下，我们甚至能够将预处理时间减少到最佳$欧(n\mathrm{日志}n)$. 由于轴平行正方形的交图等价于圆盘图${\mathrm{大号}}_{\mathrm{1个}}$- 和${\mathrm{大号}}_{\infty }$-metric 结果转移到这些指标。

为了展示我们框架的进一步应用，我们在我们的框架中重述了经典的BFS以及 Chan 和 Skrepetos [3] 的单位磁盘图的最优SSSP算法，展示了它的稳健性。