Computing a shortest path between two nodes in an undirected unweighted graph is among the most basic algorithmic tasks. Breadth first search solves this problem in linear time, which is clearly also a lower bound in the worst case. However, several works have shown how to solve this problem in sublinear time in expectation when the input graph is drawn from one of several classes of random graphs. In this work, we extend these results by giving sublinear time shortest path (and short path) algorithms for expander graphs. We thus identify a natural deterministic property of a graph (that is satisfied by typical random regular graphs) which suffices for sublinear time shortest paths. The algorithms are very simple, involving only bidirectional breadth first search and short random walks. We also complement our new algorithms by near-matching lower bounds.

翻译：计算无向无权图中两个节点之间的最短路径是最基本的算法任务之一。广度优先搜索在线性时间内解决此问题，这在最坏情况下显然是下界。然而，已有研究表明，当输入图来自若干类随机图时，可以在期望次线性时间内解决该问题。在本工作中，我们通过为扩展图提供次线性时间最短路径（和短路径）算法，扩展了这些结果。因此，我们识别出图的一种自然确定性性质（典型的随机正则图满足该性质），它足以实现次线性时间最短路径。这些算法非常简单，仅涉及双向广度优先搜索和短随机游走。我们还通过近乎匹配的下界来补充我们的新算法。