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：在无权无向图中计算两个节点之间的最短路径是最基本的算法任务之一。广度优先搜索以线性时间解决此问题，这显然也是最坏情况下的下界。然而，多项研究已表明，当输入图来自某几类随机图分布时，可以在期望意义下以次线性时间解决该问题。本文通过提出适用于展开图的次线性时间最短路径（及短路径）算法，扩展了上述结果。由此，我们识别出图的自然确定性性质（典型随机正则图满足此性质），该性质足以支撑次线性时间最短路径计算。所提出的算法极为简单，仅涉及双向广度优先搜索和短随机游走。此外，我们还通过近乎匹配的下界对算法进行补充。