In the celebrated paper of Henzinger, Klein, Rao and Subramanian (1997), it was shown that planar graphs admit a linear time single-source shortest path algorithm. Their algorithm unfortunately does not extend to Euclidean graph classes. We give criteria and prove that any Euclidean graph class satisfying the criteria admits a linear time single-source shortest path algorithm. As a main ingredient, we show that the contracted graphs of these Euclidean graph classes admit sublinear separators.

翻译：在Henzinger、Klein、Rao和Subramanian (1997)的著名论文中，证明了平面图存在线性时间的单源最短路径算法。然而，他们的算法无法推广到欧几里得图类。我们给出判定准则，并证明任何满足这些准则的欧几里得图类均存在线性时间的单源最短路径算法。作为核心要素，我们证明这些欧几里得图类的收缩图存在亚线性分离子。