Distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the distance between any two vertices $u$ and $v$ can be determined efficiently by merely inspecting the labels of $u$ and $v$, without using any other information. Similarly, routing labeling schemes label the vertices of a graph in a such a way that given the labels of a source node and a destination node, it is possible to compute efficiently the port number of the edge from the source that heads in the direction of the destination. One of important problems is finding natural classes of graphs admitting distance and/or routing labeling schemes with labels of polylogarithmic size. In this paper, we show that the class of cube-free median graphs on $n$ nodes enjoys distance and routing labeling schemes with labels of $O(\log^3 n)$ bits.

中文翻译：

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无立方体中值图的距离和路由标记方案

距离标记方案是用短标签标记图的顶点的方案，这样任何两个顶点 $u$ 和 $v$ 之间的距离可以有效地通过仅检查 $u$ 和 $v$ 的标签来确定，不使用任何其他信息。类似地，路由标记方案以这样一种方式标记图的顶点，即给定源节点和目标节点的标签，可以有效地计算从源节点到目标节点的边的端口号目的地。一个重要的问题是找到允许距离和/或路由标记方案的图的自然类别，其中标签具有多对数大小。在本文中，我们展示了 $n$ 节点上的无立方体中值图类享有距离和路由标记方案，标签为 $O(\log^3 n)$ 位。