A permutation graph is the intersection graph of a set of segments between two parallel lines. In other words, they are defined by a permutation $\pi$ on $n$ elements, such that $u$ and $v$ are adjacent if an only if $u<v$ but $\pi(u)>\pi(v)$. We consider the problem of computing the distances in such a graph in the setting of informative labeling schemes. The goal of such a scheme is to assign a short bitstring $\ell(u)$ to every vertex $u$, such that the distance between $u$ and $v$ can be computed using only $\ell(u)$ and $\ell(v)$, and no further knowledge about the whole graph (other than that it is a permutation graph). This elegantly captures the intuition that we would like our data structure to be distributed, and often leads to interesting combinatorial challenges while trying to obtain lower and upper bounds that match up to the lower-order terms. For distance labeling of permutation graphs on $n$ vertices, Katz, Katz, and Peleg [STACS 2000] showed how to construct labels consisting of $\mathcal{O}(\log^{2} n)$ bits. Later, Bazzaro and Gavoille [Discret. Math. 309(11)] obtained an asymptotically optimal bounds by showing how to construct labels consisting of $9\log{n}+\mathcal{O}(1)$ bits, and proving that $3\log{n}-\mathcal{O}(\log{\log{n}})$ bits are necessary. This however leaves a quite large gap between the known lower and upper bounds. We close this gap by showing how to construct labels consisting of $3\log{n}+\mathcal{O}(\log\log n)$ bits.

翻译：置换图是两条平行线之间一组线段相交形成的图。换言之，它们由n个元素上的一个置换$\pi$定义，使得顶点$u$和$v$相邻当且仅当$u<v$但$\pi(u)>\pi(v)$。我们在信息标记方案的框架下研究此类图中距离计算的问题。该方案的目标是为每个顶点$u$分配一个短位串$\ell(u)$，使得仅利用$\ell(u)$和$\ell(v)$即可计算$u$与$v$之间的距离，而无需关于整个图的额外知识（仅已知其为置换图）。这巧妙地体现了我们希望数据结构具有分布式的特性，并且常常在尝试获得匹配低阶项的上下界时引发有趣的组合挑战。针对n个顶点置换图的距离标记问题，Katz、Katz与Peleg [STACS 2000] 提出了构造$\mathcal{O}(\log^{2} n)$位标签的方法。随后，Bazzaro与Gavoille [Discret. Math. 309(11)] 通过展示如何构造$9\log{n}+\mathcal{O}(1)$位的标签并证明$3\log{n}-\mathcal{O}(\log{\log{n}})$位是必要的，获得了渐近最优的界。然而，已知下界与上界之间仍存在较大差距。我们通过展示如何构造$3\log{n}+\mathcal{O}(\log\log n)$位的标签来弥合这一差距。