Optimal Distance Labeling for Permutation Graphs

TL;DR

This paper presents a new distance labeling scheme for permutation graphs that significantly narrows the gap between known upper and lower bounds, achieving labels of size approximately 3 log n bits.

Contribution

The authors develop a novel labeling scheme for permutation graphs that reduces label size to near the theoretical lower bound, improving upon previous methods.

Findings

Constructed labels of size 3 log n + O(log log n) bits.

Closed the gap between existing upper and lower bounds.

Enhanced understanding of distance labeling complexity for permutation graphs.

Abstract

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.