Optimal Distance Labeling for Permutation Graphs

TL;DR

The paper investigates distance labeling for permutation graphs within the informative labeling framework, aiming to determine exact vertex distances using only local labels. It introduces a boundary-based representation that reduces distance computations to interactions among boundary points and their layered structure, enabling a sequence of label-size reductions. The main contribution is a construction of a distance labeling scheme with size \(3\log n + \mathcal{O}(\log\log n)\) bits and constant-time decoding, thereby closing the gap between the previous lower bound \(3\log n - \mathcal{O}(\log\log n)\) and the earlier \(\Theta(\log n)\) upper bounds for permutation graphs. The approach further generalizes to disconnected graphs with a modest additive overhead, and the work highlights the importance of second-order constants in labeling problems while offering a practical, scalable method for distributed distance queries on 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 $π$ on $n$ elements, such that $u$ and $v$ are adjacent if an only if $u<v$ but $π(u)>π(v)$. 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.

Figures (16)

• Figure 1: Permutation graph described by $\pi= 1 8 3 2 6 4 7 5$.
• Figure 2: Green points form bottom boundary, red points top boundary. $v$ is not on the boundary, orange lines show its quadrants. For $v$, there are four points of special interest, extreme neighbours on both boundaries.
• Figure 3: Boundary points partitioned into layers. $r,t$ are on the top boundary, but in layers 2 and 6. For any two points in layers $a$ and $b$, the distance between them is always either $|a-b|$ or $|a-b|+2$; here, $d(r,t)=4$. $v$ is not on the boundary, and any such point can be adjacent to points from at most three different layers.
• Figure 4: Geometric representation of the graph from Figure \ref{['Fig:PerGraph']}, so permutation [1,8,3,2,6,4,7,5]. Point $(4,6)$ is adjacent to $(6,5)$ and $(8,2)$, as these two points are in its bottom-right quadrant, while top-left quadrant of $(4,6)$ is empty.
• Figure 5: Green points are on the bottom boundary, red points are on the top boundary.

Theorems & Definitions (17)

• Theorem 1
• Lemma 2
• proof
• Lemma 4
• proof
• proof
• Lemma 7
• proof
• Lemma 8
• proof