Distance labeling schemes for $K_4$-free bridged graphs

TL;DR

This paper presents a 4-approximate distance labeling scheme for $K_4$-free bridged graphs, enabling efficient distance approximation with polylogarithmic label size and constant decoding time.

Contribution

It introduces the first polylogarithmic size labeling scheme providing a 4-approximate distance for this class of graphs.

Findings

Labels of size $O( ext{log}^3 n)$ are sufficient.

Decoding time is constant.

Approximate distances are within a factor of 4 of the true distance.

Abstract

$k$-Approximate distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the $k$-approximation of 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. One of the important problems is finding natural classes of graphs admitting exact or approximate distance labeling schemes with labels of polylogarithmic size. In this paper, we describe a $4$-approximate distance labeling scheme forthe class of $K_4$-free bridged graphs. This scheme uses labels of poly-logarithmic length $O(\log n^3)$ allowing a constant decoding time. Given the labels of two vertices $u$ and $v$, the decoding function returnsa value between the exact distance $d_G(u,v)$ and its quadruple $4d_G(u,v)$.