Labeling Schemes with Queries

TL;DR

This paper introduces a generalized labeling scheme with queries, showing that inspecting additional vertices can significantly reduce label sizes for various graph functions.

Contribution

It proposes the concept of labeling schemes with queries, extending traditional schemes by allowing inspection of extra vertices to improve efficiency.

Findings

Inspecting one additional vertex reduces label sizes.

The approach applies to various functions on graph pairs.

Significant improvements over known bounds are demonstrated.

Abstract

We study the question of ``how robust are the known lower bounds of labeling schemes when one increases the number of consulted labels''. Let $f$ be a function on pairs of vertices. An $f$-labeling scheme for a family of graphs $\cF$ labels the vertices of all graphs in $\cF$ such that for every graph $G\in\cF$ and every two vertices $u,v\in G$, the value $f(u,v)$ can be inferred by merely inspecting the labels of $u$ and $v$. This paper introduces a natural generalization: the notion of $f$-labeling schemes with queries, in which the value $f(u,v)$ can be inferred by inspecting not only the labels of $u$ and $v$ but possibly the labels of some additional vertices. We show that inspecting the label of a single additional vertex (one {\em query}) enables us to reduce the label size of many labeling schemes significantly.