Separating Hierarchical and General Hub Labelings

TL;DR

This paper investigates the differences between hierarchical and general hub labelings in distance oracles, showing that hierarchical labels can be significantly larger than general hub labels, with tight bounds established for hypercubes.

Contribution

It provides the first formal separation showing hierarchical labels can be polynomially larger than general hub labels, supported by tight bounds for hypercube graphs.

Findings

Hierarchical labels can be polynomially larger than general hub labels.

Tight bounds on label sizes are established for hypercube graphs.

The results highlight a fundamental gap in label efficiency.

Abstract

In the context of distance oracles, a labeling algorithm computes vertex labels during preprocessing. An $s,t$ query computes the corresponding distance from the labels of $s$ and $t$ only, without looking at the input graph. Hub labels is a class of labels that has been extensively studied. Performance of the hub label query depends on the label size. Hierarchical labels are a natural special kind of hub labels. These labels are related to other problems and can be computed more efficiently. This brings up a natural question of the quality of hierarchical labels. We show that there is a gap: optimal hierarchical labels can be polynomially bigger than the general hub labels. To prove this result, we give tight upper and lower bounds on the size of hierarchical and general labels for hypercubes.