Separating Hierarchical and General Hub Labelings

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.