Labelings vs. Embeddings: On Distributed Representations of Distances

We investigate for which metric spaces the performance of distance labeling and of ℓ ∞embeddings differ, and how significant can this difference be. Recall that a distance labeling is a distributed representation of distances in a metric space (X, d), where each point x ∈ X is assigned a succinct label, such that the distance between any two points x, y ∈ X can be approximated given only their labels. A highly structured special case is an embedding into ℓ ∞ , where each point x ∈ X is assigned a vector f (x) such that ∥f (x) -f (y)∥ ∞ is approximately d(x, y). The performance of a distance labeling or an ℓ ∞ -embedding is measured via its distortion and its label-size/dimension.

We also study the analogous question for the prioritized versions of these two measures. Here, a priority order π = (x 1 , . . . , x n ) of the point set X is given, and higher-priority points should have shorter labels. Formally, a distance labeling has prioritized label-size α(•) if every x j has label size at most α(j). Similarly, an embedding f : X → ℓ ∞ has prioritized dimension α(•) if f (x j ) is non-zero only in the first α(j) coordinates. In addition, we compare these prioritized measures to their classical (worst-case) versions.

We answer these questions in several scenarios, uncovering a surprisingly diverse range of behaviors. First, in some cases labelings and embeddings have very similar worst-case performance, but in other cases there is a huge disparity. However in the prioritized setting, we most often find a strict separation between the performance of labelings and embeddings. And finally, when comparing the classical and prioritized settings, we find that the worst-case bound for label size often "translates" to a prioritized one, but also find a surprising exception to this rule.