Labelings vs. Embeddings: On Distributed and Prioritized 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=(x₁,⋯,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.