Labelings vs. Embeddings: On Distributed and Prioritized Representations of Distances

Abstract

We investigate for which metric spaces the performance of distance labeling and of $\ell _\infty$-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 $x\in X$ is assigned a succinct label, such that the distance between any two points $x,y\in X$ can be approximated given only their labels. A highly structured special case is an embedding into $\ell _\infty$, where each point $x\in X$ is assigned a vector f(x) such that $\Vert f(x)-f(y)\Vert _\infty$ is approximately d(x, y). The performance of a distance labeling or an $\ell _\infty$-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 $\pi =(x_1,\dots ,x_n)$ of the point set X is given, and higher-priority points should have shorter labels. Formally, a distance labeling has prioritized label-size $\alpha (\,{\cdot }\,)$ if every $x_j$ has label size at most $\alpha (j)$. Similarly, an embedding $f:X\rightarrow \ell _\infty$ has prioritized dimension $\alpha (\,{\cdot }\,)$ if $f(x_j)$ is non-zero only in the first $\alpha (j)$ 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.