## Abstract

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.

Original language | English |
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Journal | Discrete and Computational Geometry |

DOIs | |

State | Accepted/In press - 2023 |

### Bibliographical note

Publisher Copyright:© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

## Keywords

- Distance labeling
- Metric embedding
- ℓ