## 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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Pages (from-to) | 849-871 |

Number of pages | 23 |

Journal | Discrete and Computational Geometry |

Volume | 71 |

Issue number | 3 |

DOIs | |

State | Published - Apr 2024 |

## Keywords

- 05C12
- 05C78
- 30L05
- 46B85
- 68R12
- Distance labeling
- Metric embedding
- ℓ

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Geometry and Topology
- Computational Theory and Mathematics