## 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 kf(x)−f(y)k_{∞} 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 their 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 a surprising exception to this rule.

Original language | English |
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Title of host publication | 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020 |

Editors | Shuchi Chawla |

Publisher | Association for Computing Machinery |

Pages | 1063-1075 |

Number of pages | 13 |

ISBN (Electronic) | 9781611975994 |

State | Published - 2020 |

Externally published | Yes |

Event | 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020 - Salt Lake City, United States Duration: 5 Jan 2020 → 8 Jan 2020 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Volume | 2020-January |

### Conference

Conference | 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020 |
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Country/Territory | United States |

City | Salt Lake City |

Period | 5/01/20 → 8/01/20 |

### Bibliographical note

Funding Information:§WeizmannInstituteofScience. Email: robert.krauthgamer@weizmann.ac.il. Work partially sup-portedbyONRAward N00014-18-1-2364, theIsraelScience Foundation grant#1086/18,andaMinervaFoundation grant.

Funding Information:

Most of the work was conducted while the author was affiliated with Ben-Gurion University of the Negev. Work supported in part by Simons Foundation, ISF grant 1817/17, and by BSF Grant 2015813. il. Work partially supported by ONR Award N00014-18-1-2364, the Israel Science Foundation grant #1086/18, and a Minerva Foundation grant.

Publisher Copyright:

Copyright © 2020 by SIAM