Labelings vs. Embeddings: On distributed representations of distances

Arnold Filtser, Lee Ad Gottlieb, Robert Krauthgamer

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

6 Scopus citations


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 = (x1, . . ., xn) 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 xj has label size at most α(j). Similarly, an embedding f : X → l has prioritized dimension α(·) if f(xj) 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 languageEnglish
Title of host publication31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020
EditorsShuchi Chawla
PublisherAssociation for Computing Machinery
Number of pages13
ISBN (Electronic)9781611975994
StatePublished - 2020
Externally publishedYes
Event31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020 - Salt Lake City, United States
Duration: 5 Jan 20208 Jan 2020

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020
Country/TerritoryUnited States
CitySalt Lake City

Bibliographical note

Publisher Copyright:
Copyright © 2020 by SIAM


§WeizmannInstituteofScience. Email: [email protected]. Work partially sup-portedbyONRAward N00014-18-1-2364, theIsraelScience Foundation grant#1086/18,andaMinervaFoundation grant. 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.

FundersFunder number
Office of Naval ResearchN00014-18-1-2364
Simons Foundation
Bonfils-Stanton Foundation2015813
Minerva Foundation
Israel Science Foundation1817/17


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