TY - JOUR

T1 - Labelings vs. Embeddings

T2 - On Distributed and Prioritized Representations of Distances

AU - Filtser, Arnold

AU - Gottlieb, Lee Ad

AU - Krauthgamer, Robert

N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023.

PY - 2024/4

Y1 - 2024/4

N2 - 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=(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 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.

AB - 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=(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 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.

KW - 05C12

KW - 05C78

KW - 30L05

KW - 46B85

KW - 68R12

KW - Distance labeling

KW - Metric embedding

KW - ℓ

UR - http://www.scopus.com/inward/record.url?scp=85171295729&partnerID=8YFLogxK

U2 - 10.1007/s00454-023-00565-2

DO - 10.1007/s00454-023-00565-2

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85171295729

SN - 0179-5376

VL - 71

SP - 849

EP - 871

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 3

ER -