TY - GEN

T1 - Labelings vs. Embeddings

T2 - 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020

AU - Filtser, Arnold

AU - Gottlieb, Lee Ad

AU - Krauthgamer, Robert

N1 - Publisher Copyright:
Copyright © 2020 by SIAM

PY - 2020

Y1 - 2020

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 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.

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 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.

UR - http://www.scopus.com/inward/record.url?scp=85084049014&partnerID=8YFLogxK

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AN - SCOPUS:85084049014

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 1063

EP - 1075

BT - 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020

A2 - Chawla, Shuchi

Y2 - 5 January 2020 through 8 January 2020

ER -