Labelings vs. Embeddings: On Distributed and Prioritized Representations of Distances

Arnold Filtser, Lee Ad Gottlieb, Robert Krauthgamer

פרסום מחקרי: פרסום בכתב עתמאמרביקורת עמיתים

1 ציטוט ‏(Scopus)

תקציר

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.

שפה מקוריתאנגלית
עמודים (מ-עד)849-871
מספר עמודים23
כתב עתDiscrete and Computational Geometry
כרך71
מספר גיליון3
מזהי עצם דיגיטלי (DOIs)
סטטוס פרסוםפורסם - אפר׳ 2024

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