Moderate Dimension Reduction for k-Center Clustering

Shaofeng H.C. Jiang, Robert Krauthgamer, Shay Sapir

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

Abstract

The Johnson-Lindenstrauss (JL) Lemma introduced the concept of dimension reduction via a random linear map, which has become a fundamental technique in many computational settings. For a set of n points in Rd and any fixed ϵ > 0, it reduces the dimension d to O(log n) while preserving, with high probability, all the pairwise Euclidean distances within factor 1 + ϵ. Perhaps surprisingly, the target dimension can be lower if one only wishes to preserve the optimal value of a certain problem on the pointset, e.g., Euclidean max-cut or k-means. However, for some notorious problems, like diameter (aka furthest pair), dimension reduction via the JL map to below O(log n) does not preserve the optimal value within factor 1 + ϵ. We propose to focus on another regime, of moderate dimension reduction, where a problem’s value is preserved within factor α > 1 using target dimension log n/poly(α). We establish the viability of this approach and show that the famous k-center problem is α-approximated when reducing to dimension O(logα2n + log k). Along the way, we address the diameter problem via the special case k = 1. Our result extends to several important variants of k-center (with outliers, capacities, or fairness constraints), and the bound improves further with the input’s doubling dimension. While our poly(α)-factor improvement in the dimension may seem small, it actually has significant implications for streaming algorithms, and easily yields an algorithm for k-center in dynamic geometric streams, that achieves O(α)-approximation using space poly(kdn1/α2 ). This is the first algorithm to beat O(n) space in high dimension d, as all previous algorithms require space at least exp(d). Furthermore, it extends to the k-center variants mentioned above.

Original languageEnglish
Title of host publication40th International Symposium on Computational Geometry, SoCG 2024
EditorsWolfgang Mulzer, Jeff M. Phillips
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Number of pages16
ISBN (Electronic)9783959773164
DOIs
StatePublished - Jun 2024
Event40th International Symposium on Computational Geometry, SoCG 2024 - Athens, Greece
Duration: 11 Jun 202414 Jun 2024

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume293
ISSN (Print)1868-8969

Conference

Conference40th International Symposium on Computational Geometry, SoCG 2024
Country/TerritoryGreece
CityAthens
Period11/06/2414/06/24

All Science Journal Classification (ASJC) codes

  • Software

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