TY - GEN
T1 - Coresets for 1-Center in ℓ1 Metrics
AU - Carmel, Amir
AU - Guo, Chengzhi
AU - Jiang, Shaofeng H.C.
AU - Krauthgamer, Robert
N1 - Publisher Copyright:
© Amir Carmel, Chengzhi Guo, Shaofeng H.-C. Jiang, and Robert Krauthgamer.
PY - 2025/2/11
Y1 - 2025/2/11
N2 - We explore the applicability of coresets – a small subset of the input dataset that approximates a predefined set of queries – to the 1-center problem in ℓ1 spaces. This approach could potentially extend to solving the 1-center problem in related metric spaces, and has implications for streaming and dynamic algorithms. We show that in ℓ1, unlike in Euclidean space, even weak coresets exhibit exponential dependency on the underlying dimension. Moreover, while inputs with a unique optimal center admit better bounds, they are not dimension independent. We then relax the guarantee of the coreset further, to merely approximate the value (optimal cost of 1-center), and obtain a dimension-independent coreset for every desired accuracy ϵ ą 0. Finally, we discuss the broader implications of our findings to related metric spaces, and show explicit implications to Jaccard and Kendall’s tau distances.
AB - We explore the applicability of coresets – a small subset of the input dataset that approximates a predefined set of queries – to the 1-center problem in ℓ1 spaces. This approach could potentially extend to solving the 1-center problem in related metric spaces, and has implications for streaming and dynamic algorithms. We show that in ℓ1, unlike in Euclidean space, even weak coresets exhibit exponential dependency on the underlying dimension. Moreover, while inputs with a unique optimal center admit better bounds, they are not dimension independent. We then relax the guarantee of the coreset further, to merely approximate the value (optimal cost of 1-center), and obtain a dimension-independent coreset for every desired accuracy ϵ ą 0. Finally, we discuss the broader implications of our findings to related metric spaces, and show explicit implications to Jaccard and Kendall’s tau distances.
KW - clustering
KW - coresets
KW - Jaccard metric
KW - k-center
KW - Kendall’s tau
KW - minimum enclosing balls
KW - ℓ1 norm
UR - http://www.scopus.com/inward/record.url?scp=85218350789&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ITCS.2025.28
DO - 10.4230/LIPIcs.ITCS.2025.28
M3 - Conference contribution
AN - SCOPUS:85218350789
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 16th Innovations in Theoretical Computer Science Conference, ITCS 2025
A2 - Meka, Raghu
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 16th Innovations in Theoretical Computer Science Conference, ITCS 2025
Y2 - 7 January 2025 through 10 January 2025
ER -