Relaxed Voronoi: A simple framework for terminal-clustering problems

Arnold Filtser, Robert Krauthgamer, Ohad Trabelsi

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

7 Scopus citations


We reprove three known algorithmic bounds for terminal-clustering problems, using a single framework that leads to simpler proofs. In this genre of problems, the input is a metric space (X, d) (possibly arising from a graph) and a subset of terminals K ⊂ X, and the goal is to partition the points X such that each part, called a cluster, contains exactly one terminal (possibly with connectivity requirements) so as to minimize some objective. The three bounds we reprove are for Steiner Point Removal on trees [Gupta, SODA 2001], for Metric 0-Extension in bounded doubling dimension [Lee and Naor, unpublished 2003], and for Connected Metric 0-Extension [Englert et al., SICOMP 2014]. A natural approach is to cluster each point with its closest terminal, which would partition X into so-called Voronoi cells, but this approach can fail miserably due to its stringent cluster boundaries. A now-standard fix, which we call the Relaxed-Voronoi framework, is to use enlarged Voronoi cells, but to obtain disjoint clusters, the cells are computed greedily according to some order. This method, first proposed by Calinescu, Karloff and Rabani [SICOMP 2004], was employed successfully to provide state-of-the-art results for terminal-clustering problems on general metrics. However, for restricted families of metrics, e.g., trees and doubling metrics, only more complicated, ad-hoc algorithms are known. Our main contribution is to demonstrate that the Relaxed-Voronoi algorithm is applicable to restricted metrics, and actually leads to relatively simple algorithms and analyses.

Original languageEnglish
Title of host publication2nd Symposium on Simplicity in Algorithms, SOSA 2019 - Co-located with the 30th ACM-SIAM Symposium on Discrete Algorithms, SODA 2019
EditorsJeremy T. Fineman, Michael Mitzenmacher
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770996
StatePublished - 1 Jan 2019
Event2nd Symposium on Simplicity in Algorithms, SOSA 2019 - San Diego, United States
Duration: 8 Jan 20199 Jan 2019

Publication series

NameOpenAccess Series in Informatics
ISSN (Print)2190-6807


Conference2nd Symposium on Simplicity in Algorithms, SOSA 2019
Country/TerritoryUnited States
CitySan Diego


  • Clustering
  • Doubling dimension
  • Relaxed voronoi
  • Steiner point removal
  • Zero extension

ASJC Scopus subject areas

  • Geography, Planning and Development
  • Modeling and Simulation


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