Noisy Voronoi: a Simple Framework for Terminal-Clustering Problems
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We reprove three known (algorithmic) bounds for terminal-clustering problems, using a single framework that leads to simpler proofs. The input in terminal-clustering problems is a metric space (X,d) (possibly arising from a graph) and a subset K⊂ X of terminals, and the goal is to partition the points X, such that each part, called 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for Steiner Point Removal on trees [Gupta, SODA 2001], Metric 0-Extension for bounded doubling dimension [Lee and Naor, unpublished 2003], and Connected Metric 0-Extension [Englert et al., SICOMP 2014].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 is to enlarge each Voronoi cell computed in some order to obtain disjoint clusters, which defines the Noisy-Voronoi algorithm. 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 Noisy-Voronoi algorithm is not only applicable to restricted metrics, but actually leads to relatively simple algorithms and analyses.
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