  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 $K\subset X$, 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unpublished 2003], and for Connected Metric $0$-Extension [Englert et al.,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.
