Fully Dynamic $k$-Center Clustering Made Simple

• Main Authors: Bhattacharya, Sayan, Costa, Martín, Lattanzi, Silvio, Parotsidis, Nikos
• Format: Journal Article
• Language: English
• Published: 15-10-2024
• Subjects: Computer Science - Data Structures and Algorithms
• DOI: 10.48550/arxiv.2410.11470

In this paper, we consider the \emph{metric $k$-center} problem in the fully
dynamic setting, where we are given a metric space $(V,d)$ evolving via a
sequence of point insertions and deletions and our task is to maintain a subset
$S \subseteq V$ of at most $k$ points that minimizes the objective $\max_{x \in
V} \min_{y \in S}d(x, y)$. We want to design our algorithm so that we minimize
its \emph{approximation ratio}, \emph{recourse} (the number of changes it makes
to the solution $S$) and \emph{update time} (the time it takes to handle an
update).
We give a simple algorithm for dynamic $k$-center that maintains a
$O(1)$-approximate solution with $O(1)$ amortized recourse and $\tilde O(k)$
amortized update time, \emph{obtaining near-optimal approximation, recourse and
update time simultaneously}. We obtain our result by combining a variant of the
dynamic $k$-center algorithm of Bateni et al.~[SODA'23] with the dynamic
sparsifier of Bhattacharya et al.~[NeurIPS'23].