Dynamic clustering to minimize the sum of radii

TL;DR

This paper introduces a dynamic clustering approach that efficiently maintains near-optimal solutions in environments where clients arrive and depart, focusing on minimizing the sum of radii and center costs.

Contribution

The paper presents a novel data structure for dynamic sum-of-radii clustering with constant-factor approximation and logarithmic update time in bounded doubling dimension metric spaces.

Findings

Maintains solutions within a constant factor of optimal.

Achieves logarithmic worst-case update time.

Applicable in dynamic environments with client arrivals and departures.

Abstract

In this paper, we study the problem of opening centers to cluster a set of clients in a metric space so as to minimize the sum of the costs of the centers and of the cluster radii, in a dynamic environment where clients arrive and depart, and the solution must be updated efficiently while remaining competitive with respect to the current optimal solution. We call this dynamic sum-of-radii clustering problem. We present a data structure that maintains a solution whose cost is within a constant factor of the cost of an optimal solution in metric spaces with bounded doubling dimension and whose worst-case update time is logarithmic in the parameters of the problem.