Kinetic Clustering of Points on the Line

TL;DR

This paper addresses the problem of clustering moving points on a line by proposing efficient algorithms for two classical measures, including a polynomial-time solution and an approximation algorithm with a specific ratio.

Contribution

It introduces polynomial-time algorithms for clustering points on a line based on diameter minimization and provides a (2.71 + epsilon)-approximation for the maximum diameter measure.

Findings

Polynomial-time algorithms for sum of diameters

A (2.71 + epsilon)-approximation for maximum diameter

Improved methods for clustering moving points

Abstract

The problem of clustering a set of points moving on the line consists of the following: given positive integers n and k, the initial position and the velocity of n points, find an optimal k-clustering of the points. We consider two classical quality measures for the clustering: minimizing the sum of the clusters diameters and minimizing the maximum diameter of a cluster. For the former, we present polynomial-time algorithms under some assumptions and, for the latter, a (2.71 + epsilon)-approximation.