Minimizing the Total Movement for Movement to Independence Problem on a Line

TL;DR

This paper introduces an efficient algorithm to minimize total movement when repositioning points on a line to ensure a minimum distance, solving an open problem for this specific cost measure.

Contribution

The paper presents the first $O(n \, ext{log}(n))$ algorithm for the movement to independence problem on a line with the total movement cost measure.

Findings

Achieved an $O(n \, ext{log}(n))$ algorithm for the problem.

Solved the open problem for the total movement cost measure in one dimension.

Established the problem's computational complexity and provided an optimal solution.

Abstract

Given a positive real value $\delta$, a set $P$ of points along a line and a distance function $d$, in the movement to independence problem, we wish to move the points to new positions on the line such that for every two points $p_{i},p_{j} \in P$, we have $d(p_{i},p_{j}) \geq \delta$ while minimizing the sum of movements of all points. This measure of the cost for moving the points was previously unsolved in this setting. However for different cost measures there are algorithms of $O(n \log(n))$ or of $O(n)$. We present an $O(n \log(n))$ algorithm for the points on a line and thus conclude the setting in one dimension.