Gathering by Repulsion

TL;DR

This paper presents efficient algorithms for placing and activating repulsion actuators in convex environments to gather particles at a single point, optimizing for minimal actuators and computational time.

Contribution

It introduces new algorithms with improved time complexity for locating actuators and coordinating their activation to achieve particle gathering.

Findings

O(n^2) algorithm for all actuator locations

O(n) algorithm for a single actuator location

O(n) algorithm for optimal actuator placement

Abstract

We consider a repulsion actuator located in an $n$-sided convex environment full of point particles. When the actuator is activated, all the particles move away from the actuator. We study the problem of gathering all the particles to a point. We give an $O(n^2)$ time algorithm to compute all the actuator locations that gather the particles to one point with one activation, and an $O(n)$ time algorithm to find a single such actuator location if one exists. We then provide an $O(n)$ time algorithm to place the optimal number of actuators whose sequential activation results in the gathering of the particles when such a placement exists.