Min-Sum Uniform Coverage Problem by Autonomous Mobile Robots

TL;DR

This paper presents deterministic algorithms for autonomous robots to achieve minimal total movement while uniformly covering line segments and circles, characterizing solvable configurations and optimal solutions under the Look-Compute-Move model.

Contribution

It introduces algorithms for uniform coverage with minimal total movement and characterizes initial configurations where the problem is unsolvable.

Findings

Optimal algorithms for line segment coverage minimizing total distance

Complete characterization of initial configurations for circle coverage

Identification of unsolvable configurations under the robot model

Abstract

We study the \textit{min-sum uniform coverage} problem for a swarm of $n$ mobile robots on a given finite line segment and on a circle having finite positive radius, where the circle is given as an input. The robots must coordinate their movements to reach a uniformly spaced configuration that minimizes the total distance traveled by all robots. The robots are autonomous, anonymous, identical, and homogeneous, and operate under the \textit{Look-Compute-Move} (LCM) model with \textit{non-rigid} motion controlled by a fair asynchronous scheduler. They are oblivious and silent, possessing neither persistent memory nor a means of explicit communication. In the \textbf{line-segment setting}, the \textit{min-sum uniform coverage} problem requires placing the robots at uniformly spaced points along the segment so as to minimize the total distance traveled by all robots. In the \textbf{circle setting} for this problem, the robots have to arrange themselves uniformly around the given circle to form a regular $n$-gon. There is no fixed orientation or designated starting vertex, and the goal is to minimize the total distance traveled by all the robots. We present a deterministic distributed algorithm that achieves uniform coverage in the line-segment setting with minimum total movement cost. For the circle setting, we characterize all initial configurations for which the \textit{min-sum uniform coverage} problem is deterministically unsolvable under the considered robot model. For all the other remaining configurations, we provide a deterministic distributed algorithm that achieves uniform coverage while minimizing the total distance traveled. These results characterize the deterministic solvability of min-sum coverage for oblivious robots and achieve optimal cost whenever solvable.