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.

翻译：本文研究了一群$n$个移动机器人在给定有限线段及具有有限正半径的圆上的\textit{最小和均匀覆盖}问题，其中圆作为输入给出。机器人必须协调其运动以达到均匀间隔的配置，从而最小化所有机器人移动的总距离。这些机器人是自主、匿名、相同且同质的，并在由公平异步调度器控制的\textit{非刚性}运动下的\textit{观察-计算-移动}（LCM）模型中运行。它们是无记忆且静默的，既没有持久存储也没有显式通信手段。在\textbf{线段场景}中，\textit{最小和均匀覆盖}问题要求将机器人放置在线段上均匀间隔的点处，以最小化所有机器人移动的总距离。在\textbf{圆场景}中，机器人必须围绕给定圆均匀排列形成正$n$边形。不存在固定方向或指定起始顶点，目标是最小化所有机器人移动的总距离。我们提出了一种确定性分布式算法，在线段场景中以最小总移动成本实现均匀覆盖。对于圆场景，我们刻画了在所考虑的机器人模型下\textit{最小和均匀覆盖}问题确定性不可解的所有初始配置。对于所有其他剩余配置，我们提供了一种确定性分布式算法，在最小化总移动距离的同时实现均匀覆盖。这些结果刻画了无记忆机器人最小和覆盖问题的确定性可解性，并在可解时实现了最优成本。