We consider the fundamental problem of arranging a set of n autonomous robots (points) on a real plane according to an arbitrary given pattern. Each robot operates in a, largely oblivious, look-compute-move step. We measure the time complexity of our algorithms in “epochs,” which denotes the minimum time interval in which every robot performs at least one look-compute-move step. We present a framework for the pattern formation problem in which leader election is key. Where n robots can elect a leader in $T_{\mathit{LE}}(n)$ epochs, we show that pattern formation can be solved in $O(T_{\mathit{LE}}(n))$ epochs on the semi-synchronous model using robots that either are transparent (that is, the classical oblivious robots model where complete visibility is guaranteed at all times) or have lights with a constant number of colors (that is, the robots with lights model where robots are not transparent but the colors of the lights are persistent between steps). We also prove that, for some cases, the $O(T_{\mathit{LE}}(n))$ epochs are optimal on the semi-synchronous model.

Our results on the semi-synchronous model indicate that transparency and lights compensate for each other in the pattern formation problem. The proposed method runs in $O(T_{LE}(n)+\log n)$ epochs on the asynchronous model of robots with lights. This translates to an algorithm on the asynchronous model that runs in $O(\log n)$ randomized epochs with high probability. Our algorithms do not assume agreement on direction or orientation of the robots such as global coordinate system, chirality, or one-axis agreement. Robot motions are rigid in the sense that when a robot moves it reaches its destination point at the completion of that move. The proposed algorithms are randomized and guarantee the claimed time bounds with high probability. The randomization is used only for leader election.

Finally, we provide lower bounds regarding optimality of our results as well as impossibility of designing an asynchronous algorithm in the classical model.

我们考虑根据任意给定模式在真实平面上布置一组n 个自主机器人（点）的基本问题。每个机器人都在一个基本上不经意的、看起来-计算-移动的步骤中运行。我们以“epochs”来衡量我们算法的时间复杂度，它表示每个机器人执行至少一个look-compute-move 步骤的最小时间间隔。我们提出了一个模式形成问题的框架，其中领导者选举是关键。n个机器人可以在哪里选举领导者${吨}_{\mathit{乐}}(n)$epochs，我们证明了模式的形成可以解决$○({吨}_{\mathit{乐}}(n))$在半同步模型上使用透明的机器人（即始终保证完全可见性的经典无意识机器人模型）或具有恒定颜色数量的灯光（即具有灯光的机器人模型机器人不是透明的，但灯光的颜色在步骤之间是持久的）。我们还证明，在某些情况下，$○({吨}_{\mathit{乐}}(n))$epochs 在半同步模型上是最优的。

我们在半同步模型上的结果表明，透明度和光线在图案形成问题中相互补偿。建议的方法运行在$○({吨}_{\mathrm{大号}乙}(n)+\mathrm{日志}n)$带灯机器人异步模型的时代。这转化为运行在异步模型上的算法$○(\mathrm{日志}n)$具有高概率的随机时期。我们的算法不假设机器人的方向或方向一致，例如全局坐标系、手性或单轴一致。从某种意义上说，机器人运动是刚性的，即当机器人移动时，它会在完成该移动时到达其目的地点。所提出的算法是随机的，并以高概率保证声称的时间范围。随机化仅用于领导者选举。

最后，我们提供了关于我们的结果的最优性以及在经典模型中设计异步算法的不可能的下限。