Move and Time Optimal Arbitrary Pattern Formation by Asynchronous Robots on Infinite Grid
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The Arbitrary Pattern Formation (Apf) is a widely studied in distributed computing for swarm robots. This problem asks to design a distributed algorithm that allows a team of identical, autonomous mobile robots to form any arbitrary pattern given as input. This paper considers that the robots are operating on a two-dimensional infinite grid. Robots are initially positioned on distinct grid points forming an asymmetric configuration (no two robots have the same snapshot). They operate under a fully asynchronous scheduler and do not have any access to a global coordinate system, but they will align the axes of their local coordinate systems along the grid lines. The previous work dealing with Apf problem solved it in O(𝒟^2k) robot movements under similar conditions, where 𝒟 is the side of the smallest square that can contain both initial and target configuration and, k is the number of robots. Let 𝒟'=max{𝒟,k}. This paper presents two algorithms of Apf on an infinite grid. The first algorithm solves the Apf problem using O(𝒟') asymptotically move optimal. The second algorithm solves the Apf problem in O(𝒟') epochs, which we show is asymptotically optimal.
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