Efficient formation path planning on large graphs

For the task of transferring a group of robots from one formation to another on a connected graph with unit edge lengths, we provide an efficient hierarchical algorithm that can complete goal assignment and path planning for 10,000 robots on a 250,000 vertex grid in under one second. In the extreme, our algorithm can handle up to one million robots on a grid with one billion vertices in approximately 30 minutes. Perhaps more importantly, we prove that with high probability, the algorithm supplies paths with total distance within a constant multiple of the optimal total distance. Furthermore, our hierarchical method also allows these paths to be scheduled with a tight completion time guarantee. In practice, our implementation yields a total path distance less than two times of the true optimum and a much shorter completion time.