When considering motion planning for a swarm of n labeled robots, we need to rearrange a given start configuration into a desired target configuration via a sequence of parallel, collision-free moves. The objective is to reach the new configuration in a minimum amount of time. Problems of this type have been considered before, with recent notable results achieving constant stretch for parallel reconfiguration: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of d, the total duration of an overall schedule can be bounded to \(\mathcal {O}(d)\) , which is optimal up to constant factors. An important constraint for coordinated reconfiguration is to keep the swarm connected after each time step. In previous work, constant stretch could only be achieved if disconnected reconfiguration is allowed, or for scaled configurations of unlabeled robots; on the other hand, the existence of non-constant lower bounds on the stretch factor was unknown. We resolve these major open problems by (1) establishing a lower bound of \(\Omega (\sqrt{n})\) for connected, labeled reconfiguration and, most importantly, by (2) proving that for scaled arrangements, constant stretch for connected, labeled reconfiguration can be achieved. In addition, we show that (3) it is NP-complete to decide whether a makespan of 2 can be achieved, while it is possible to check in polynomial time whether a schedule of makespan 1 exists.
