  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 robot motions. The objective is to
reach the new configuration in a minimum amount of time; an important
constraint is to keep the swarm connected at all times. Problems of this type
have been considered before, with recent notable results achieving constant
stretch for not necessarily connected 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. However, constant
stretch could only be achieved if disconnected reconfiguration is allowed, or
for scaled configurations (which arise by increasing all dimensions of a given
object by the same multiplicative factor) of unlabeled robots.
  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 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 makespan of 1 can be achieved.
