  We consider the problem of connected coordinated motion planning for a large
collective of simple, identical robots: From a given start grid configuration
of robots, we need to reach a desired target configuration via a sequence of
parallel, collision-free robot motions, such that the set of robots induces a
connected grid graph at all integer times. The objective is to minimize the
makespan of the motion schedule, i.e., to reach the new configuration in a
minimum amount of time. We show that this problem is NP-complete, even for
deciding 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. On the algorithmic
side, we establish simultaneous constant-factor approximation for two
fundamental parameters, by achieving constant stretch for constant scale.
Scaled shapes (which arise by increasing all dimensions of a given object by
the same multiplicative factor) have been considered in previous seminal work
on self-assembly, often with unbounded or logarithmic scale factors; we provide
methods for a generalized scale factor, bounded by a constant. Moreover, our
algorithm achieves a constant stretch factor: If mapping the start
configuration to the target configuration requires a maximum Manhattan distance
of $d$, then the total duration of our overall schedule is $\mathcal{O}(d)$,
which is optimal up to constant factors.
