  We present a number of breakthroughs for coordinated motion planning, in
which the objective is to reconfigure a swarm of labeled convex objects by a
combination of parallel, continuous, collision-free translations into a given
target arrangement. Problems of this type can be traced back to the classic
work of Schwartz and Sharir (1983), who gave a method for deciding the
existence of a coordinated motion for a set of disks between obstacles; their
approach is polynomial in the complexity of the obstacles, but exponential in
the number of disks. Other previous work has largely focused on {\em
sequential} schedules, in which one robot moves at a time.
  We provide constant-factor approximation algorithms for minimizing the
execution time of a coordinated, {\em parallel} motion plan for a swarm of
robots in the absence of obstacles, provided some amount of separability.
  Our algorithm achieves {\em constant stretch factor}: If all robots are at
most $d$ units from their respective starting positions, the total duration of
the overall schedule is $O(d)$. Extensions include unlabeled robots and
different classes of robots. We also prove that finding a plan with minimal
execution time is NP-hard, even for a grid arrangement without any stationary
obstacles. On the other hand, we show that for densely packed disks that cannot
be well separated, a stretch factor $\Omega(N^{1/4})$ may be required. On the
positive side, we establish a stretch factor of $O(N^{1/2})$ even in this case.
