  We prove that by successively combining subassemblies, we can achieve
sublinear construction times for "staged" assembly of micro-scale objects from
a large number of tiny particles, for vast classes of shapes; this is a
significant advance in the context of programmable matter and self-assembly for
building high-yield micro-factories.The underlying model has particles moving
under the influence of uniform external forces until they hit an obstacle;
particles bond when forced together with a compatible particle. Previous work
considered sequential composition of objects, resulting in construction time
that is linear in the number N of particles, which is inefficient for large N.
Our progress implies critical speedup for constructible shapes; for convex
polyominoes, even a constant construction time is possible. We also show that
our construction process can be used for pipelining, resulting in an amortized
constant production time.
