  In this paper we present algorithms for collective construction systems in
which a large number of autonomous mobile robots trans- port modular building
elements to construct a desired structure. We focus on building block
structures subject to some physical constraints that restrict the order in
which the blocks may be attached to the structure. Specifically, we determine a
partial ordering on the blocks such that if they are attached in accordance
with this ordering, then (i) the structure is a single, connected piece at all
intermediate stages of construction, and (ii) no block is attached between two
other previously attached blocks, since such a space is too narrow for a robot
to maneuver a block into it. Previous work has consider this problem for
building 2D structures without holes. Here we extend this work to 2D structures
with holes. We accomplish this by modeling the problem as a graph orientation
problem and describe an O(n^2) algorithm for solving it. We also describe how
this partial ordering may be used in a distributed fashion by the robots to
coordinate their actions during the building process.
