  This paper addresses practical implementation of summing out, expanding, and
reordering of messages in Local Conditioning (LC) for undirected networks. In
particular, incoming messages conditioned on potentially different subsets of
the receiving node's relevant set must be expanded to be conditioned on this
relevant set, then reordered so that corresponding columns of the conditioned
matrices can be fused through element-wise multiplication. An outgoing message
is then reduced by summing out loop cutset nodes that are upstream of the
outgoing edge. The emphasis on implementation is the primary contribution over
the theoretical justification of LC given in Fay et al. Nevertheless, the
complexity of Local Conditioning in grid networks is still no better than that
of Clustering.
