  We present a topological framework for finding low-flop algorithms for
evaluating element stiffness matrices associated with multilinear forms for
finite element methods posed over straight-sided affine domains. This framework
relies on phrasing the computation on each element as the contraction of each
collection of reference element tensors with an element-specific geometric
tensor. We then present a new concept of complexity-reducing relations that
serve as distance relations between these reference element tensors. This
notion sets up a graph-theoretic context in which we may find an optimized
algorithm by computing a minimum spanning tree. We present experimental results
for some common multilinear forms showing significant reductions in operation
count and also discuss some efficient algorithms for building the graph we use
for the optimization.
