The algebraic expression for the tree-level scattering amplitude in
terms of Feynman diagrams is itself a tree. The much slower growth of
the set of 1POWs compared to the set of Feynman diagrams shows that this
representation is extremely redundant. In this case, Directed
Acyclical Graphs (DAGs) provide a more efficient representation, as
illustrated by a trivial example
where one multiplication is saved. The replacement of expression
trees by equivalent DAGs is part of the repertoire of optimizing
compilers, known as common subexpression elimination.
Unfortunately, this approach fails in practice for all interesting
expressions appearing in quantum field theory, because of the
combinatorial growth of space and time required to find an almost
optimal factorization.
However, the recursive definition in equation (??)
allows to construct the DAG of the 1POWs in equation (2)
directly [1], without having to construct and
factorize the Feynman diagrams explicitely.
As mentioned above, there is more than one consistent prescription for
constructing the set of keystones [1]. The symbolic
expressions constructed by O'Mega contain the symbolic equivalents of
the numerical expressions computed by [4] (maximally
symmetric keystones) and [5] (maximally asymmetric
keystones) as special cases.
