  We design a low complexity distributed compression scheme for computing
arbitrary functions of sources with discrete alphabets. We use a helper-based
method that extends the definition of the G{\'a}cs-K{\"o}rner-Witsenhausen
(GKW) common information to functional common information. The helper relaxes
the combinatorial structure of GKW by partitioning the joint source
distribution into nests imposed by the function, which ensures hierarchical
cooperation between the sources for effectively distributed computing. By
contrasting our approach's performance with existing efficient techniques, we
demonstrate the rate savings in recovering function and source data.
  Permutation invariant functions are prevalent in learning and combinatorial
optimization fields and most recently applied to graph neural networks. We
consider the efficient compression for computing permutation invariant
functions in a network with two sources and one decoder. We use a bipartite
graph representation, where two disjoint sets of vertices (parts) denote the
individual source graphs and the edge weights capture the joint source
distribution. We compress bipartite graphs by creating connected components
determined by the function's distribution, accounting for the edge symmetries,
and eliminating the low probability edges. We separately encode those edges and
send them as refinements. Our approach can substitute high complexity joint
decoding techniques and inform neural networks to reduce the computing time and
reduce complexity.
