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\emph{Reductions} combine collections of input values with an associative(and usually also commutative) operator to produce either a single, or acollection of outputs. They are ubiquitous in computing, especially with bigdata and deep learning. When the \emph{same} input value contributes tomultiple output values, there is a tremendous opportunity for reducing (punintended) the computational effort. This is called \emph{simplification}.\emph{Polyhedral reductions} are reductions where the input and output datacollections are (dense) multidimensional arrays (i.e., \emph{tensors}),accessed with linear/affine functions of the indices. % \emph{generalizedtensor contractions} Gautam and Rajopadhye \cite{sanjay-popl06} showed howpolyhedral reductions could be simplified automatically (through compile timeanalysis) and optimally (the resulting program had minimum asymptoticcomplexity). Yang, Atkinson and Carbin \cite{yang2020simplifying} extended thisto the case when (some) input values depend on (some) outputs. Specifically,they showed how the optimal simplification problem could be formulated as abilinear programming problem, and for the case when the reduction operatoradmits an inverse, they gave a heuristic solution that retained optimality. Inthis note, we show that simplification of dependent reductions can beformulated as a simple extension of the Gautam-Rajopadhye backtracking searchalgorithm.

On Simplifying Dependent Polyhedral Reductions