Reductions combine multiple input values with an associative operator to produce a single (or multiple) result(s). When the same input value contributes to multiple outputs, there is an opportunity to reuse partial results, enabling reduction simplification. Simplification produces a program with lower asymptotic complexity. It is well known that reductions in polyhedral programs may be simplified automatically but previous methods are incapable of exploiting all available reuse. This paper resolves this long standing open problem, thereby attaining minimal asymptotic complexity in the simplified program.

翻译：规约通过结合律算子将多个输入值合并，生成单个（或多个）结果。当同一输入值贡献给多个输出时，存在重用部分结果的机会，从而实现规约简化。简化可生成具有更低渐近复杂度的程序。众所周知，多面体程序中的规约可自动简化，但现有方法无法充分利用所有可用的重用机会。本文解决了这一长期存在的开放问题，从而在简化程序中实现了最小的渐近复杂度。