Simplification of Polyhedral Reductions in Practice

TL;DR

This paper presents a comprehensive compiler implementation for reduction simplification that reuses partial results, improving efficiency and discovering new algorithms across various applications.

Contribution

It introduces the first complete, automated reduction simplification method in a compiler, demonstrating practical benefits and uncovering new algorithms.

Findings

Reduces computational effort in real-world applications

Discovers alternative algorithms without changing asymptotic complexity

Reveals key algorithmic improvements through simplification

Abstract

Reductions combine collections of inputs with an associative (and here, also commutative) operator to produce collections of outputs. When the same value contributes to multiple outputs, there is an opportunity to reuse partial results, enabling reduction simplification. We provide the first complete push-button implementation of reduction simplification in a compiler. We evaluate its effectiveness on a range of real-world applications, and show that simplification rediscovers several key results in algorithmic improvement across multiple domains, previously only obtained through clever manual human analysis and effort. We also discover alternate, previously unknown algorithms, albeit without improving the asymptotic complexity.