Beyond Time Complexity: Data Movement Complexity Analysis for Matrix Multiplication

TL;DR

This paper extends the Data Movement Distance framework to analyze data movement in matrix multiplication, demonstrating its ability to differentiate algorithms based on memory behavior and deriving precise cache miss models.

Contribution

It applies data movement analysis to multiple matrix multiplication variants, bridging theory and practice with analytical cache miss models.

Findings

Data Movement Distance conforms with microarchitectural trends.

It differentiates algorithms with same time complexity but different memory behavior.

Provides the first analytical cache miss ratio model for recursive matrix multiplication.

Abstract

Data movement is becoming the dominant contributor to the time and energy costs of computation across a wide range of application domains. However, time complexity is inadequate to analyze data movement. This work expands upon Data Movement Distance, a recently proposed framework for memory-aware algorithm analysis, by 1) demonstrating that its assumptions conform with microarchitectural trends, 2) applying it to six variants of matrix multiplication, and 3) showing it to be capable of asymptotically differentiating algorithms with the same time complexity but different memory behavior, as well as locality optimized vs. non-optimized versions of the same algorithm. In doing so, we attempt to bridge theory and practice by combining the operation count analysis used by asymptotic time complexity with per-operation data movement cost resulting from hierarchical memory structure. Additionally, this paper derives the first fully precise, fully analytical form of recursive matrix multiplication's miss ratio curve on LRU caching systems. Our results indicate that the Data Movement Distance framework is a powerful tool going forward for engineers and algorithm designers to understand the algorithmic implications of hierarchical memory.