Tightening I/O Lower Bounds through the Hourglass Dependency Pattern

TL;DR

This paper introduces a new method for deriving tighter lower bounds on data movement (I/O) complexity in algorithms by identifying a common dependency pattern called the hourglass, improving previous bounds.

Contribution

It identifies the hourglass dependency pattern in linear algebra kernels and uses it to mathematically prove tighter I/O lower bounds, enhancing the IOLB tool.

Findings

Tighter I/O lower bounds for linear algebra kernels

Identification of the hourglass dependency pattern

Improved bounds by a parametric ratio

Abstract

When designing an algorithm, one cares about arithmetic/computational complexity, but data movement (I/O) complexity plays an increasingly important role that highly impacts performance and energy consumption. For a given algorithm and a given I/O model, scheduling strategies such as loop tiling can reduce the required I/O down to a limit, called the I/O complexity, inherent to the algorithm itself. The objective of I/O complexity analysis is to compute, for a given program, its minimal I/O requirement among all valid schedules. We consider a sequential execution model with two memories, an infinite one, and a small one of size S on which the computations retrieve and produce data. The I/O is the number of reads and writes between the two memories. We identify a common "hourglass pattern" in the dependency graphs of several common linear algebra kernels. Using the properties of this pattern, we mathematically prove tighter lower bounds on their I/O complexity, which improves the previous state-of-the-art bound by a parametric ratio. This proof was integrated inside the IOLB automatic lower bound derivation tool.