I/O complexity and pebble games with partial computations

TL;DR

This paper introduces a new pebble game variant that models DAGs with arbitrary in-degrees using partial computations, addressing limitations of existing models and analyzing the complexity of optimal strategies.

Contribution

It proposes a novel pebble game model accommodating arbitrary in-degree DAGs and analyzes the NP-completeness of finding optimal strategies in this setting.

Findings

Deciding optimal pebbling strategies is NP-complete even for simple DAGs.

The new model allows direct representation of graphs with arbitrary in-degree.

Approximation algorithms are discussed for specific cases.

Abstract

Optimizing data movements during program executions is essential for achieving high performance in modern computing systems. This has been classically modeled with the Red-Blue Pebble Game and its variants. In existing models, it is typically assumed that the number of red pebbles, i.e., the size of the fast memory, is larger than the maximum in-degree in the computational directed acyclic graph (DAG). Graphs that do not satisfy this constraint need to be first transformed appropriately, which is not a trivial task for general graphs. In this work we propose a Pebble Game variant to model DAGs with arbitrary in-degrees, by allowing partial computations. In the new model, we show that it is NP-complete to decide whether there exists an optimal pebbling strategy with cost $k$, even for single-level DAGs and when only two words fit in the fast memory. Approximation algorithms for special cases are also outlined.