The Impact of Partial Computations on the Red-Blue Pebble Game

TL;DR

This paper extends the red-blue pebble game to include partial computations, demonstrating that such steps can significantly reduce I/O costs and providing new tools for analyzing computational lower bounds.

Contribution

It introduces the partial-computing red-blue pebble game (PRBP), analyzes its properties, and adapts existing tools to derive I/O lower bounds and NP-hardness results for this extended model.

Findings

Allowing partial computations can reduce I/O costs by up to a linear factor.

Deciding if partial computations reduce costs is NP-hard.

Lower bounds on I/O costs are established for key computational tasks.

Abstract

We study an extension of the well-known red-blue pebble game (RBP) with partial computation steps, inspired by the recent work of Sobczyk. While the original RBP assumes that we need to have all the inputs of an operation in fast memory at the same time, in many concrete computations, the inputs can be aggregated one by one into the final output value. These partial computation steps can enable pebbling strategies with much smaller I/O cost, and in settings where such a step-by-step aggregation is possible, this extended red-blue pebble game offers a much more realistic cost model. We establish the fundamental properties of this partial-computing red-blue pebble game (PRBP), and compare it to the original RBP. We begin with some simple examples where allowing partial computations can decrease the optimal I/O cost. It is also shown that the cost can decrease by up to a linear factor this way, but in general, it is NP-hard to decide whether partial computations allow for a smaller cost in a specific DAG. We then discuss how $S$-partitions, a crucial tool for deriving I/O lower bounds in RBP, can be adapted to the PRBP model. These new tools are then used to establish lower bounds on the I/O cost of some prominent computational tasks. Finally, we also adapt a hardness result from RBP, showing that the optimum cost is still NP-hard to approximate in PRBP to any reasonable factor.