I/O complexity and pebble games with partial computations

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 the 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 graph (e.g. an arithmetic circuit). This assumption can be restrictive for many real applications, especially when dealing with "big data" in Machine Learning and Scientific Computing. In this work we study a generalization of the original Red-Blue Pebble Game to allow arbitrary in-degrees, that can be larger than the size of the fast memory. The objective is to minimize the I/O operations by allowing the computation of partial results in the fast memory. We show that this variant of the problem is NP-complete, even for the special case where the computational graph consists of a single level, and only two words fit in the fast memory. Approximation algorithms for a couple of special cases are also outlined.