Automated Lower Bounds on the I/O Complexity of Computation Graphs
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by
Saachi Jain, Matei Zaharia
2019
Abstract
We consider the problem of finding lower bounds on the I/O complexity of
arbitrary computations. Executions of complex computations can be formalized as
an evaluation order over the underlying computation graph. In this paper, we
present two novel methods to find I/O lower bounds for an arbitrary computation
graph. In the first, we bound the I/O using the eigenvalues of the graph
Laplacian. This spectral bound is not only efficiently computable, but also can
be computed in closed form for graphs with known spectra. In our second method,
we leverage a novel Integer Linear Program that directly solves for the optimal
evaluation order; we solve this ILP on constant sized sub-graphs of the
original computation graph to find I/O lower bounds. We apply our spectral
method to compute closed-form analytical bounds on two computation graphs
(hypercube and Fast Fourier Transform). We further empirically validate our
methods on four computation graphs, and find that our methods provide tighter
bounds than current empirical methods and behave similarly to previously
published I/O bounds.
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