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Register a new userSpectral Lower Bounds on the I/O Complexity of Computation Graphs
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Added by
Saachi Jain
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We consider the problem of finding lower bounds on the I/O complexity of arbitrary computations in a two level memory hierarchy. Executions of complex computations can be formalized as an evaluation order over the underlying computation graph. However, prior methods for finding I/O lower bounds leverage the graph structures for specific problems (e.g matrix multiplication) which cannot be applied to arbitrary graphs. In this paper, we first present a novel method to bound the I/O of any computation graph using the first few eigenvalues of the graphs Laplacian. We further extend this bound to the parallel setting. This spectral bound is not only efficiently computable by power iteration, but can also be computed in closed form for graphs with known spectra. We apply our spectral method to compute closed-form analytical bounds on two computation graphs (the Bellman-Held-Karp algorithm for the traveling salesman problem and the Fast Fourier Transform), as well as provide a probabilistic bound for random Erdos Renyi graphs. We empirically validate our bound on four computation graphs, and find that our method provides tighter bounds than current empirical methods and behaves similarly to previously published I/O bounds.
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This paper initiates the study of I/O algorithms (minimizing cache misses) from the perspective of fine-grained complexity (conditional polynomial lower bounds). Specifically, we aim to answer why sparse graph problems are so hard, and why the Longest Common Subsequence problem gets a savings of a factor of the size of cache times the length of a cache line, but no more. We take the reductions and techniques from complexity and fine-grained complexity and apply them to the I/O model to generate new (conditional) lower bounds as well as faster algorithms. We also prove the existence of a time hierarchy for the I/O model, which motivates the fine-grained reductions. Using fine-grained reductions, we give an algorithm for distinguishing 2 vs. 3 diameter and radius that runs in $O(|E|^2/(MB))$ cache misses, which for sparse graphs improves over the previous $O(|V|^2/B)$ running time. We give new reductions from radius and diameter to Wiener index and median. We show meaningful reductions between problems that have linear-time solutions in the RAM model. The reductions use low I/O complexity (typically $O(n/B)$), and thus help to finely capture the relationship between I/O linear time $Theta(n/B)$ and RAM linear time $Theta(n)$. We generate new I/O assumptions based on the difficulty of improving sparse graph problem running times in the I/O model. We create conjectures that the current best known algorithms for Single Source Shortest Paths (SSSP), diameter, and radius are optimal. From these I/O-model assumptions, we show that many of the known reductions in the word-RAM model can naturally extend to hold in the I/O model as well (e.g., a lower bound on the I/O complexity of Longest Common Subsequence that matches the best known running time). Finally, we prove an analog of the Time Hierarchy Theorem in the I/O model.
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