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Space-Time Tradeoffs in Memory Hierarchies
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Graph expansion and communication costs of fast matrix multiplication
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We study the space and the access complexity of computations represented by Computational Directed Acyclic Graphs (CDAGs) in hierarchical memory systems. First, we present a unifying framework for proving lower bounds on the space complexity, which captures most of the bounds known in the literature for relevant CDAGs, previously proved through ad-hoc arguments. Then, we expose a close relationship between the notions of space and access complexity, where the latter represents the minimum number of accesses performed by any computation of a CDAG at a given level of the memory hierarchy. Specifically, we present two general techniques to derive bounds on the access complexity of a CDAG based on the space complexity of certain subgraphs. One technique, simpler to apply, provides only lower bounds, while the other provides (almost) matching lower and upper bounds and improves upon previous well-known result by Hong and Kung.