Cache and I/O efficent functional algorithms

ABSTRACT

The widely studied I/O and ideal-cache models were developed to account for the large difference in costs to access memory at different levels of the memory hierarchy. Both models are based on a two level memory hierarchy with a fixed size primary memory(cache) of size M, an unbounded secondary memory organized in blocks of size B. The cost measure is based purely on the number of block transfers between the primary and secondary memory. All other operations are free. Many algorithms have been analyzed in these models and indeed these models predict the relative performance of algorithms much more accurately than the standard RAM model. The models, however, require specifying algorithms at a very low level requiring the user to carefully lay out their data in arrays in memory and manage their own memory allocation.

In this paper we present a cost model for analyzing the memory efficiency of algorithms expressed in a simple functional language. We show how some algorithms written in standard forms using just lists and trees (no arrays) and requiring no explicit memory layout or memory management are efficient in the model. We then describe an implementation of the language and show provable bounds for mapping the cost in our model to the cost in the ideal-cache model. These bound imply that purely functional programs based on lists and trees with no special attention to any details of memory layout can be as asymptotically as efficient as the carefully designed imperative I/O efficient algorithms. For example we describe an O(n_B logM/Bn_B)cost sorting algorithm, which is optimal in the ideal cache and I/O models.

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Index Terms

• Software and its engineering

  • Software notations and tools

    • Formal language definitions

      • Semantics

• Theory of computation

  • Semantics and reasoning

    • Program semantics