Lorenzo De Stefani
ABSTRACT
Almost asymptotically tight lower bounds are derived for the Input/Output (I/O) complexity IOA(n, M) of a general class of hybrid algorithms computing the product of two integers, each represented with n digits in a given base s, in a two-level storage hierarchy with M words of fast memory, with different digits stored in different memory words. The considered hybrid algorithms combine the Toom-Cook-k (or Toom-k) fast integer multiplication approach with computational complexity Θ(cknlogk (2k-1)), and "standard" integer multiplication algorithms which compute Ω(n2) digit multiplications. We present an Ω((n/max(M,n0))logk (2k-1) (max(1,n_0/M))2M) lower bound for the I/O complexity of a class of "uniform, non-stationary" hybrid algorithms, where n0 denotes the threshold size of sub-problems which are computed using standard algorithms with algebraic complexity Ω(n2). As a special case, our result yields an asymptotically tight Ω(n2/M) lower bound for the I/O complexity of any standard integer multiplication algorithm. As some sequential hybrid algorithms from this class exhibit I/O cost within a O(k2) multiplicative term of the corresponding lower bounds, the proposed lower bounds are almost asymptotically tight and indeed tight for constant values of k. By extending these results to a distributed memory model with P processors, we obtain both memory-dependent and memory-independent I/O lower bounds for parallel versions of hybrid integer multiplication algorithms. All the lower bounds are derived for the more general class of "non-uniform, non-stationary" hybrid algorithms that allow recursive calls to have a different structure, even when computing sub-problems with the same input size, and to use different versions of Toom-k.
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Index Terms
- Brief Announcement: On the I/O Complexity of Sequential and Parallel Hybrid Integer Multiplication Algorithms
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