On the I/O complexity of hybrid algorithms for Integer Multiplication

Almost asymptotically tight lower bounds are derived for the Input/Output (I/O) complexity $IO_\mathcal{A}\left(n,M\right)$ 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 $\Theta\left(c_kn^{\log_k \left(2k-1\right)}\right)$, and "standard" integer multiplication algorithms which compute $\Omega\left(n^2\right)$ digit multiplications. We present an $\Omega\left(\left(n/\max\{M,n_0\}\right)^{\log_k \left(2k-1\right)}\left(\max\{1,n_0/M\}\right)^2M\right)$ lower bound for the I/O complexity of a class of "uniform, non-stationary" hybrid algorithms, where $n_0$ denotes the threshold size of sub-problems which are computed using standard algorithms with algebraic complexity $\Omega\left(n^2\right)$. As a special case, our result yields an asymptotically tight $\Theta\left(n^2/M\right)$ 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 $\mathcal{O}\left(k^2\right)$ 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 $n_0$ 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.