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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 of a general class of hybrid
algorithms computing the product of two integers, each represented with
digits in a given base , in a two-level storage hierarchy with words of
fast memory, with different digits stored in different memory words. The
considered hybrid algorithms combine the Toom-Cook- (or Toom-) fast
integer multiplication approach with computational complexity
, and "standard" integer
multiplication algorithms which compute digit
multiplications. We present an
lower bound for the
I/O complexity of a class of "uniform, non-stationary" hybrid algorithms, where
denotes the threshold size of sub-problems which are computed using
standard algorithms with algebraic complexity . As a
special case, our result yields an asymptotically tight
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
multiplicative term of the corresponding lower bounds, the proposed lower
bounds are almost asymptotically tight and indeed tight for constant values of
. By extending these results to a distributed memory model with
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.Comment: arXiv admin note: substantial text overlap with arXiv:1904.1280