1 research outputs found
Communication-Optimal Parallel Standard and Karatsuba Integer Multiplication in the Distributed Memory Model
We present COPSIM a parallel implementation of standard integer
multiplication for the distributed memory setting, and COPK a parallel
implementation of Karatsuba's fast integer multiplication algorithm for a
distributed memory setting. When using processors, each equipped
with a local non-shared memory, to compute the product of tho -digits
integer numbers, under mild conditions, our algorithms achieve optimal speedup
of the computational time. That is,
for COPSIM, and for COPK.
The total amount of memory required across the processors is
, that is, within a constant factor of the minimum
space required to store the input values. We rigorously analyze the
Input/Output (I/O) cost of the proposed algorithms. We show that their
bandwidth cost (i.e., the number of memory words sent or received by at least
one processors) matches asymptotically corresponding known I/O lower bounds,
and their latency (i.e., the number of messages sent or received in the
algorithm's critical execution path) is asymptotically within a multiplicative
factor of the corresponding
known I/O lower bounds. Hence, our algorithms are asymptotically optimal with
respect to the bandwidth cost and almost asymptotically optimal with respect to
the latency cost