O(1) sum

[H. S. Teoh via Digitalmars-d-learn]

On Mon, Jun 12, 2017 at 06:16:06PM +0000, Stefan Koch via Digitalmars-d-learn wrote:
> On Monday, 12 June 2017 at 06:15:07 UTC, Biotronic wrote:
[...]
> > On a multi-core system we can do better:
> > 
> > auto nums = iota(10_000_000.0f);
> > auto sum = taskPool.reduce!"a + b"(nums);
> > 
> > Given arbitrary parallelism (yeah, right!), this will be O(log(N)).
> > For real-world systems, it might give a speed-up for large arrays,
> > but won't reduce the big-O complexity. Of course, there will also be
> > overhead to such a solution, so there is a limit to how much one'd
> > actually benefit from it.
> > 
> > --
> >   Biotronic
> 
> Biotronic how do you arrive at  O(log(N)) ??
> And which logarithm  ?

His stated presupposition is arbitrary parallelism, which I assume means
arbitrary number of CPUs or cores that can run in parallel, so then you
can divide the array of N elements into N/2 pairs, sum each pair in
parallel, which gives you N/2 subtotals after one iteration, then you
recursively repeat this on the subtotals until you're left with the
final total. The complexity would be O(log_2(N)) iterations, assuming
that the constant factor hidden by the big-O covers the overhead of
managing the parallel summing operations across the arbitrary number of
cores.

You can also get logarithms of a different base if you divided the
initial array, say, into triplets or j-tuplets, for some constant j.
Then you'd get O(log_j(N)). (Of course, with a slightly larger constant
factor, assuming that each CPU core only has binary summation
instructions. But if your instruction set has multiple-summation
instructions you may be able to get a higher j at little or no
additional cost. Assuming you can produce a machine with an unlimited
number of cores in the first place.)

Of course, his comment "yeah, right!" indicates that he's aware that
this is an unrealistic scenario. :-)


T

-- 
Notwithstanding the eloquent discontent that you have just respectfully expressed at length against my verbal capabilities, I am afraid that I must unfortunately bring it to your attention that I am, in fact, NOT verbose.