On Wed, 17 Nov 2021 19:48:25 GMT, kabutz <d...@openjdk.java.net> wrote:
>> BigInteger currently uses three different algorithms for multiply. The
>> simple quadratic algorithm, then the slightly better Karatsuba if we exceed
>> a bit count and then Toom Cook 3 once we go into the several thousands of
>> bits. Since Toom Cook 3 is a recursive algorithm, it is trivial to
>> parallelize it. I have demonstrated this several times in conference talks.
>> In order to be consistent with other classes such as Arrays and Collection,
>> I have added a parallelMultiply() method. Internally we have added a
>> parameter to the private multiply method to indicate whether the calculation
>> should be done in parallel.
>>
>> The performance improvements are as should be expected. Fibonacci of 100
>> million (using a single-threaded Dijkstra's sum of squares version)
>> completes in 9.2 seconds with the parallelMultiply() vs 25.3 seconds with
>> the sequential multiply() method. This is on my 1-8-2 laptop. The final
>> multiplications are with very large numbers, which then benefit from the
>> parallelization of Toom-Cook 3. Fibonacci 100 million is a 347084 bit number.
>>
>> We have also parallelized the private square() method. Internally, the
>> square() method defaults to be sequential.
>>
>> Some benchmark results, run on my 1-6-2 server:
>>
>>
>> Benchmark (n) Mode Cnt Score
>> Error Units
>> BigIntegerParallelMultiply.multiply 1000000 ss 4 51.707
>> ± 11.194 ms/op
>> BigIntegerParallelMultiply.multiply 10000000 ss 4 988.302
>> ± 235.977 ms/op
>> BigIntegerParallelMultiply.multiply 100000000 ss 4 24662.063
>> ± 1123.329 ms/op
>> BigIntegerParallelMultiply.parallelMultiply 1000000 ss 4 49.337
>> ± 26.611 ms/op
>> BigIntegerParallelMultiply.parallelMultiply 10000000 ss 4 527.560
>> ± 268.903 ms/op
>> BigIntegerParallelMultiply.parallelMultiply 100000000 ss 4 9076.551
>> ± 1899.444 ms/op
>>
>>
>> We can see that for larger calculations (fib 100m), the execution is 2.7x
>> faster in parallel. For medium size (fib 10m) it is 1.873x faster. And for
>> small (fib 1m) it is roughly the same. Considering that the fibonacci
>> algorithm that we used was in itself sequential, and that the last 3
>> calculations would dominate, 2.7x faster should probably be considered quite
>> good on a 1-6-2 machine.
>
> kabutz has updated the pull request incrementally with one additional commit
> since the last revision:
>
> Removed JVM flags from benchmark
I looked more closely and now understand a bit more about the threshold. It
would be useful to have some implementation notes detailing approximately when
the parallel execution kicks in. For `a*a` it's `>=
TOOM_COOK_SQUARE_THRESHOLD(216)` and for `a*b` it's `>=
TOOM_COOK_THRESHOLD(240)`, so we could refer to certain bit lengths.
The branching factor is 4 (well 3 i guess, but its easier to think in powers of
2). It might be reasonable to assume the problem gets split equally in 4 parts.
I don't know in practice what the depth of recursion might, its hard to see
this getting completely out of control, but we could always keep track of the
depth and cut off in proportion to the # runtime processors if need be.
Given the existing degree of specialization, the minimal changes to code, and
the fact that the creation of recursive tasks is in the noise this PR looks
quite reasonable.
I think we can simplify the creation of the recursive tasks (assuming we don't
track the depth):
private static final class RecursiveOp {
private static <T> RecursiveTask<T> exec(RecursiveTask<T> op, boolean
parallel) {
if (parallel) {
op.fork();
} else {
op.invoke();
}
return op;
}
private static RecursiveTask<BigInteger> multiply(BigInteger a,
BigInteger b, boolean parallel) {
var op = new RecursiveTask<BigInteger>() {
@Override
protected BigInteger compute() {
return a.multiply(b, true, parallel);
}
};
return exec(op, parallel);
}
private static RecursiveTask<BigInteger> square(BigInteger a, boolean
parallel) {
var op = new RecursiveTask<BigInteger>() {
@Override
protected BigInteger compute() {
return a.square(true, parallel);
}
};
return exec(op, parallel);
}
}
-------------
PR: https://git.openjdk.java.net/jdk/pull/6409
