RFR: JDK-8277175 : Add a parallel multiply method to BigInteger [v9]

[kabutz]

On Fri, 28 Jan 2022 18:59:56 GMT, kabutz <duke at 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:
> 
>   Benchmark for testing the effectiveness of BigInteger.parallelMultiply()

I have added a benchmark for checking performance difference between sequential and parallel multiply of very large Mersenne primes using BigInteger. We want to measure real time, user time, system time and the amount of memory allocated. To calculate this, we create our own thread factory for the common ForkJoinPool and then use that to measure user time, cpu time and bytes allocated.

We use reflection to discover all methods that match "*ultiply", and use them to multiply two very large Mersenne primes together.

### Results on a 1-6-2 machine running Ubuntu linux

Memory allocation increased from 83.9GB to 84GB, for both the sequential and parallel versions. This is an increase of just 0.1%. On this machine, the parallel version was 3.8x faster in latency (real time), but it used 2.7x more CPU resources.

Testing multiplying Mersenne primes of 2^57885161-1 and 2^82589933-1

#### openjdk version "18-internal" 2022-03-15

BigInteger.parallelMultiply()
real  0m6.288s
user  1m3.010s
sys   0m0.027s
mem   84.0GB
BigInteger.multiply()
real  0m23.682s
user  0m23.530s
sys   0m0.004s
mem   84.0GB


#### openjdk version "1.8.0_302"

BigInteger.multiply()
real  0m25.657s
user  0m25.390s
sys   0m0.001s
mem   83.9GB


#### openjdk version "9.0.7.1"

BigInteger.multiply()
real  0m24.907s
user  0m24.700s
sys   0m0.001s
mem   83.9GB


#### openjdk version "10.0.2" 2018-07-17

BigInteger.multiply()
real  0m24.632s
user  0m24.380s
sys   0m0.004s
mem   83.9GB


#### openjdk version "11.0.12" 2021-07-20 LTS

BigInteger.multiply()
real  0m22.114s
user  0m21.930s
sys   0m0.001s
mem   83.9GB


#### openjdk version "12.0.2" 2019-07-16

BigInteger.multiply()
real  0m23.015s
user  0m22.830s
sys   0m0.000s
mem   83.9GB


#### openjdk version "13.0.9" 2021-10-19

BigInteger.multiply()
real  0m23.548s
user  0m23.350s
sys   0m0.005s
mem   83.9GB


#### openjdk version "14.0.2" 2020-07-14

BigInteger.multiply()
real  0m22.918s
user  0m22.530s
sys   0m0.131s
mem   83.9GB



#### openjdk version "15.0.5" 2021-10-19

BigInteger.multiply()
real  0m22.038s
user  0m21.750s
sys   0m0.003s
mem   83.9GB


#### openjdk version "16.0.2" 2021-07-20

BigInteger.multiply()
real  0m23.049s
user  0m22.760s
sys   0m0.006s
mem   83.9GB


#### openjdk version "17" 2021-09-14

BigInteger.multiply()
real  0m22.580s
user  0m22.310s
sys   0m0.001s
mem   83.9GB

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PR: https://git.openjdk.java.net/jdk/pull/6409