RFR: JDK-8277175 : Add a parallel multiply method to BigInteger [v5]

[kabutz]

On Wed, 24 Nov 2021 15:20:42 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:
> 
>   Made forkOrInvoke() method protected to avoid strange compiler error

Based on the excellent feedback in this thread, I have added a depth threshold for the parallelMultiply(). This is based on log2(processors)


(int)Math.ceil(Math.log(Runtime.getRuntime().availableProcessors()) / Math.log(2))


and can be overridden with `-Djava.math.BigInteger.parallelForkThreshold=num`

We can also influence this number with `-XX:ActiveProcessorCount=num`, but that would have a wider impact.

Here are results for running the benchmark `BigIntegerParallelMultiply` out of the box on my 1-6-2 server:


Benchmark                                          (n)  Mode  Cnt      Score      Error  Units
BigIntegerParallelMultiply.multiply            1000000    ss    4     77.314 ±   35.690  ms/op
BigIntegerParallelMultiply.multiply           10000000    ss    4    938.650 ±   69.232  ms/op
BigIntegerParallelMultiply.multiply          100000000    ss    4  24811.339 ± 1457.943  ms/op
BigIntegerParallelMultiply.parallelMultiply    1000000    ss    4     67.550 ±   39.059  ms/op
BigIntegerParallelMultiply.parallelMultiply   10000000    ss    4    489.738 ±  140.965  ms/op
BigIntegerParallelMultiply.parallelMultiply  100000000    ss    4   8814.412 ±  542.567  ms/op


Setting ActiveProcessorCount to 1 means that we do not fork at all:


Benchmark                                          (n)  Mode  Cnt      Score      Error  Units                
BigIntegerParallelMultiply.multiply            1000000    ss    4     72.752 ±   26.336  ms/op                
BigIntegerParallelMultiply.multiply           10000000    ss    4    949.417 ±   19.686  ms/op            
BigIntegerParallelMultiply.multiply          100000000    ss    4  24530.183 ± 1319.775  ms/op             
BigIntegerParallelMultiply.parallelMultiply    1000000    ss    4     71.492 ±   21.860  ms/op      
BigIntegerParallelMultiply.parallelMultiply   10000000    ss    4    926.025 ±   41.539  ms/op             
BigIntegerParallelMultiply.parallelMultiply  100000000    ss    4  23947.256 ±   90.076  ms/op      


Similarly, when we set `-Djava.math.BigInteger.parallelForkThreshold=0` we do not fork either:


Benchmark                                          (n)  Mode  Cnt      Score      Error  Units
BigIntegerParallelMultiply.multiply            1000000    ss    4     73.121 ±   30.315  ms/op
BigIntegerParallelMultiply.multiply           10000000    ss    4    931.979 ±   34.988  ms/op
BigIntegerParallelMultiply.multiply          100000000    ss    4  24779.179 ± 5167.362  ms/op
BigIntegerParallelMultiply.parallelMultiply    1000000    ss    4     72.280 ±   32.011  ms/op
BigIntegerParallelMultiply.parallelMultiply   10000000    ss    4    957.168 ±   37.957  ms/op
BigIntegerParallelMultiply.parallelMultiply  100000000    ss    4  25032.332 ±  994.664  ms/op


On the 1-6-2 machine, the maximum parallel forked tasks that we get with our Fibonacci algorithm is 468 (4 x 9 x 13). The parallel recursion depth is 4 in this case. Note that before this change we could get millions of forks, depending on the size of the numbers multiplied.

Once we get to Toom-Cook 3, we are multiplying very large numbers. Also, for convenience we create RecursiveTasks whether we fork or invoke directly. This means that object allocation goes up slightly (less than 1%) when doing sequential multiply() on very large numbers.

-------------

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