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

> IIRC you are not overly concerned about the additional object creation of `RecursiveOp` instances?
> 
> If that changes the operation method could return `Object` and choose to perform the operation directly returning the `BigInteger` result or returning the particular `RecursiveOp`.
> 
> ```java
> private static Object multiply(BigInteger a, BigInteger b, boolean parallel, int depth) {
>     if (isParallel(parallel, depth)) {
>         return new RecursiveMultiply(a, b, parallel, depth).fork();
>     } else {
>         // Also called by RecursiveMultiply.compute()
>         return RecursiveMultiply.compute(a, b, false, depth);
>     }
> }
> ```
> 
> Then we could have another method on `RecursiveOp` that pattern matches e.g.:
> 
> ```java
>   static BigInteger join(Object o) {
>     // replace with pattern match switch when it exits preview
>     if (o instanceof BigInteger b) {
>         return b;
>     } else if (o instanceof RecursiveOp r) {
>         return r.join();
>     } else {
>         throw new InternalError("Cannot reach here);
>     }
>   }
> ```
> 
> That seems simple enough it might be worth doing anyway.

If we need a demonstration as to why pattern matching switch is necessary, then I guess we could do this. Each RecursiveOp instance is 40 bytes. To calculate Fibonacci (1_000_000_000) we create 7_324_218 tasks, thus we are allocating an additional 292_968_720 bytes of memory. In addition, I believe that calling invoke() allocates some more bytes. According to the GC logs, we allocate an additional 2.33 GB of memory. That might sound like a lot, but it takes 2.25 TB to calculate Fibonacci of 1 billion using our algorithm. The additional memory allocated is thus roughly 0.1%. The performance of the old sequential multiply and the new one, with the additional object creation, seems equivalent.

I would thus recommend that we keep it the way it is at the moment, with the new RecursiveOp task creation. Considering the volume of objects that we will be allocating once we get to Toom Cook 3, a 0.1% reduction in object allocation will not be noticed.


Old BigInteger#multiply()
Fibonacci   memory          bytes 
      100   11.5KB          11808
       1k  119.0KB         121856
      10k    1.2MB        1238552
     100k   13.0MB       13634608
       1m  177.5MB      186104688
      10m    3.3GB     3574666840
     100m   85.6GB    91866740256
    1000m    2.3TB  2475468459952

New BigInteger#multiply()
Fibonacci   memory          bytes  increase %
      100   11.5KB          11808  0.0
       1k  119.0KB         121856  0.0
      10k    1.2MB        1238552  0.0
     100k   13.0MB       13649176  0.1067
       1m  177.6MB      186197472  0.0498
      10m    3.3GB     3577835016  0.088
     100m   85.6GB    91960170576  0.1017
    1000m    2.3TB  2477979788992  0.1014

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

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