Alice, Bob and Mallory: Finding primes in parallel

Finding primes in parallel

Jonas Elfström — Thu, 14 Jan 2010 22:55:00 +0100

Justin Etheredge has been blogging about his challenge to find prime numbers with LINQ. He later used AsParallel() (coming in .NET 4) to speed things up and then followed that up with a post about using The Sieve Of Eratosthenes.

As you can see in the comments of those posts I tried to speed the Sieve of Eratosthenes up by using Parallel.For in the inner loop. I also tried AsParallel() in the LINQ expression but it made no difference in either case. At most it got 5% faster. I'm not sure but it could be that because SoE is very memory intense we could have a scaling issue and maybe also memory bandwidth exhaustion. This is mere speculation.

I then searched for other algorithms and found The Sieve of Atkin. It uses less memory than SoE so I thought I'd give it a try.

I set the limit to 20,000,000 and then benchmarked it. It timed in on 2.48s so actually worse than the 2.2s that SoE took. Not good! Then I added Parallel.For in the loop that did most of the work and lo and behold, it scaled! I have two cores in my machine (T7200@2.0GHz) and the average runtime went down to 1.26s. That's almost linear and surprisingly good! If you happen have a quad core (or more) and feel like trying it out then please contact me. It would be interesting to see if it scales further.

static List<int> FindPrimesBySieveOfAtkins(int max)
{
    var isPrime = new BitArray((int)max+1, false);
    var sqrt = (int)Math.Sqrt(max);

    Parallel.For(1, sqrt, x =>
    {
        var xx = x * x;
        for (int y = 1; y <= sqrt; y++)
        {
            var yy = y * y;
            var n = 4 * xx + yy;
            if (n <= max && (n % 12 == 1 || n % 12 == 5))
                isPrime[n] = !isPrime[n];

            n = 3 * xx + yy;
            if (n <= max && n % 12 == 7)
                isPrime[n] = !isPrime[n];

            n = 3 * xx - yy;
            if (x > y && n <= max && n % 12 == 11)
                isPrime[n] = !isPrime[n];
        }
    });

    var primes = new List<int>() { 2, 3 };
    for (int n = 5; n <= sqrt; n++)
    {
        if (isPrime[n])
        {
            primes.Add(n);
            int nn = n * n;
            for (int k = nn; k <= max; k += nn)
                isPrime[k] = false;
        }
    }

    for (int n = sqrt + 1; n <= max; n++)
        if (isPrime[n])
            primes.Add(n);

    return primes;
}

This is C# 4.0 code, compiled in Visual C# 2010 Express Beta 2.

Edit 2010-01-20

Indications are that this does in fact not scale very good on a quad core. It's even worse, it seems it scales good on my old T7200 but not on a dual core E6320. I don't know why but of course the shared state of the isPrime BitArray is a huge problem and maybe it could be that differences in CPU architecture (FSB speed, caches and so on) in the E6320 is an explanation. Average execution time on the E6320 was 1290ms in a single thread and 1064ms in two.

If you want to try this in an older version of C# than 4.0 then check out this post.

A reader asked how I timed the executions. Here's how.

var steps = new List<long>();
var watch = new Stopwatch();

for (int i = 0; i < 10; i++) 
{
    watch.Reset();
    watch.Start();
    var primes = FindPrimesBySieveOfAtkins(20000000);
    watch.Stop();
    Console.WriteLine(watch.ElapsedMilliseconds.ToString());
    steps.Add(watch.ElapsedMilliseconds);
}
Console.WriteLine("Average: " + steps.Average().ToString());

"Finding primes in parallel" by Jonas Elfström

Wed, 20 Jan 2010 09:11:22 +0100

http://coding-time.blogspot.com/2008/03/implement-your-own-parallelfor-in-c.html - makes it possible to run FindPrimesBySieveOfAtkins unchanged in C# 2.0-3.5.

"Finding primes in parallel" by Svish

Tue, 19 Jan 2010 18:33:16 +0100

So should maybe use the Set method instead then? Or doesn’t make much difference perhaps…

Thanks for the link. Will check it out :)

"Finding primes in parallel" by Jonas Elfström

Sat, 16 Jan 2010 16:11:21 +0100

“I was wondering if it could be done in a nice way without using Parallel.”

Check out: http://www.codeproject.com/KB/dotnet/PoorMansParallelForEach.aspx

"Finding primes in parallel" by Jonas Elfström

Fri, 15 Jan 2010 17:24:41 +0100

If Holterman is correct then my usage is thread safe: http://stackoverflow.com/questions/1213997/is-there-a-generic-type-safe-bitarray-in-net/1214686#1214686

"Finding primes in parallel" by Jonas Elfström

Fri, 15 Jan 2010 16:23:15 +0100

That’s for “long, ulong, double, and decimal”. Read/write of booleans is atomic. I’m just not sure that isPrime[n] = !isPrime[n]; is the same as Boolean test = false; test = !test; which would be atomic.

"Finding primes in parallel" by Svish

Fri, 15 Jan 2010 15:50:13 +0100

From that link you posted: “Aside from the library functions designed for that purpose, there is no guarantee of atomic read-modify-write, such as in the case of increment or decrement.” – Wouldn’t that mean that it is not thread-safe? or?

"Finding primes in parallel" by Svish

Fri, 15 Jan 2010 15:47:57 +0100

But that would remove the parallelism :p I was wondering if it could be done in a nice way without using Parallel.For, but still have the parallelism. (So do whatever Parallel.For does yourself)

"Finding primes in parallel" by Jonas Elfström

Fri, 15 Jan 2010 13:33:16 +0100

Replace the Parallel.For with for (int x = 1; x <= sqrt; x++) and remove ); from row 24 and you should be good to go.

If I understand http://bit.ly/8lZagW correctly the isPrime[n] = !isPrime[n]; is an atomic operation but I have to investigate the matter of thread safety further. Thanks!

"Finding primes in parallel" by Svish

Fri, 15 Jan 2010 11:03:18 +0100

Are BitArrays thread-safe, or how does that work?

Is it possible to write a version of this without the Parrallel.For method? Well, I suppose it is possible of course, but would it be a big mess? :p