c# - Why is there huge performance hit in 2048x2048 versus 2047x2047 array multiplication?

Question

I am making some matrix multiplication benchmarking, as previously mentioned in Why is MATLAB so fast in matrix multiplication?

Now I've got another issue, when multiplying two 2048x2048 matrices, there is a big difference between C# and others. When I try multiply only 2047x2047 matrices, it seems normal. Added some others for comparsion too.

1024x1024 - 10 seconds.

1027x1027 - 10 seconds.

2047x2047 - 90 seconds.

2048x2048 - 300 seconds.

2049x2049 - 91 seconds. (update)

2500x2500 - 166 seconds

That is three and a half minute difference for the 2k by 2k case.

using 2dim arrays

//Array init like this
int rozmer = 2048;
float[,] matice = new float[rozmer, rozmer];

//Main multiply code
for(int j = 0; j < rozmer; j++)
{
   for (int k = 0; k < rozmer; k++)
   {
     float temp = 0;
     for (int m = 0; m < rozmer; m++)
     {
       temp = temp + matice1[j,m] * matice2[m,k];
     }
     matice3[j, k] = temp;
   }
 }

Answer 1

This probably has do with conflicts in your L2 cache.

Cache misses on matice1 are not the problem because they are accessed sequentially. However for matice2 if a full column fits in L2 (i.e when you access matice2[0, 0], matice2[1, 0], matice2[2, 0] ... etc, nothing gets evicted) than there is no problem with cache misses with matice2 either.

Now to go deeper in how caches works, if byte address of your variable is X, than the cache line for it would be (X >> 6) & (L - 1). Where L is total number of cache lines in your cache. L is always power of 2. The six comes from fact that 2^6 == 64 bytes is standard size of cache line.

Now what does this mean? Well it means that if I have address X and address Y and (X >> 6) - (Y >> 6) is divisible by L (i.e. some large power of 2), they will be stored in the same cacheline.

Now to go back to your problem what is the difference between 2048 and 2049,

when 2048 is your size:

if you take &matice2[x, k] and &matice2[y, k] the difference (&matice2[x, k] >> 6) - (&matice2[y,k] >> 6) will be divisible by 2048 * 4 (size of float). So a large power of 2.

Thus depending on size of your L2 you will have a lot of cache line conflicts, and only utilize small portion of your L2 to store a column, thus you wont actually be able to store full column in your cache, thus you will get bad performance.

When size is 2049, then the difference is 2049 * 4 which is not power of 2 thus you will have less conflicts and your column will safely fit into your cache.

Now to test this theory there are couple things you can do:

Allocate your array matice2 array like this matice2 [razmor, 4096], and run with razmor = 1024, 1025 or any size, and you should see very bad performance compared to what you had before. This is because you forcefully align all columns to conflict with each other.

Then try matice2 [razmor, 4097] and run it with any size and you should see much better performance.