OpenCL Matrix multiplication: inner product versus outer product - opencl

Question

I'm hoping everyone is familiar with the standard "naive" method of multiplying two (n x n square for simplicity) matrices. In C this is:
for(int i = 0; i < n; ++i)
for(int j = 0; j < n; ++j)
for(int k = 0; k < n; ++k)
C[i*n + j] += A[i*n + k] * B[k*n + j];
The above method computes the dot (inner) product of a row of A with a column of B and is easy to implement in OpenCL as follows:
__kernel void matmul_ocl(
__global const float *A,
__global const float *B,
__global float *C,
const int n
)
{
const int row = get_global_id(1); // row
const int col = get_global_id(0); // col
for(int i = 0; i < n; i++)
C[row*n + col] += A[row*n + i]*B[i*n + col];
}
Interchanging the two inner-most loops of the original C implementation results in a method that computes outer products, i.e., it computes rank-1 updates of the rows of the C matrix:
for(int i = 0; i < n; ++i)
for(int k = 0; k < n; ++k)
for(int j = 0; j < n; ++j)
C[i*n + j] += A[i*n + k] * B[k*n + j];
Does anybody know how to properly implement the above outer-product method in OpenCL? I have two of my attempts pasted below but I just can't seem to nail it
Attempt 1
__kernel void matmul_ocl(
__global const float *A,
__global const float *B,
__global float *C,
const int n
)
{
const int row = get_global_id(1); // row
const int col = get_global_id(0); // col
__local float r;
r = A[row*n + col];
barrier(CLK_LOCAL_MEM_FENCE);
for(int i = 0; i < n; ++i)
C[row*n + i] += r * B[col*n + i];
}
Attempt 2
#define TS 1
__kernel void matmul_ocl(
__global const float *A,
__global const float *B,
__global float *C,
int n)
{
// Thread coordinates
const int row = get_local_id(1); // row
const int col = get_local_id(0); // col
// Group tile coordinates
const int by = get_group_id(1); // row
const int bx = get_group_id(0); // col
A += TS*by + TS*bx*n + n*row + (col);
B += TS*by*n + n*row + (col);
C += TS*bx*n + n*(row) + col;
__global const float *Blast = B + n;
float c[2] = {0.0f,0.0f};
float* cptr = &c[0];
__local float bs[2];
do
{
bs[0] = B[0];
bs[1] = B[n];
barrier(CLK_LOCAL_MEM_FENCE);
*cptr += A[0] * bs[0];
*cptr++ += A[0] * bs[1];
B++;
barrier(CLK_LOCAL_MEM_FENCE);
} while( B < Blast );
C[0] += c[0];
C[1] += c[1];
}

Answer 1

The OpenCL implementation of the common algorithm maps the outer two loops to the OpenCL NDRange implicit loops. This works because the outer two loops can be safely run in parallel.
There are a few problems with Attempt 1:
The __local variable r is assigned different values from multiple work-items simultaneously. There is a race condition here, the value of r is undefined. This could be fixed by just making r a private variable instead.
The more serious problem is that there is a race condition in the assignment of C. Every value of col (NDRange dimension 0) will be running its own loop over i in parallel.
There isn't a simple way around the second issue. The loop over k (in the transposed version) cannot be run in parallel. You can only map either the outer loop or the inner loop to a single dimensional NDRange in OpenCL.