Efficient layout and reduction of virtual 2d data (abstract)

Question

I use C++ and CUDA/C and want to write code for a specific problem and I ran into a quite tricky reduction problem.

My experience in parallel programming isn't negligible but quite limited and I cannot totally forsee the specificity of this problem. I doubt there is a convenient or even "easy" way to handle the problems I am facing but perhaps I am wrong. If there are any resources (i.e. articles, books, web-links, ...) or key-words covering this or similar problems, please let me know.

I tried to generalize the whole case as good as possible and keep it abstract instead of posting too much code.

The Layout ...

I have a system of N inital elements and N result elements. (I'll use N=8 for example but N can be any integral value greater than three.)

static size_t const N = 8;
double init_values[N], result[N];

I need to calculate almost every (not all i'm afraid) unique permutation of the init-values without self-interference.

This means calculation f(init_values[0],init_values[1]), f(init_values[0],init_values[2]), ..., f(init_values[0],init_values[N-1]), f(init_values[1],init_values[2]), ..., f(init_values[1],init_values[N-1]), ... and so on.

This is in fact a virtual triangular matrix which has the shape seen in the following illustration.

 P     0    1    2    3    4    5    6    7
   |---------------------------------------
  0|   x 
   |
  1|   0    x
   |
  2|   1    2    x 
   |
  3|   3    4    5    x
   |
  4|   6    7    8    9    x
   |
  5|  10   11   12   13   14    x
   |
  6|  15   16   17   18   19   20    x
   |
  7|  21   22   23   24   25   26   27    x

Each element is a function of the respective column and row elements in init_values.

P[i] (= P[row(i)][col(i]) = f(init_values[col(i)], init_values[row(i)])

i.e.

P[11] (= P[5][1]) = f(init_values[1], init_values[5])

There are (N*N-N)/2 = 28 possible, unique combinations (Note: P[1][5]==P[5][1], so we only have a lower (or upper) triangular matrix) using the example N = 8.

The basic problem

The result array is computed from P as a sum of the row elements minus the sum of the respective column elements. For example the result at position 3 will be calculated as a sum of row 3 minus the sum of column three.

result[3] = (P[3]+P[4]+P[5]) - (P[9]+P[13]+P[18]+P[24])
result[3] = sum_elements_row(3) - sum_elements_column(3)

I tried to illustrate it in a picture with N = 4.

As a consequence the following is true:

• N-1 operations (potential concurrent writes) will be performed on each result[i]
• result[i] will have N-(i+1) writes from subtractions and i additions
• Outgoing from each P[i][j] there will be a subtraction to r[j] and a addition to r[i]

This is where the main problems come into place:

• Using one thread to compute each P and updating the result directly will result in multiple kernels trying to write to the same result location (N-1 threads each).
• Storing the whole matrix P for a subsequent reduction step on the other hand is very expensive in terms of memory consumption and therefore impossible for very large systems.

The idea of having a unqiue, shared result vector for each thread-block is impossible, too. (N of 50k makes 2.5 billion P elements and therefore [assuming a maximum number of 1024 threads per block] a minimal number of 2.4 million blocks consuming over 900GiB of memory if each block has its own result array with 50k double elements.)

I think I could handle reduction for a more static behaviour but this problem is rather dynamic in terms of potential concurrent memory write-access. (Or is it possible to handle it by some "basic" type of reduction?)

Adding some complications ...

Unfortunatelly, depending on (arbitrary user) input, which is independant of the initial values, some elements of P need to be skipped. Let's assume we need to skip permutations P[6], P[14] and P[18]. Therefore we have 24 combinations left, which need to be calculated.

How to tell the kernel which values need to be skipped? I came up with three approaches, each having notable downsides if N is very large (like several ten thousands of elements).

1. Store all combinations ...

... with their respective row and column index struct combo { size_t row,col; };, that need to be calculated in a vector<combo> and operate on this vector. (used by the current implementation)

std::vector<combo> elements;
// somehow fill
size_t const M = elements.size();
for (size_t i=0; i<M; ++i)
{
    // do the necessary computations using elements[i].row and elements[i].col  
}

This solution consumes is consuming lots of memory since only "several" (may even be ten thousands of elements but that's not much in contrast to several billion in total) but it avoids

• indexation computations
• finding of removed elements

for each element of P which is the downside of the second approach.

2. Operate on all elements of P and find removed elements

If I want to operate on each element of P and avoid nested loops (which i couldn't reproduce very well in cuda) I need to do something like this:

size_t M = (N*N-N)/2;
for (size_t i=0; i<M; ++i)
{
   // calculate row indices from `i`
   double tmp = sqrt(8.0*double(i+1))/2.0 + 0.5;
   double row_d = floor(tmp);
   size_t current_row = size_t(row_d);
   size_t current_col = size_t(floor(row_d*(ict-row_d)-0.5));
   // check whether the current combo of row and col is not to be removed
   if (!removes[current_row].exists(current_col))
   {
     // do the necessary computations using current_row and current_col
   }
}

The vector removes is very small in contrast to the elements vector in the first example but the additional computations to obtain current_row, current_col and the if-branch are very inefficient. (Remember we're still talking about billions of evaluations.)

3. Operate on all elements of P and remove elements afterwards

Another idea I had was to calculate all valid and invalid combinations independently. But unfortunately, due to summation errors the following statement is true:

calc_non_skipped() != calc_all() - calc_skipped()

Is there a convenient, known, high performance way to get the desired results from the initial values?

I know that this question is rather complicated and perhaps limited in relevance. Nevertheless, I hope some illuminative answers will help me to solve my problems.

---

The current implementation

Currently this is implemented as CPU Code with OpenMP. I first set up a vector of the above mentioned combos storing every P that needs to be computed and pass it to a parallel for loop. Each thread is provided with a private result vector and a critical section at the end of the parallel region is used for a proper summation.

Answer 1

First, I was puzzled for a moment why (N**2 - N)/2 yielded 27 for N=7 ... but for indices 0-7, N=8, and there are 28 elements in P. Shouldn't try to answer questions like this so late in the day. :-)

But on to a potential solution: Do you need to keep the array P for any other purpose? If not, I think you can get the result you want with just two intermediate arrays, each of length N: one for the sum of the rows and one for the sum of the columns.

Here's a quick-and-dirty example of what I think you're trying to do (subroutine direct_approach()) and how to achieve the same result using the intermediate arrays (subroutine refined_approach()):

#include <cstdlib>
#include <cstdio>

const int N = 7;
const float input_values[N] = { 3.0F, 5.0F, 7.0F, 11.0F, 13.0F, 17.0F, 23.0F };
float P[N][N];      // Yes, I'm wasting half the array.  This way I don't have to fuss with mapping the indices.
float result1[N] = { 0.0F, 0.0F, 0.0F, 0.0F, 0.0F, 0.0F, 0.0F };
float result2[N] = { 0.0F, 0.0F, 0.0F, 0.0F, 0.0F, 0.0F, 0.0F };

float f(float arg1, float arg2)
{
    // Arbitrary computation
    return (arg1 * arg2);
}

float compute_result(int index)
{
    float row_sum = 0.0F;
    float col_sum = 0.0F;
    int row;
    int col;

    // Compute the row sum
    for (col = (index + 1); col < N; col++)
    {
        row_sum += P[index][col];
    }

    // Compute the column sum
    for (row = 0; row < index; row++)
    {
        col_sum += P[row][index];
    }

    return (row_sum - col_sum);
}

void direct_approach()
{
    int row;
    int col;

    for (row = 0; row < N; row++)
    {
        for (col = (row + 1); col < N; col++)
        {
            P[row][col] = f(input_values[row], input_values[col]);
        }
    }

    int index;

    for (index = 0; index < N; index++)
    {
        result1[index] = compute_result(index);
    }
}

void refined_approach()
{
    float row_sums[N];
    float col_sums[N];
    int index;

    // Initialize intermediate arrays
    for (index = 0; index < N; index++)
    {
        row_sums[index] = 0.0F;
        col_sums[index] = 0.0F;
    }

    // Compute the row and column sums
    // This can be parallelized by computing row and column sums
    //  independently, instead of in nested loops.
    int row;
    int col;

    for (row = 0; row < N; row++)
    {
        for (col = (row + 1); col < N; col++)
        {
            float computed = f(input_values[row], input_values[col]);
            row_sums[row] += computed;
            col_sums[col] += computed;
        }
    }

    // Compute the result
    for (index = 0; index < N; index++)
    {
        result2[index] = row_sums[index] - col_sums[index];
    }
}

void print_result(int n, float * result)
{
    int index;

    for (index = 0; index < n; index++)
    {
        printf("  [%d]=%f\n", index, result[index]);
    }
}

int main(int argc, char * * argv)
{
    printf("Data reduction test\n");

    direct_approach();

    printf("Result 1:\n");
    print_result(N, result1);

    refined_approach();

    printf("Result 2:\n");
    print_result(N, result2);

    return (0);
}

Parallelizing the computation is not so easy, since each intermediate value is a function of most of the inputs. You can compute the sums individually, but that would mean performing f(...) multiple times. The best suggestion I can think of for very large values of N is to use more intermediate arrays, computing subsets of the results, then summing the partial arrays to yield the final sums. I'd have to think about that one when I'm not so tired.

To cope with the skip issue: If it's a simple matter of "don't use input values x, y, and z", you can store x, y, and z in a do_not_use array and check for those values when looping to compute the sums. If the values to be skipped are some function of row and column, you can store those as pairs and check for the pairs.

Hope this gives you ideas for your solution!

Update, now that I'm awake: Dealing with "skip" depends a lot on what data needs to be skipped. Another possibility for the first case - "don't use input values x, y, and z" - a much faster solution for large data sets would be to add a level of indirection: create yet another array, this one of integer indices, and store only the indices of the good inputs. F'r instance, if invalid data is in inputs 2 and 5, the valid array would be:

int valid_indices[] = { 0, 1, 3, 4, 6 };

Interate over the array valid_indices, and use those indices to retrieve the data from your input array to compute the result. On the other paw, if the values to skip depend on both indices of the P array, I don't see how you can avoid some kind of lookup.

Back to parallelizing - No matter what, you'll be dealing with (N**2 - N)/2 computations of f(). One possibility is to just accept that there will be contention for the sum arrays, which would not be a big issue if computing f() takes substantially longer than the two additions. When you get to very large numbers of parallel paths, contention will again be an issue, but there should be a "sweet spot" balancing the number of parallel paths against the time required to compute f().

If contention is still an issue, you can partition the problem several ways. One way is to compute a row or column at a time: for a row at a time, each column sum can be computed independently and a running total can be kept for each row sum.

Another approach would be to divide the data space and, thus, the computation into subsets, where each subset has its own row and column sum arrays. After each block is computed, the independent arrays can then be summed to produce the values you need to compute the result.

Answer 2

This probably will be one of those naive and useless answers, but it also might help. Feel free to tell me that I'm utterly and completely wrong and I have misunderstood the whole affair.

So... here we go!

The Basic Problem

It seems to me that you can define you result function a little differently and it will lift at least some contention off your intermediate values. Let's suppose that your P matrix is lower-triangular. If you (virtually) fill the upper triangle with the negative of the lower values (and the main diagonal with all zeros,) then you can redefine each element of your result as the sum of a single row: (shown here for N=4, and where -i means the negative of the value in the cell marked as i)

 P     0    1    2    3 
   |--------------------
  0|   x   -0   -1   -3
   |
  1|   0    x   -2   -4
   |
  2|   1    2    x   -5
   |
  3|   3    4    5    x

If you launch independent threads (executing the same kernel) to calculate the sum of each row of this matrix, each thread will write a single result element. It seems that your problem size is large enough to saturate your hardware threads and keep them busy.

The caveat, of course, is that you'll be calculating each f(x, y) twice. I don't know how expensive that is, or how much the memory contention was costing you before, so I cannot judge whether this is a worthwhile trade-off to do or not. But unless f was really really expensive, I think it might be.

Skipping Values

You mention that you might have tens of thousands elements of the P matrix that you need to ignore in your calculations (effectively skip them.)

To work with the scheme I've proposed above, I believe you should store the skipped elements as (row, col) pairs, and you have to add the transposed of each coordinate pair too (so you'll have twice the number of skipped values.) So your example skip list of P[6], P[14] and P[18] becomes P(4,0), P(5,4), P(6,3) which then becomes P(4,0), P(5,4), P(6,3), P(0,4), P(4,5), P(3,6).

Then you sort this list, first based on row and then column. This makes our list to be P(0,4), P(3,6), P(4,0), P(4,5), P(5,4), P(6,3).

If each row of your virtual P matrix is processed by one thread (or a single instance of your kernel or whatever,) you can pass it the values it needs to skip. Personally, I would store all these in a big 1D array and just pass in the first and last index that each thread would need to look at (I would also not store the row indices in the final array that I passed in, since it can be implicitly inferred, but I think that's obvious.) In the example above, for N = 8, the begin and end pairs passed to each thread will be: (note that the end is one past the final value needed to be processed, just like STL, so an empty list is denoted by begin == end)

Thread 0: 0..1
Thread 1: 1..1 (or 0..0 or whatever)
Thread 2: 1..1
Thread 3: 1..2
Thread 4: 2..4
Thread 5: 4..5
Thread 6: 5..6
Thread 7: 6..6

Now, each thread goes on to calculate and sum all the intermediate values in a row. While it is stepping through the indices of columns, it is also stepping through this list of skipped values and skipping any column number that comes up in the list. This is obviously an efficient and simple operation (since the list is sorted by column too. It's like merging.)

Pseudo-Implementation

I don't know CUDA, but I have some experience working with OpenCL, and I imagine the interfaces are similar (since the hardware they are targeting are the same.) Here's an implementation of the kernel that does the processing for a row (i.e. calculates one entry of result) in pseudo-C++:

double calc_one_result (
    unsigned my_id, unsigned N, double const init_values [],
    unsigned skip_indices [], unsigned skip_begin, unsigned skip_end
)
{
    double res = 0;
    for (unsigned col = 0; col < my_id; ++col)
        // "f" seems to take init_values[column] as its first arg
        res += f (init_values[col], init_values[my_id]);
    for (unsigned row = my_id + 1; row < N; ++row)
        res -= f (init_values[my_id], init_values[row]);
    // At this point, "res" is holding "result[my_id]",
    // including the values that should have been skipped
    
    unsigned i = skip_begin;
    // The second condition is to check whether we have reached the
    // middle of the virtual matrix or not
    for (; i < skip_end && skip_indices[i] < my_id; ++i)
    {
        unsigned col = skip_indices[i];
        res -= f (init_values[col], init_values[my_id]);
    }
    for (; i < skip_end; ++i)
    {
        unsigned row = skip_indices[i];
        res += f (init_values[my_id], init_values[row]);
    }
    
    return res;
}

Note the following:

1. The semantics of init_values and function f are as described by the question.

2. This function calculates one entry in the result array; specifically, it calculates result[my_id], so you should launch N instances of this.

3. The only shared variable it writes to is result[my_id]. Well, the above function doesn't write to anything, but if you translate it to CUDA, I imagine you'd have to write to that at the end. However, no one else writes to that particular element of result, so this write will not cause any contention of data race.

4. The two input arrays, init_values and skipped_indices are shared among all the running instances of this function.

5. All accesses to data are linear and sequential, except for the skipped values, which I believe is unavoidable.

6. skipped_indices contain a list of indices that should be skipped in each row. It's contents and structure are as described above, with one small optimization. Since there was no need, I have removed the row numbers and left only the columns. The row number will be passed into the function as my_id anyways and the slice of the skipped_indices array that should be used by each invocation is determined using skip_begin and skip_end.

   For the example above, the array that is passed into all invocations of calc_one_result will look like this:[4, 6, 0, 5, 4, 3].

7. As you can see, apart from the loops, the only conditional branch in this code is skip_indices[i] < my_id in the third for-loop. Although I believe this is innocuous and totally predictable, even this branch can be easily avoided in the code. We just need to pass in another parameter called skip_middle that tells us where the skipped items cross the main diagonal (i.e. for row #my_id, the index at skipped_indices[skip_middle] is the first that is larger than my_id.)

In Conclusion

I'm by no means an expert in CUDA and HPC. But if I have understood your problem correctly, I think this method might eliminate any and all contentions for memory. Also, I don't think this will cause any (more) numerical stability issues.

The cost of implementing this is:

• Calling f twice as many times in total (and keeping track of when it is called for row < col so you can multiply the result by -1.)
• Storing twice as many items in the list of skipped values. Since the size of this list is in the thousands (and not billions!) it shouldn't be much of a problem.
• Sorting the list of skipped values; which again due to its size, should be no problem.

(UPDATE: Added the Pseudo-Implementation section.)