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Ranking Algorithm

  1. Jun 22, 2015 #1
    Not sure if this is the correct place to post this type of question, so I will relocate if need be. What I'm interested in doing is taking an array of numbers and assigning a rank based on ascending order. Explicitly, the ranking of an array would look like

    1.534 - 15
    -0.887 - 3
    -0.489 - 5
    0.887 - 13
    1.15 - 14
    0.157 - 9
    -1.15 - 2
    0 - 8
    0.319 - 10
    -0.319 - 6
    -1.534 - 1
    -0.157 - 7
    0.489 - 11
    0.674 - 12
    -0.674 - 4

    So I don't actually want to re-order the array, rather assign a number based on the position's size. I'm coding this in C. It is easy to get the min and max, but how would I go about the rest of the ranking in an efficient manner? Looking ahead, I would like to do this on an array of length 25000!

    Thanks in advance
     
  2. jcsd
  3. Jun 22, 2015 #2
    I'm thinking of something like

    sort array and store in new array sorted
    for i from 0 to size(sorted)
    for j from 0 to size(sorted)
    if sorted(i) == unsorted(j) then rank(i) = j

    where unsorted is the original array. Not sure how efficient this is, but it should work.
     
  4. Jun 22, 2015 #3

    Svein

    User Avatar
    Science Advisor

    In C, you would most likely do it with pointers. Create a matching array with pointers to the elements in the first array and sort them (check your libraries for quicksort).
     
  5. Jun 22, 2015 #4

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    That's not very efficient, and there's a problem if you have duplicate elements.

    First things first:
    • This is C, not Fortran. The obvious value in C for the smallest value of ranking is zero, not one. Zero works much nicer in C than does one.
    • Do you need to know how the array was organized prior to sorting? If you don't, just sort the array and be done with it. In that case, the index numbers *are* the rankings.
    Assuming that maintaining the original ordering is important, I'll outline a solution. Oftentimes, it's handy to sort the indices rather than the values. Doing this maintains the original structure. You'll need a comparison function:
    Code (Text):
    static const double* array_to_be_sorted;
    void set_array_to_be_sorted (const double* arr)
    {
       array_to_be_sorted = arr;
    }

    int compare_by_index (const void* a, const void* b)
    {
       double da = array_to_be_sorted[*(size_t*)a];
       double db = array_to_be_sorted[*(size_t*)b];
       return (da > db) - (da < db);
    }
    Note that the above uses the global (but static, meaning hidden from the linker) variable array_to_be_sorted. In C++, this need for a global vanishes if one uses a functor or a lambda.

    With this, you can sort the array by indices:
    Code (Text):
       size_t* sorted = malloc (len*sizeof(size_t));
       for (size_t idx = 0; idx < len; idx++) { sorted[idx] = idx; }
       set_array_to_be_sorted (arr);
       qsort (sorted, len, sizeof(size_t), compare_by_index);
     
    With this, the array sorted gives the index of the elements in arr had that array itself been sorted. You can use these to find the zero-based ranking of elements in arr via
    Code (Text):
       size_t* rank = malloc (len*sizeof(size_t));
       for (size_t idx = 0; idx < len; idx++) { rank[sorted[idx]] = idx; }
     
  6. Jun 22, 2015 #5

    wle

    User Avatar

    You can also avoid it by sorting an array of pointers, like Svein suggested.

    I could use the practice, so I typed a working example:

    Code (C):
    #include <stdio.h>
    #include <stdlib.h>


    typedef int (*qsort_comp)(const void *, const void *);

    int dptr_comp(const double **px, const double **py)
    {
        double x = **px, y = **py;
        return (x > y) - (x < y);
    }

    int rank_elts(const double *base, size_t *rank, size_t n_elts)
    {
        const double **elts = malloc(sizeof(base) * n_elts);

        if (!elts)
            return 0;

        for (size_t i = 0; i < n_elts; ++i)
            elts[i] = base + i;

        qsort(elts, n_elts, sizeof(*elts), (qsort_comp)dptr_comp);

        for (size_t i = 0; i < n_elts; ++i)
            rank[elts[i] - base] = i + 1;

        free(elts);
        return 1;
    }


    /* Try it out. */

    int main(int argc, char **argv)
    {
        double my_data[] = { 1.534, -0.887, -0.489,  0.887,  1.15,
                             0.157, -1.15,   0.0,    0.319, -0.319,
                            -1.534, -0.157,  0.489,  0.674, -0.674};
       
        size_t rank[sizeof(my_data) / sizeof(*my_data)];
        size_t n_elts = sizeof(my_data) / sizeof(*my_data);

        if ( !rank_elts(my_data, rank, n_elts) ) {
            if (argc > 0)
                fprintf(stderr, "%s: ", argv[0]);

            fputs("error: unable to allocate memory.\n", stderr);
            return EXIT_FAILURE;
        }

        for (size_t i = 0; i < n_elts; ++i)
            printf("%6.3f => %2zu\n", my_data[i], rank[i]);

        return 0;
    }
    This uses that &my_data[n] - &my_data[0] == n in order to recover the index.
     
  7. Jun 23, 2015 #6
    Thanks for the responses! I'll be testing out how quick these are on a 25000x128 size this week and updating with issues/progress.
     
  8. Jun 23, 2015 #7

    D H

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    Staff Emeritus
    Science Advisor

    If you want to speed things up a bit, take my approach but use an array of `int` for the indices rather than an array of `size_t`, and change the types of the loop indices from `size_t` to `int`. On a 64 bit machine, an array of pointers will take twice the size as an array of integers. With 3200000 (25000x128) elements, you won't be able to have all of your data in level 2 cache. That reduction in size will result in fewer cache misses. While using an array of pointers (wle's suggestion) does eliminate the need for a global variable, it comes at the expense of double indirection and increased memory consumption.
     
  9. Jun 24, 2015 #8
    Alright boys, what did I do wrong?

    Code (C):
    #include <stdio.h>
    #include <stdlib.h>

    static const double* array_to_be_sorted;

    void set_array_to_be_sorted(const double* arr)
    {
      array_to_be_sorted = arr;
    }

    int compare_by_index(const void* a, const void* b)
    {
      double da = array_to_be_sorted[*(int*)a];
      double db = array_to_be_sorted[*(int*)b];
      return (da > db) - (da < db);
    }

    int main(int argc, char **argv)
    {
      int len = 15;
      double arr[] = { 1.534, -0.887, -0.489, 0.887,  1.15,
      0.157, -1.14,  0.0,  0.319, -0.319,
      -1.534, -0.157,  0.489, 0.674, -0.674 };
      int* sorted = malloc(len*sizeof(int));
      int* rank = malloc(len*sizeof(int));

      for (int idx = 0; idx < len; idx++)
      {
      sorted[idx] = idx;
      }
      set_array_to_be_sorted(arr);
      qsort(sorted, len, sizeof(int), compare_by_index);
      for (int idx = 0; idx < len; idx++)
      {
      rank[sorted[idx]] = idx;
      }

      for (int idx = 0; idx < len; idx++)
      {
      printf("\t%f\t%d\t%d\n", arr[idx], sorted[idx], rank[idx]);
      }

      return 0;
    }
    Gives output

    Code (Text):

      1.534000  10  14
      -0.887000  6  2
      -0.489000  1  4
      0.887000  14  12
      1.150000  2  13
      0.157000  9  8
      -1.140000  11  1
      0.000000  7  7
      0.319000  5  9
      -0.319000  8  5
      -1.534000  12  0
      -0.157000  13  6
      0.489000  3  10
      0.674000  4  11
      -0.674000  0  3
     
     
  10. Jun 24, 2015 #9

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    The only thing you did wrong was to pay attention to the array sorted. That array is but an intermediary that helps your get what you want, which is the ranking. Taking your exact output, and discarding the middle item, and prettying things up a bit yields

    Code (Text):

      1.534000   14  
     -0.887000    2  
     -0.489000    4  
      0.887000   12  
      1.150000   13  
      0.157000    8  
     -1.140000    1  
      0.000000    7  
      0.319000    9  
     -0.319000    5  
     -1.534000    0  
     -0.157000    6  
      0.489000   10  
      0.674000   11  
     -0.674000    3
     
    That's exactly what you want.
     
  11. Jun 24, 2015 #10
    Brain fart! I'm going to attempt to modify things a bit to fit in my overall program. I will update if I run into any issues.

    Thanks again for the help.
     
  12. Jun 29, 2015 #11
    DH,

    I have successfully implemented your routine! I modified the above code to be a subroutine taking in a NxK matrix and displaying the sorted columns individually with the following results:

    Code (Text):

    Time to sort:
    Number of iterations: 25000
    Number of variables: 128
    Total time: 0.481050 (s)
     
    (I have verified the results as well.) The next step for me is to reorder the original column based off of this new rank order and iterate the "guts" of my program a certain number of times. I'm very happy with the sorting time of your method! Thanks again!

    I will post my (probably naive) method of reordering the original column sometime this week. Hopefully y'all can help me out with another slick way of doing it!
     
  13. Jun 29, 2015 #12
    OK after doing some scribbling on paper, here is a brute force method I've come up with

    Code (Text):

    read in R, R*, m, n to subroutine sort
    /* looking to resort columns of R based off of rank of columns of R */
    /* m is number of rows, n is number of columns (R & R* are mxn) */
    for i = 0, n
       get rank of column i for R, R* with current code (store in rankR, rankR*, respectively)
       for j = 0, m
          for k = 0, m
             if (rankR(k) == rankR*(j))
                 tmp(j) = R(rankR(k)) /* can use sorted array as tmp to save memory since it is available after ranking is completed */
             endif
          end k for
       end j for
       R = tmp
    end i for
     
    Is there possibly a slicker way to do this? I had no idea qsort existed, so there may be something clever that I don't even know about!

    Thanks in advance.[/code]
     
  14. Jun 29, 2015 #13

    wle

    User Avatar

    Assuming rankR and rankR* contain the same integer ranks, shouldn't this bit of your pseudocode:
    Code (Text):
          for k = 0, m
             if (rankR(k) == rankR*(j))
                 tmp(j) = R(rankR(k)) /* can use sorted array as tmp to save memory since it is available after ranking is completed */
             endif
          end k for
    just reduce to this:
    Code (Text):
          tmp(j) = R(rankR*(j))
    ?

    It's not very clear from your explanations what you actually want to do. Guessing from your pseudocode, it looks like you've got some matrix R and you want to sort its columns so they're in the same order as the columns of another matrix R*. So for example if R*(0,i) is the 5th smallest element in the ith column of R* then, after sorting, you want R(0,i) to be the 5th smallest element in the ith column of R. Is this correct?

    Incidentally, if you're using C then you should seriously consider storing your data to be sorted by row instead of by column. In general in C, the array elements a[j][k] and a[j][k+1] are adjacent to one another in memory; the array elements a[j][k] and a[j+1][k] are not. a[j] also is (or evaluates to) a pointer to the first element of the jth row, so you can write code like
    Code (C):

    double *v = R[j];
    v[k] = some_value;  // Same thing as R[j][k] = some_value;
    foo(v);             // Call a function that operates on a row.
                        //   Same thing as foo(R[j]);
     
     
  15. Jun 29, 2015 #14
    wle,

    You're exactly right with the row-major bit. I'm so used to Fortran HPC that I often mix these up. I have a Pearsons correlation routine in this program that could possibly be sped up by simply transposing the data.

    This is correct.
     
  16. Jun 29, 2015 #15

    wle

    User Avatar

    In this case, is there a reason you can't simply sort both arrays? What you want to do will involve sorting two arrays anyway.

    If you specifically want to sort an array so it's in the same order as another array, then the first point is that the ranks aren't what you want in the end, so you can delete the final loop that assigns them (the loop over rank[sorted[idx]] = idx; from DH's post). You just need a shorter function that orders the indices of a given array:
    Code (C):
    void sort_indices(const double *arr, size_t *ind, size_t len)
    {
       for (size_t j = 0; j < len; ++j)
           ind[j] = j;

       set_array_to_be_sorted(arr);
       qsort(ind, len, sizeof(ind), compare_by_index);
    }
    This sorts an array of indices so that, for example, if ind[0] == 5, it means that arr[5] is the smallest element of the array.

    Now let's say you've got two arrays, a and b, and you want to sort a so that its order matches b. You'll want to generate two corresponding arrays of sorted indices, ind_a and ind_b, and use them to relocate the elements of a. You can do this by running tmp[ind_b[j]] = a[ind_a[j]] in a loop* and then copying the contents of tmp to a:
    Code (C):
    void matching_sort(double *a, const double *b, size_t len)
    {
        size_t *ind_a = malloc(len * sizeof(size_t));
        size_t *ind_b = malloc(len * sizeof(size_t));
        double *tmp = malloc(len * sizeof(*a));

        sort_indices(a, ind_a, len);
        sort_indices(b, ind_b, len);

        for (size_t j = 0; j < len; ++j)
            tmp[ind_b[j]] = a[ind_a[j]];

        for (size_t j = 0; j < len; ++j)
            a[j] = tmp[j];

        free(ind_a);
        free(ind_b);
        free(tmp);
    }

    Finally, since it's actually two tables of numbers you've got, you can simply sort each row in a loop:
    Code (C):
    int main(void)
    {
        /* Generate/read your data and store it by row in arrays r and s. */

        for (size_t i = 0; i < number_of_rows; ++i)
            matching_sort(r[i], s[i], row_length);

        /* The rows of r are now ordered the same way as the rows of s. */

        return 0;
    }

    A few remarks:
    1. I've omitted error handling for simplicity, to avoid cluttering the example code above. In particular, matching_sort() tries to allocate temporary storage, which can fail (malloc() returns a null pointer to signal this). It's a good habit to explicitly test for and handle this sort of thing (see my post #5 above for example).
    2. The way I've structured this code, matching_sort() allocates temporary storage every time you call it on a new row of data, which might significantly affect performance. It's possible to avoid this (e.g., make ind_a, ind_b, and tmp global variables or make matching_sort() take them as additional parameters, and allocate the necessary storage just once in advance, or just write matching_sort() to operate on the entire table) if you find that this slows your code more than you're willing to accept.
    3. You can change the size_ts to ints like DH suggested, though my impression is that this is a bit of a hack and a potential bug (I might be wrong and maybe DH can comment on this, but my understanding is that size_t is guaranteed to be large enough to hold any (nonnegative) array index, while there's no such guarantee with int, though in practice this shouldn't be an issue with the array sizes you want to manipulate).


    * a[ind_a[j]] is the jth smallest element of a (counting from j = 0) before we've sorted it. b[ind_b[j]] is similarly the jth smallest element of b. If the end goal is for a to be ordered the same way as b, this means that, after reordering, we want a[ind_b[j]] to be the jth smallest element of a, i.e., we want to move a[ind_a[j]] ##\mapsto## a[ind_b[j]].
     
    Last edited: Jun 29, 2015
  17. Jun 30, 2015 #16
    wle,

    This looks great. Per your row-major suggestion, I'm going to run back through my code and transpose the data off the bat so that I'm working with 128x25000 in the end rather than 25000x128. I don't know why I didn't think of that before! I'm hoping to get this rank and sort routine runtime down as low as possible since they may be performed upwards of a million times.

    I will let you know when I get through with that and onto your suggestions above. For the time being, the general idea of what I'm doing is this:

    Code (Text):
    input matrix A
    do while some criteria is not met
       perform some linear algebra stuff to calculate A*
       reshuffle A based on rank of A*
       check criteria
    end do
     
  18. Jun 30, 2015 #17
    wle,

    I actually do need these rank values (for A*) to calculate the Spearman rho values between rows (I successfully converted to row major) as the Spearmans rank correlation matrix is part of the criteria for convergence.
     
  19. Jun 30, 2015 #18
    wle,

    I'm having some trouble implementing your method. I'm reading in the R and R* matrices and selecting their rows to operate the sort on, but am running into a segfault somehow. Here is my current code

    Code (C):
    #include <stdio.h>

    #include <stdlib.h> // qsort
    #include <mkl.h>
    #include "params.h"

    static const REAL* array_to_be_sorted;

    int compare_by_indx(const void* a, const void* b)
    {
       REAL da = array_to_be_sorted[*(int*)a];
       REAL db = array_to_be_sorted[*(int*)b];
       return (da > db) - (da < db);
    }

    void set_array_to_be_sorted(const REAL* arr)
    {
       array_to_be_sorted = arr;
    }

    void sort_indices(const REAL *arr, int *ind, int len)
    {
       for (int i = 0; i < len; i++)
       {
          ind[i] = i;
       }
       set_array_to_be_sorted(arr);

       printf("Printing first row of R\n");
       for (int i = 0; i < len; i++)
       {
          printf("%f ", arr[i]);
       }

       qsort(ind, len, sizeof(ind), compare_by_indx);
       printf("finished that too\n");
    }

    void matching_sort(REAL *r_data, REAL *rstarb_data, int m_dim, int n_dim)
    {
       int *ind_a = mkl_malloc(n_dim * sizeof(int), 64);
       int *ind_b = mkl_malloc(n_dim * sizeof(int), 64);
       REAL *tmp = mkl_malloc(n_dim * sizeof(REAL), 64);
       REAL *a = mkl_malloc(n_dim * sizeof(REAL), 64);
       REAL *b = mkl_malloc(n_dim * sizeof(REAL), 64);

       if (ind_a == NULL || ind_b == NULL || tmp == NULL || a == NULL || b == NULL)
       {
          printf("\nERROR: Cannot allocate memory for matrices in MATCHING_SORT. Aborting...\n\n");
          mkl_free(ind_a);
          mkl_free(ind_b);
          mkl_free(tmp);
          mkl_free(a);
          mkl_free(b);
          exit(0);
       }
       else // Zero out arrays
       {
          for (int i = 0; i < n_dim; i++)
          {
             ind_a[i] = 0;
             ind_b[i] = 0;
             tmp[i] = 0.0;
             ind_a[i] = 0.0;
             ind_b[i] = 0.0;
          }
       }

       for (int i = 0; i < m_dim; i++)
       {
          int indx = i * n_dim;
          for (int j = 0; j < n_dim; j++)
          {
             a[j] = r_data[indx + j];
             b[j] = rstarb_data[indx + j];
             printf("%f %f\n", a[j], b[j]);
          }

          sort_indices(a, ind_a, n_dim);
          sort_indices(b, ind_b, n_dim);

          for (int j = 0; j < n_dim; j++)
          {
             tmp[ind_b[j]] = a[ind_a[j]];
          }

          for (int j = 0; j < n_dim; j++)
          {
             r_data[j] = tmp[j];
          }
       }

       mkl_free(a);
       mkl_free(b);
       mkl_free(ind_a);
       mkl_free(ind_b);
       mkl_free(tmp);
    }
     
    and with these debug statements, I get the following output

    Code (Text):
    1.534000 1.534000
    -0.887000 -0.887000
    -0.489000 -0.489000
    0.887000 0.887000
    1.150000 1.150000
    0.157000 0.157000
    -1.150000 -1.150000
    0.000000 0.000000
    0.319000 0.319000
    -0.319000 -0.319000
    -1.534000 -1.534000
    -0.157000 -0.157000
    0.489000 0.489000
    0.674000 0.674000
    -0.674000 -0.674000
    Printing first row of R
    Segmentation fault (core dumped)
     
    Do you have any idea why I'm getting the segfault?
     
  20. Jun 30, 2015 #19
    wle,

    I fixed the error by basically inlining your sort_indices from my already working ranking routine. This produces what I need

    Code (C):
    #include <stdio.h>

    #include <stdlib.h> // qsort
    #include <mkl.h>
    #include "params.h"

    static REAL* array_to_be_ranked;

    int compare_by_index(const void* a, const void* b)
    {
       REAL da = array_to_be_ranked[*(int*)a];
       REAL db = array_to_be_ranked[*(int*)b];
       return (da > db) - (da < db);
    }

    void set_array_to_be_ranked(REAL* arr)
    {
       array_to_be_ranked = arr;
    }

    void matching_sort(REAL *r_data, REAL *rstarb_data, int m_dim, int n_dim)
    {
       int *ind_a = mkl_malloc(n_dim * sizeof(int), 64);
       int *ind_b = mkl_malloc(n_dim * sizeof(int), 64);
       int *rank_a = mkl_malloc(n_dim * sizeof(int), 64);
       int *rank_b = mkl_malloc(n_dim * sizeof(int), 64);
       REAL *tmp = mkl_malloc(n_dim * sizeof(REAL), 64);
       REAL *a = mkl_malloc(n_dim * sizeof(REAL), 64);
       REAL *b = mkl_malloc(n_dim * sizeof(REAL), 64);

       if (ind_a == NULL || ind_b == NULL || tmp == NULL || a == NULL || b == NULL)
       {
          printf("\nERROR: Cannot allocate memory for matrices in MATCHING_SORT. Aborting...\n\n");
          mkl_free(ind_a);
          mkl_free(ind_b);
          mkl_free(tmp);
          mkl_free(a);
          mkl_free(b);
          exit(0);
       }
       else // Zero out arrays
       {
          for (int i = 0; i < n_dim; i++)
          {
             ind_a[i] = 0;
             ind_b[i] = 0;
             tmp[i] = 0.0;
             a[i] = 0.0;
             b[i] = 0.0;
          }
       }

       for (int i = 0; i < m_dim; i++)
       {
          int indx = i * n_dim;
          for (int j = 0; j < n_dim; j++)
          {
             ind_a[j] = j;
             ind_b[j] = j;
             a[j] = r_data[indx + j];
             b[j] = rstarb_data[indx + j];
          }

          set_array_to_be_ranked(a);
          qsort(ind_a, n_dim, sizeof(int), compare_by_index);

          set_array_to_be_ranked(b);
          qsort(ind_b, n_dim, sizeof(int), compare_by_index);

          for (int j = 0; j < n_dim; j++)
          {
             tmp[ind_b[j]] = a[ind_a[j]];
          }

          for (int j = 0; j < n_dim; j++)
          {
             r_data[indx + j] = tmp[j];
          }
       }

       mkl_free(ind_a);
       mkl_free(ind_b);
       mkl_free(tmp);
       mkl_free(a);
       mkl_free(b);
    }
    I have modified all of my routines (with the exception of this one which I will do later) to read in the arrays they used to allocate to save on memory cost per your suggestion. Once I get that done, put everything in a do-while loop to terminate on a criteria, and verify the results with the white paper I'm testing, it'll be time for me to go to town on vectorization, parallelization, and other optimizations.

    You guys have been a great help, I really appreciate it. If you're interested, I'll give initial timings for the 25000x128 case and update with speedups from the optimizations I plan on making.
     
  21. Jul 4, 2015 #20

    wle

    User Avatar

    I think that was my fault. My guess is that the problem was probably the point where you call qsort() in your sort_indices() function:
    The argument sizeof(ind) should have been sizeof(*ind) or sizeof(ind[0]). On my PC (x86-64 architecture), this error doesn't show up if ind is of type size_t *, but an int is 4 bytes while an int * is 8 bytes, which would cause qsort() to think the array of integers to be sorted is spread over twice as much memory as it really is. Using sizeof(int), like in your updated code, also fixes this, though it means that you'll have to remember to also change the argument to qsort() if you later decide to change the type of integer you use in your arrays.

    (It's not obvious that the crash occurs earlier since standard output is buffered, meaning that just because you called printf() in the immediately preceding loop doesn't mean the data is immediately displayed. You can call fflush(stdout) to force currently buffered data to be displayed.)


    Well the first thing to consider when optimising is whether it's worth it at all. By the sound of things you've already spent at least a week or two developing this code. How often and how long is it actually going to run? If the unoptimised version runs in, say, a couple of hours or even a day or two, is it really a good use of your time to spend many hours or days optimising it?

    That said, one thing you can certainly do easily is remove redundant code, since this has the practical benefit of shortening your code (and hence making it clearer) in addition to speeding it up a bit. In your version of matching_sort(),
    1. You declare two pointers, int *rank_a and int *rank_b and allocate memory to them, but you never seem to actually use them.
    2. This bit of code, which zero-initialises a lot of memory, is redundant:
      You later overwrite all of this memory before doing anything important with it anyway.
    3. You also declare and allocate memory for pointers REAL *a and REAL *b which seem unnecessary to me. Since the calls to qsort() don't modify your arrays of REAL numbers, you don't need to copy rows of your initial data. So you can delete all the code involving a and b and change the code using qsort() and the subsequent allocation to tmp to use r_data and rstarb_data directly:
      Code (C):
            set_array_to_be_ranked(r_data + indx);
            qsort(ind_a, n_dim, sizeof(int), compare_by_index);

            set_array_to_be_ranked(rstarb_data + indx);
            qsort(ind_b, n_dim, sizeof(int), compare_by_index);

            for (int j = 0; j < n_dim; j++)
            {
               tmp[ind_b[j]] = r_data[indx + ind_a[j]];
            }
      Alternatively, you can just set a = r_data + indx and b = rstarb_data + indx (so that for example a[j] becomes a synonym for r_data[indx + j]) and don't use mkl_malloc() on or copy to a or b, if you find this clearer.
    4. A minor detail, but ++i is a simpler instruction than i++ (semantically, the latter needs to save a copy of the initial value somewhere). In practice your compiler will probably produce identical code if you don't use the return value, but older or less sophisticated compilers might not. I've just checked that both GCC and Clang produce identical code for ++i and i++ (even without -On optimisations), but PCC (an admittedly older and less developed compiler) doesn't unless you explicitly turn optimisation on.
     
    Last edited: Jul 4, 2015
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