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# Sorting

This chapter describes functions for sorting data, both directly and indirectly (using an index). All the functions use the heapsort algorithm. Heapsort is an O(N \log N) algorithm which operates in-place. It does not require any additional storage and provides consistent performance. The running time for its worst-case (ordered data) is not significantly longer than the average and best cases. Note that the heapsort algorithm does not preserve the relative ordering of equal elements -- it is an unstable sort. However the resulting order of equal elements will be consistent across different platforms when using these functions.

## Sorting objects

The following function provides a simple alternative to the standard library function `qsort`. It is intended for systems lacking `qsort`, not as a replacement for it. The function `qsort` should be used whenever possible, as it will be faster and can provide stable ordering of equal elements. Documentation for `qsort` is available in the GNU C Library Reference Manual.

The functions described in this section are defined in the header file ``gsl_heapsort.h'`.

Function: void gsl_heapsort (void * array, size_t count, size_t size, gsl_comparison_fn_t compare)

  This function sorts the count elements of the array array, each of size size, into ascending order using the comparison function compare. The type of the comparison function is defined by,

      int (*gsl_comparison_fn_t) (const void * a,
                                  const void * b)

  A comparison function should return a negative integer if the first argument is less than the second argument, `0` if the two arguments are equal and a positive integer if the first argument is greater than the second argument.

  For example, the following function can be used to sort doubles into ascending numerical order.

      int
      compare_doubles (const double * a,
                       const double * b)
      {
          if (*a > *b)
             return 1;
          else if (*a < *b)
             return -1;
          else
             return 0;
      }

  The appropriate function call to perform the sort is,

      gsl_heapsort (array, count, sizeof(double),
                    compare_doubles);

  Note that unlike `qsort` the heapsort algorithm cannot be made into a stable sort by pointer arithmetic. The trick of comparing pointers for equal elements in the comparison function does not work for the heapsort algorithm. The heapsort algorithm performs an internal rearrangement of the data which destroys its initial ordering.

Function: int gsl_heapsort_index (size_t * p, const void * array, size_t count, size_t size, gsl_comparison_fn_t compare)

  This function indirectly sorts the count elements of the array array, each of size size, into ascending order using the comparison function compare. The resulting permutation is stored in p, an array of length n. The elements of p give the index of the array element which would have been stored in that position if the array had been sorted in place. The first element of p gives the index of the least element in array, and the last element of p gives the index of the greatest element in array. The array itself is not changed.

## Sorting vectors

The following functions will sort the elements of an array or vector, either directly or indirectly. They are defined for all real and integer types using the normal suffix rules. For example, the `float` versions of the array functions are `gsl_sort_float` and `gsl_sort_float_index`. The corresponding vector functions are `gsl_sort_vector_float` and `gsl_sort_vector_float_index`. The prototypes are available in the header files ``gsl_sort_float.h'` ``gsl_sort_vector_float.h'`. The complete set of prototypes can be included using the header files ``gsl_sort.h'` and ``gsl_sort_vector.h'`.

There are no functions for sorting complex arrays or vectors, since the ordering of complex numbers is not uniquely defined. To sort a complex vector by magnitude compute a real vector containing the the magnitudes of the complex elements, and sort this vector indirectly. The resulting index gives the appropriate ordering of the original complex vector.

Function: void gsl_sort (double * data, size_t stride, size_t n)
  This function sorts the n elements of the array data with stride stride into ascending numerical order.

Function: void gsl_sort_vector (gsl_vector * v)
  This function sorts the elements of the vector v into ascending numerical order.

Function: int gsl_sort_index (size_t * p, const double * data, size_t stride, size_t n)
  This function indirectly sorts the n elements of the array data with stride stride into ascending order, storing the resulting permutation in p. The array p must be allocated to a sufficient length to store the n elements of the permutation. The elements of p give the index of the array element which would have been stored in that position if the array had been sorted in place. The array data is not changed.

Function: int gsl_sort_vector_index (gsl_permutation * p, const gsl_vector * v)
  This function indirectly sorts the elements of the vector v into ascending order, storing the resulting permutation in p. The elements of p give the index of the vector element which would have been stored in that position if the vector had been sorted in place. The first element of p gives the index of the least element in v, and the last element of p gives the index of the greatest element in v. The vector v is not changed.

## Selecting the k-th smallest or largest elements

The functions described in this section select the k-th smallest or largest elements of a data set of size N. The routines use an O(kN) direct insertion algorithm which is suited to subsets that are small compared with the total size of the dataset. For example, the routines are useful for selecting the 10 largest values from one million data points, but not for selecting the largest 100,000 values. If the subset is a significant part of the total dataset it may be faster to sort all the elements of the dataset directly with an O(N \log N) algorithm and obtain the smallest or largest values that way.

Function: void gsl_sort_smallest (double * dest, size_t k, const double * src, size_t stride, size_t n)
  This function copies the k-th smallest elements of the array src, of size n and stride stride, in ascending numerical order in dest. The size of the subset k must be less than or equal to n. The data src is not modified by this operation.

Function: void gsl_sort_largest (double * dest, size_t k, const double * src, size_t stride, size_t n)
  This function copies the k-th largest elements of the array src, of size n and stride stride, in descending numerical order in dest. The size of the subset k must be less than or equal to n. The data src is not modified by this operation.

Function: void gsl_sort_vector_smallest (double * dest, size_t k, const gsl_vector * v)
Function: void gsl_sort_vector_largest (double * dest, size_t k, const gsl_vector * v)
  These functions copy the k-th smallest or largest elements of the vector v into the array dest. The size of the subset k must be less than or equal to the length of the vector v.

The following functions find the indices of the k-th smallest or largest elements of a dataset,

Function: void gsl_sort_smallest_index (size_t * p, size_t k, const double * src, size_t stride, size_t n)
  This function stores the indices of the k-th smallest elements of the array src, of size n and stride stride, in the array p. The indices are chosen so that the corresponding data is in ascending numerical order. The size of the subset k must be less than or equal to n. The data src is not modified by this operation.

Function: void gsl_sort_largest_index (size_t * p, size_t k, const double * src, size_t stride, size_t n)
  This function stores the indices of the k-th largest elements of the array src, of size n and stride stride, in the array p. The indices are chosen so that the corresponding data is in descending numerical order. The size of the subset k must be less than or equal to n. The data src is not modified by this operation.

Function: void gsl_sort_vector_smallest_index (size_t * p, size_t k, const gsl_vector * v)
Function: void gsl_sort_vector_largest_index (size_t * p, size_t k, const gsl_vector * v)
  These functions store the indices of k-th smallest or largest elements of the vector v in the array p. The size of the subset k must be less than or equal to the length of the vector v.

## Computing the rank

The rank of an element is its order in the sorted data. The rank is the inverse of the index permutation, p. It can be computed using the following algorithm,

    for (i = 0; i < p->size; i++)
    {
        size_t pi = p->data[i];
        rank->data[pi] = i;
    }

This can be computed directly from the function `gsl_permutation_invert(rank,p)`.

The following function will print the rank of each element of the vector v,

    void
    print_rank (gsl_vector * v)
    {
      size_t i;
      size_t n = v->size;
      gsl_permutation * perm = gsl_permutation_alloc(n);
      gsl_permutation * rank = gsl_permutation_alloc(n);

      gsl_sort_vector_index (perm, v);
      gsl_permutation_invert (rank, perm);

      for (i = 0; i < n; i++)
       {
        double vi = gsl_vector_get(v, i);
        printf("element = %d, value = %g, rank = %d\n",
                i, vi, rank->data[i]);
       }

      gsl_permutation_free (perm);
      gsl_permutation_free (rank);
    }

## Examples

The following example shows how to use the permutation p to print the elements of the vector v in ascending order,

    gsl_sort_vector_index (p, v);

    for (i = 0; i < v->size; i++)
    {
        double vpi = gsl_vector_get(v, p->data[i]);
        printf("order = %d, value = %g\n", i, vpi);
    }

The next example uses the function `gsl_sort_smallest` to select the 5 smallest numbers from 100000 uniform random variates stored in an array,

    #include <gsl/gsl_rng.h>
    #include <gsl/gsl_sort_double.h>

    int
    main (void)
    {
      const gsl_rng_type * T;
      gsl_rng * r;

      int i, k = 5, N = 100000;

      double * x = malloc (N * sizeof(double));
      double * small = malloc (k * sizeof(double));

      gsl_rng_env_setup();

      T = gsl_rng_default;
      r = gsl_rng_alloc (T);

      for (i = 0; i < N; i++)
        {
          x[i] = gsl_rng_uniform(r);
        }

      gsl_sort_smallest (small, k, x, 1, N);

      printf("%d smallest values from %d\n", k, N);

      for (i = 0; i < k; i++)
        {
          printf ("%d: %.18f\n", i, small[i]);
        }
      return 0;
    }

The output lists the 5 smallest values, in ascending order,

    $ ./a.out
    5 smallest values from 100000
    0: 0.000005466630682349
    1: 0.000012384494766593
    2: 0.000017581274732947
    3: 0.000025131041184068
    4: 0.000031369971111417

## References and Further Reading

The subject of sorting is covered extensively in Knuth's Sorting and Searching,

- Donald E. Knuth, The Art of Computer Programming: Sorting and Searching (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.

The Heapsort algorithm is described in the following book,

- Robert Sedgewick, Algorithms in C, Addison-Wesley, ISBN 0201514257.

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