10. Combinations

This chapter describes functions for creating and manipulating combinations. A combination c is represented by an array of k integers in the range 0 .. n-1, where each value c_i is from the range 0 .. n-1 and occurs at most once. The combination c corresponds to indices of k elements chosen from an n element vector. Combinations are useful for iterating over all k-element subsets of a set.

The functions described in this chapter are defined in the header file `gsl_combination.h'.

10.1 The Combination struct

A combination is stored by a structure containing three components, the values of n and k, and a pointer to the combination array. The elements of the combination array are all of type size_t, and are stored in increasing order. The gsl_combination structure looks like this,

typedef struct
{
  size_t n;
  size_t k;
  size_t *data;
} gsl_combination;

10.2 Combination allocation

Function: gsl_combination * gsl_combination_alloc (size_t n, size_t k)

This function allocates memory for a new combination with parameters n, k. The combination is not initialized and its elements are undefined. Use the function gsl_combination_calloc if you want to create a combination which is initialized to the lexicographically first combination. A null pointer is returned if insufficient memory is available to create the combination.

Function: gsl_combination * gsl_combination_calloc (size_t n, size_t k)

This function allocates memory for a new combination with parameters n, k and initializes it to the lexicographically first combination. A null pointer is returned if insufficient memory is available to create the combination.

Function: void gsl_combination_init_first (gsl_combination * c)

This function initializes the combination c to the lexicographically first combination, i.e. (0,1,2,...,k-1).

Function: void gsl_combination_init_last (gsl_combination * c)

This function initializes the combination c to the lexicographically last combination, i.e. (n-k,n-k+1,...,n-1).

Function: void gsl_combination_free (gsl_combination * c)

This function frees all the memory used by the combination c.

Function: int gsl_combination_memcpy (gsl_combination * dest, const gsl_combination * src)

This function copies the elements of the combination src into the combination dest. The two combinations must have the same sizes.

10.3 Accessing combination elements

The following function can be used to access combinations elements.

Function: size_t gsl_combination_get (const gsl_combination * c, const size_t i)

This function returns the value of the i-th element of the combination c. If i lies outside the allowed range of 0 to k-1 then the error handler is invoked and 0 is returned.

10.4 Combination properties

Function: size_t gsl_combination_n (const gsl_combination * c)

This function returns the n parameter of the combination c.

Function: size_t gsl_combination_k (const gsl_combination * c)

This function returns the k parameter of the combination c.

Function: size_t * gsl_combination_data (const gsl_combination * c)

This function returns a pointer to the array of elements in the combination c.

Function: int gsl_combination_valid (gsl_combination * c)

This function checks that the combination c is valid. The k elements should contain numbers from range 0 .. n-1, each number at most once. The numbers have to be in increasing order.

10.5 Combination functions

Function: int gsl_combination_next (gsl_combination * c)

This function advances the combination c to the next combination in lexicographic order and returns GSL_SUCCESS. If no further combinations are available it returns GSL_FAILURE and leaves c unmodified. Starting with the first combination and repeatedly applying this function will iterate through all possible combinations of a given order.

Function: int gsl_combination_prev (gsl_combination * c)

This function steps backwards from the combination c to the previous combination in lexicographic order, returning GSL_SUCCESS. If no previous combination is available it returns GSL_FAILURE and leaves c unmodified.

10.6 Reading and writing combinations

The library provides functions for reading and writing combinations to a file as binary data or formatted text.

Function: int gsl_combination_fwrite (FILE * stream, const gsl_combination * c)

This function writes the elements of the combination c to the stream stream in binary format. The function returns GSL_EFAILED if there was a problem writing to the file. Since the data is written in the native binary format it may not be portable between different architectures.

Function: int gsl_combination_fread (FILE * stream, gsl_combination * c)

This function reads into the combination c from the open stream stream in binary format. The combination c must be preallocated with correct values of n and k since the function uses the size of c to determine how many bytes to read. The function returns GSL_EFAILED if there was a problem reading from the file. The data is assumed to have been written in the native binary format on the same architecture.

Function: int gsl_combination_fprintf (FILE * stream, const gsl_combination * c, const char *format)

This function writes the elements of the combination c line-by-line to the stream stream using the format specifier format, which should be suitable for a type of size_t. On a GNU system the type modifier Z represents size_t, so "%Zu\n" is a suitable format. The function returns GSL_EFAILED if there was a problem writing to the file.

Function: int gsl_combination_fscanf (FILE * stream, gsl_combination * c)

This function reads formatted data from the stream stream into the combination c. The combination c must be preallocated with correct values of n and k since the function uses the size of c to determine how many numbers to read. The function returns GSL_EFAILED if there was a problem reading from the file.

10.7 Examples

The example program below prints all subsets of the set {0,1,2,3} ordered by size. Subsets of the same size are ordered lexicographically.

#include <stdio.h>
#include <gsl/gsl_combination.h>

int 
main (void) 
{
  gsl_combination * c;
  size_t i;

  printf ("All subsets of {0,1,2,3} by size:\n") ;
  for (i = 0; i <= 4; i++)
    {
      c = gsl_combination_calloc (4, i);
      do
        {
          printf ("{");
          gsl_combination_fprintf (stdout, c, " %u");
          printf (" }\n");
        }
      while (gsl_combination_next (c) == GSL_SUCCESS);
      gsl_combination_free (c);
    }

  return 0;
}

Here is the output from the program,

bash$ ./a.out 
All subsets of {0,1,2,3} by size:
{ }
{ 0 }
{ 1 }
{ 2 }
{ 3 }
{ 0 1 }
{ 0 2 }
{ 0 3 }
{ 1 2 }
{ 1 3 }
{ 2 3 }
{ 0 1 2 }
{ 0 1 3 }
{ 0 2 3 }
{ 1 2 3 }
{ 0 1 2 3 }

All 16 subsets are generated, and the subsets of each size are sorted lexicographically.

10.8 References and Further Reading

Further information on combinations can be found in,

• Donald L. Kreher, Douglas R. Stinson, Combinatorial Algorithms: Generation, Enumeration and Search, 1998, CRC Press LLC, ISBN 084933988X
• Donald E. Knuth, The Art of Computer Programming: Combinatorial Algorithms (Vol 4, pre-fascicle 2c) http://www-cs-faculty.stanford.edu/~knuth/fasc2c.ps.gz

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