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This chapter describes functions for creating and manipulating
permutations. A permutation p is represented by an array of
n integers in the range 0 .. n-1, where each value
p_i occurs once and only once. The application of a permutation
p to a vector v yields a new vector v' where
v'_i = v_{p_i}.
For example, the array (0,1,3,2) represents a permutation
which exchanges the last two elements of a four element vector.
The corresponding identity permutation is (0,1,2,3).
Note that the permutations produced by the linear algebra routines
correspond to the exchange of matrix columns, and so should be considered
as applying to row-vectors in the form v' = v P rather than
column-vectors, when permuting the elements of a vector.
The functions described in this chapter are defined in the header file
`gsl_permutation.h'.
A permutation is stored by a structure containing two components, the size
of the permutation and a pointer to the permutation array. The elements
of the permutation array are all of type size_t. The
gsl_permutation structure looks like this,
typedef struct
{
size_t size;
size_t * data;
} gsl_permutation;
- Function: gsl_permutation * gsl_permutation_alloc (size_t n)
-
This function allocates memory for a new permutation of size n.
The permutation is not initialized and its elements are undefined. Use
the function
gsl_permutation_calloc if you want to create a
permutation which is initialized to the identity. A null pointer is
returned if insufficient memory is available to create the permutation.
- Function: gsl_permutation * gsl_permutation_calloc (size_t n)
-
This function allocates memory for a new permutation of size n and
initializes it to the identity. A null pointer is returned if
insufficient memory is available to create the permutation.
- Function: void gsl_permutation_init (gsl_permutation * p)
-
This function initializes the permutation p to the identity, i.e.
(0,1,2,...,n-1).
- Function: void gsl_permutation_free (gsl_permutation * p)
-
This function frees all the memory used by the permutation p.
- Function: int gsl_permutation_memcpy (gsl_permutation * dest, const gsl_permutation * src)
-
This function copies the elements of the permutation src into the
permutation dest. The two permutations must have the same size.
The following functions can be used to access and manipulate
permutations.
- Function: size_t gsl_permutation_get (const gsl_permutation * p, const size_t i)
-
This function returns the value of the i-th element of the
permutation p. If i lies outside the allowed range of 0 to
n-1 then the error handler is invoked and 0 is returned.
- Function: int gsl_permutation_swap (gsl_permutation * p, const size_t i, const size_t j)
-
This function exchanges the i-th and j-th elements of the
permutation p.
- Function: size_t gsl_permutation_size (const gsl_permutation * p)
-
This function returns the size of the permutation p.
- Function: size_t * gsl_permutation_data (const gsl_permutation * p)
-
This function returns a pointer to the array of elements in the
permutation p.
- Function: int gsl_permutation_valid (gsl_permutation * p)
-
This function checks that the permutation p is valid. The n
elements should contain each of the numbers 0 .. n-1 once and only
once.
- Function: void gsl_permutation_reverse (gsl_permutation * p)
-
This function reverses the elements of the permutation p.
- Function: int gsl_permutation_inverse (gsl_permutation * inv, const gsl_permutation * p)
-
This function computes the inverse of the permutation p, storing
the result in inv.
- Function: int gsl_permutation_next (gsl_permutation * p)
-
This function advances the permutation p to the next permutation
in lexicographic order and returns
GSL_SUCCESS. If no further
permutations are available it returns GSL_FAILURE and leaves
p unmodified. Starting with the identity permutation and
repeatedly applying this function will iterate through all possible
permutations of a given order.
- Function: int gsl_permutation_prev (gsl_permutation * p)
-
This function steps backwards from the permutation p to the
previous permutation in lexicographic order, returning
GSL_SUCCESS. If no previous permutation is available it returns
GSL_FAILURE and leaves p unmodified.
- Function: int gsl_permute (const size_t * p, double * data, size_t stride, size_t n)
-
This function applies the permutation p to the array data of
size n with stride stride.
- Function: int gsl_permute_inverse (const size_t * p, double * data, size_t stride, size_t n)
-
This function applies the inverse of the permutation p to the
array data of size n with stride stride.
- Function: int gsl_permute_vector (const gsl_permutation * p, gsl_vector * v)
-
This function applies the permutation p to the elements of the
vector v, considered as a row-vector acted on by a permutation
matrix from the right, v' = v P. The j-th column of the
permutation matrix P is given by the p_j-th column of the
identity matrix. The permutation p and the vector v must
have the same length.
- Function: int gsl_permute_vector_inverse (const gsl_permutation * p, gsl_vector * v)
-
This function applies the inverse of the permutation p to the
elements of the vector v, considered as a row-vector acted on by
an inverse permutation matrix from the right, v' = v P^T. Note
that for permutation matrices the inverse is the same as the transpose.
The j-th column of the permutation matrix P is given by
the p_j-th column of the identity matrix. The permutation p
and the vector v must have the same length.
The library provides functions for reading and writing permutations to a
file as binary data or formatted text.
- Function: int gsl_permutation_fwrite (FILE * stream, const gsl_permutation * p)
-
This function writes the elements of the permutation p 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_permutation_fread (FILE * stream, gsl_permutation * p)
-
This function reads into the permutation p from the open stream
stream in binary format. The permutation p must be
preallocated with the correct length since the function uses the size of
p 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_permutation_fprintf (FILE * stream, const gsl_permutation * p, const char *format)
-
This function writes the elements of the permutation p
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_permutation_fscanf (FILE * stream, gsl_permutation * p)
-
This function reads formatted data from the stream stream into the
permutation p. The permutation p must be preallocated with
the correct length since the function uses the size of p to
determine how many numbers to read. The function returns
GSL_EFAILED if there was a problem reading from the file.
The example program below creates a random permutation by shuffling and
finds its inverse.
#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_permutation.h>
int
main (void)
{
const size_t N = 10;
const gsl_rng_type * T;
gsl_rng * r;
gsl_permutation * p = gsl_permutation_alloc (N);
gsl_permutation * q = gsl_permutation_alloc (N);
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
printf("initial permutation:");
gsl_permutation_init (p);
gsl_permutation_fprintf (stdout, p, " %u");
printf("\n");
printf(" random permutation:");
gsl_ran_shuffle (r, p->data, N, sizeof(size_t));
gsl_permutation_fprintf (stdout, p, " %u");
printf("\n");
printf("inverse permutation:");
gsl_permutation_invert (q, p);
gsl_permutation_fprintf (stdout, q, " %u");
printf("\n");
return 0;
}
Here is the output from the program,
bash$ ./a.out
initial permutation: 0 1 2 3 4 5 6 7 8 9
random permutation: 1 3 5 2 7 6 0 4 9 8
inverse permutation: 6 0 3 1 7 2 5 4 9 8
The random permutation p[i] and its inverse q[i] are
related through the identity p[q[i]] = i, which can be verified
from the output.
The next example program steps forwards through all possible 3-rd order
permutations, starting from the identity,
#include <stdio.h>
#include <gsl/gsl_permutation.h>
int
main (void)
{
gsl_permutation * p = gsl_permutation_alloc (3);
gsl_permutation_init (p);
do
{
gsl_permutation_fprintf (stdout, p, " %u");
printf("\n");
}
while (gsl_permutation_next(p) == GSL_SUCCESS);
return 0;
}
Here is the output from the program,
bash$ ./a.out
0 1 2
0 2 1
1 0 2
1 2 0
2 0 1
2 1 0
All 6 permutations are generated in lexicographic order. To reverse the
sequence, begin with the final permutation (which is the reverse of the
identity) and replace gsl_permutation_next with
gsl_permutation_prev.
The subject of permutations 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.
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