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,
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.
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.
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.
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.
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.
This function combines the two permutations pa and pb into a
single permutation p, where p = pa . pb. The permutation
p is equivalent to applying pb first and then pa.
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.
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.
A permutation can be represented in both linear and cyclic notations.
The functions described in this section can be used to convert between
the two forms.
The linear notation is an index mapping, and has already been described
above. The cyclic notation represents a permutation as a series of
circular rearrangements of groups of elements, or cycles.
Any permutation can be decomposed into a combination of cycles. For
example, under the cycle (1 2 3), 1 is replaced by 2, 2 is replaced by 3
and 3 is replaced by 1 in a circular fashion. Cycles of different sets
of elements can be combined independently, for example (1 2 3) (4 5)
combines the cycle (1 2 3) with the cycle (4 5), which is an exchange of
elements 4 and 5. A cycle of length one represents an element which is
unchanged by the permutation and is referred to as a singleton.
The cyclic notation for a permutation is not unique, but can be
rearranged into a unique canonical form by a reordering of
elements. The library uses the canonical form defined in Knuth's
Art of Computer Programming (Vol 1, 3rd Ed, 1997) Section 1.3.3,
p.178.
The procedure for obtaining the canonical form given by Knuth is,
Write all singleton cycles explicitly
Within each cycle, put the smallest number first
Order the cycles in decreasing order of the first number in the cycle.
For example, the linear representation (2 4 3 0 1) is represented as (1
4) (0 2 3) in canonical form. The permutation corresponds to an
exchange of elements 1 and 4, and rotation of elements 0, 2 and 3.
The important property of the canonical form is that it can be
reconstructed from the contents of each cycle without the brackets. In
addition, by removing the brackets it can be considered as a linear
representation of a different permutation. In the example given above
the permutation (2 4 3 0 1) would become (1 4 0 2 3). This mapping
between linear permutations defined by the canonical form has many
important uses in the theory of permutations.
Function: int gsl_permutation_linear_to_canonical(gsl_permutation * q, const gsl_permutation * p)
This function computes the canonical form of the permutation p and
stores it in the output argument q.
Function: int gsl_permutation_canonical_to_linear(gsl_permutation * p, const gsl_permutation * q)
This function converts a permutation q in canonical form back into
linear form storing it in the output argument p.
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.
For the definition of the canonical form see,
Donald E. Knuth, The Art of Computer Programming: Fundamental
Algorithms (Vol 1, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.
Section 1.3.3, An Unusual Correspondence, p.178-179.