permutation.texi

math library from gnu

@end deftypefun

@node Reading and writing permutations
@section Reading and writing permutations

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

@deftypefun int gsl_permutation_fwrite (FILE * @var{stream}, const gsl_permutation * @var{p})
This function writes the elements of the permutation @var{p} to the
stream @var{stream} in binary format.  The function returns
@code{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.
@end deftypefun

@deftypefun int gsl_permutation_fread (FILE * @var{stream}, gsl_permutation * @var{p})
This function reads into the permutation @var{p} from the open stream
@var{stream} in binary format.  The permutation @var{p} must be
preallocated with the correct length since the function uses the size of
@var{p} to determine how many bytes to read.  The function returns
@code{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.
@end deftypefun

@deftypefun int gsl_permutation_fprintf (FILE * @var{stream}, const gsl_permutation * @var{p}, const char * @var{format})
This function writes the elements of the permutation @var{p}
line-by-line to the stream @var{stream} using the format specifier
@var{format}, which should be suitable for a type of @var{size_t}. 
In ISO C99 the type modifier @code{z} represents @code{size_t}, so
@code{"%zu\n"} is a suitable format.@footnote{In versions of the 
GNU C library prior to the ISO C99 standard, 
the type modifier @code{Z} was used instead.}
The function returns @code{GSL_EFAILED} if there was a problem writing
to the file.
@end deftypefun

@deftypefun int gsl_permutation_fscanf (FILE * @var{stream}, gsl_permutation * @var{p})
This function reads formatted data from the stream @var{stream} into the
permutation @var{p}.  The permutation @var{p} must be preallocated with
the correct length since the function uses the size of @var{p} to
determine how many numbers to read.  The function returns
@code{GSL_EFAILED} if there was a problem reading from the file.
@end deftypefun

@node Permutations in cyclic form
@section Permutations in cyclic form

A permutation can be represented in both @dfn{linear} and @dfn{cyclic}
notations.  The functions described in this section convert between the
two forms.  The linear notation is an index mapping, and has already
been described above.  The cyclic notation expresses a permutation as a
series of circular rearrangements of groups of elements, or
@dfn{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 @dfn{singleton}.

It can be shown that every permutation can be decomposed into
combinations of cycles.  The decomposition is not unique, but can always
be rearranged into a standard @dfn{canonical form} by a reordering of
elements.  The library uses the canonical form defined in Knuth's
@cite{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,

@enumerate
@item Write all singleton cycles explicitly
@item Within each cycle, put the smallest number first
@item Order the cycles in decreasing order of the first number in the cycle.
@end enumerate

@noindent
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 has
many applications in the theory of permutations.

@deftypefun int gsl_permutation_linear_to_canonical (gsl_permutation * @var{q}, const gsl_permutation * @var{p})
This function computes the canonical form of the permutation @var{p} and
stores it in the output argument @var{q}.
@end deftypefun

@deftypefun int gsl_permutation_canonical_to_linear (gsl_permutation * @var{p}, const gsl_permutation * @var{q})
This function converts a permutation @var{q} in canonical form back into
linear form storing it in the output argument @var{p}.
@end deftypefun

@deftypefun size_t gsl_permutation_inversions (const gsl_permutation * @var{p})
This function counts the number of inversions in the permutation
@var{p}.  An inversion is any pair of elements that are not in order.
For example, the permutation 2031 has three inversions, corresponding to
the pairs (2,0) (2,1) and (3,1).  The identity permutation has no
inversions.
@end deftypefun

@deftypefun size_t gsl_permutation_linear_cycles (const gsl_permutation * @var{p})
This function counts the number of cycles in the permutation @var{p}, given in linear form.
@end deftypefun

@deftypefun size_t gsl_permutation_canonical_cycles (const gsl_permutation * @var{q})
This function counts the number of cycles in the permutation @var{q}, given in canonical form.
@end deftypefun


@node Permutation Examples
@section Examples
The example program below creates a random permutation (by shuffling the
elements of the identity) and finds its inverse.

@example
@verbatiminclude examples/permshuffle.c
@end example

@noindent
Here is the output from the program,

@example
$ ./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
@end example

@noindent
The random permutation @code{p[i]} and its inverse @code{q[i]} are
related through the identity @code{p[q[i]] = i}, which can be verified
from the output.

The next example program steps forwards through all possible third order
permutations, starting from the identity,

@example
@verbatiminclude examples/permseq.c
@end example

@noindent
Here is the output from the program,

@example
$ ./a.out 
 0 1 2
 0 2 1
 1 0 2
 1 2 0
 2 0 1
 2 1 0
@end example

@noindent
The permutations are generated in lexicographic order.  To reverse the
sequence, begin with the final permutation (which is the reverse of the
identity) and replace @code{gsl_permutation_next} with
@code{gsl_permutation_prev}.

@node Permutation References and Further Reading
@section References and Further Reading

The subject of permutations is covered extensively in Knuth's
@cite{Sorting and Searching},

@itemize @asis
@item
Donald E. Knuth, @cite{The Art of Computer Programming: Sorting and
Searching} (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.
@end itemize

@noindent
For the definition of the @dfn{canonical form} see,

@itemize @asis
@item
Donald E. Knuth, @cite{The Art of Computer Programming: Fundamental
Algorithms} (Vol 1, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.
Section 1.3.3, @cite{An Unusual Correspondence}, p.178--179.
@end itemize