perm.rdoc

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Last Update: Sun Nov 14 14:53:48 -0800 2010

Permutations

Contents:

  1. Permuation allocations
  2. Methods
    1. Accessing permutation elements
    2. Permuation properties
    3. Permuation functions
    4. Reading and writing permutations
  3. Permutations in cyclic form
  4. Applying Permutations

Permuation allocations


  • GSL::Permutation.alloc(n)

    These functions create a new permutation of size n. The permutation is not initialized and its elements are undefined. Use GSL::Permutation.calloc if you want to create a permutation which is initialized to the identity.


  • GSL::Permutation.calloc(n)

    This creates a new permutation of size n and initializes it to the identity.

Methods


  • GSL::Permutation#init()

    This initializes the permutation to the identity, i.e. (0,1,2,…,n-1).


  • GSL::Permutation.memcpy(dest, src)

    This method copies the elements of the permutation src into the permutation dest. The two permutations must have the same size.


  • GSL::Permutation#clone

    This creates a new permutation with the same elements of self.

Accessing permutation elements


  • GSL::Permutation#get(i)

    Returns the value of the i-th element of the permutation.


  • GSL::Permutation#swap(i, j)

    This exchanges the i-th and j-th elements of the permutation.

Permutation properties


  • GSL::Permutation#size

    Returns the size of the permutation.


  • GSL::Permutation#valid

    This checks that the permutation self is valid. The n elements should contain each of the numbers 0 .. n-1 once and only once.


  • GSL::Permutation#valid?

    This returns true if the permutation self is valid, and false otherwise.

Permutation functions


  • GSL::Permutation#reverse

    This reverses the elements of the permutation self.


  • GSL::Permutation#inverse

    This computes the inverse of the permutation self, and returns as a new permutation.


  • GSL::Permutation#next

    This method advances the permutation self to the next permutation in lexicographic order and returns GSL::SUCCESS. If no further permutations are available it returns GSL::FAILURE and leaves self unmodified. Starting with the identity permutation and repeatedly applying this function will iterate through all possible permutations of a given order.


  • GSL::Permutation#prev

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

Reading and writing permutations


  • GSL::Permutation#fwrite(io)
  • GSL::Permutation#fwrite(filename)
  • GSL::Permutation#fread(io)
  • GSL::Permutation#fread(filename)
  • GSL::Permutation#fprintf(io, format = "%u\n")
  • GSL::Permutation#fprintf(filename, format = "%u\n")
  • GSL::Permutation#fscanf(io)
  • GSL::Permutation#fscanf(filename)

Permutations in cyclic Form

A permutation can be represented in both linear and 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 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.

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 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,

  1. Write all singleton cycles explicitly
  2. Within each cycle, put the smallest number first
  3. 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 has many applications in the theory of permutations.


  • GSL::Permutation#linear_to_canonical
  • GSL::Permutation#to_canonical

    Computes the canonical form of the permutation self and returns it as a new GSL::Permutation.


  • GSL::Permutation#canonical_to_linear
  • GSL::Permutation#to_linear

    Converts a permutation self in canonical form back into linear form and returns it as a new GSL::Permutation.


  • GSL::Permutation#inversions

    Counts the number of inversions in the permutation self. 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.


  • GSL::Permutation#linear_cycles

    Counts the number of cycles in the permutation self, given in linear form.


  • GSL::Permutation#canonical_cycles

    Counts the number of cycles in the permutation self, given in canonical form.

Applying Permutations


  • GSL::Permutation::permute(v)

    Applies the permutation self 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 self and the vector v must have the same length.


  • GSL::Permutation::permute_inverse(v)

    Applies the inverse of the permutation self 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 self and the vector v must have the same length.


  • GSL::Permutation.mul(pa, pb)

    Combines the two permutations pa and pb into a single permutation p and returns it. The permutation p is equivalent to applying pb first and then pa.

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