# Source code for qutip.graph

```"""
This module contains a collection of graph theory routines used mainly
to reorder matrices for iterative steady state solvers.
"""

'reverse_cuthill_mckee', 'maximum_bipartite_matching',
'weighted_bipartite_matching']

import numpy as np
import scipy.sparse as sp
from qutip.cy.graph_utils import (
_reverse_cuthill_mckee, _maximum_bipartite_matching,
_weighted_bipartite_matching)
import warnings

def _deprecate():
warnings.warn(("qutip graph functions are deprecated."
DeprecationWarning)

[docs]def graph_degree(A):
"""
Returns the degree for the nodes (rows) of a symmetric graph in sparse CSR
or CSC format, or a qobj.

Parameters
----------
A : qobj, csr_matrix, csc_matrix
Input quantum object or csr_matrix.

Returns
-------
degree : array
Array of integers giving the degree for each node (row).
"""
_deprecate()
if not (sp.isspmatrix_csc(A) or sp.isspmatrix_csr(A)):
raise TypeError('Input must be CSC or CSR sparse matrix.')
return np.asarray(_node_degrees(A.indices, A.indptr, A.shape[0]))

"""
Breadth-First-Search (BFS) of a graph in CSR or CSC matrix format starting
from a given node (row).  Takes Qobjs and CSR or CSC matrices as inputs.

This function requires a matrix with symmetric structure.  Use A+trans(A)
if original matrix is not symmetric or not sure.

Parameters
----------
A : csc_matrix, csr_matrix
Input graph in CSC or CSR matrix format
start : int
Staring node for BFS traversal.

Returns
-------
order : array
Order in which nodes are traversed from starting node.
levels : array
Level of the nodes in the order that they are traversed.

"""
_deprecate()
if not (sp.isspmatrix_csc(A) or sp.isspmatrix_csr(A)):
raise TypeError('Input must be CSC or CSR sparse matrix.')
num_rows = A.shape[0]
start = int(start)
order, levels = _breadth_first_search(A.indices, A.indptr, num_rows, start)
# since maybe not all nodes are in search, check for unused entires in
# arrays
return order[order != -1], levels[levels != -1]

def column_permutation(A):
"""
Finds the non-symmetric column permutation of A such that the columns
are given in ascending order according to the number of nonzero entries.
This is sometimes useful for decreasing the fill-in of sparse LU
factorization.

Parameters
----------
A : csc_matrix
Input sparse CSC sparse matrix.

Returns
-------
perm : array
Array of permuted row and column indices.
"""
_deprecate()
if not sp.isspmatrix_csc(A):
A = sp.csc_matrix(A)
count = np.diff(A.indptr)
perm = np.argsort(count)
return perm

[docs]def reverse_cuthill_mckee(A, sym=False):
"""
Returns the permutation array that orders a sparse CSR or CSC matrix in
Reverse-Cuthill McKee ordering. Since the input matrix must be symmetric,
this routine works on the matrix A+Trans(A) if the sym flag is set to False
(Default).

It is assumed by default (*sym=False*) that the input matrix is not
symmetric. This is because it is faster to do A+Trans(A) than it is to
check for symmetry for a generic matrix. If you are guaranteed that the
matrix is symmetric in structure (values of matrix element do not matter)
then set *sym=True*

Parameters
----------
A : csc_matrix, csr_matrix
Input sparse CSC or CSR sparse matrix format.
sym : bool {False, True}
Flag to set whether input matrix is symmetric.

Returns
-------
perm : array
Array of permuted row and column indices.

Notes
-----
This routine is used primarily for internal reordering of Lindblad
superoperators for use in iterative solver routines.

References
----------
E. Cuthill and J. McKee, "Reducing the Bandwidth of Sparse Symmetric
Matrices", ACM '69 Proceedings of the 1969 24th national conference,
(1969).
"""
_deprecate()
if not (sp.isspmatrix_csc(A) or sp.isspmatrix_csr(A)):
raise TypeError('Input must be CSC or CSR sparse matrix.')
nrows = A.shape[0]
if not sym:
A = A + A.transpose()
return _reverse_cuthill_mckee(A.indices, A.indptr, nrows)

[docs]def maximum_bipartite_matching(A, perm_type='row'):
"""
Returns an array of row or column permutations that removes nonzero
elements from the diagonal of a nonsingular square CSC sparse matrix. Such
a permutation is always possible provided that the matrix is nonsingular.
This function looks at the structure of the matrix only.

The input matrix will be converted to CSC matrix format if necessary.

Parameters
----------
A : sparse matrix
Input matrix

perm_type : str {'row', 'column'}
Type of permutation to generate.

Returns
-------
perm : array
Array of row or column permutations.

Notes
-----
This function relies on a maximum cardinality bipartite matching algorithm
based on a breadth-first search (BFS) of the underlying graph[1]_.

References
----------
I. S. Duff, K. Kaya, and B. Ucar, "Design, Implementation, and
Analysis of Maximum Transversal Algorithms", ACM Trans. Math. Softw.
38, no. 2, (2011).
"""
_deprecate()
nrows = A.shape[0]
if A.shape[0] != A.shape[1]:
raise ValueError(
'Maximum bipartite matching requires a square matrix.')
if sp.isspmatrix_csr(A) or sp.isspmatrix_coo(A):
A = A.tocsc()
elif not sp.isspmatrix_csc(A):
raise TypeError("matrix must be in CSC, CSR, or COO format.")

if perm_type == 'column':
A = A.transpose().tocsc()
perm = _maximum_bipartite_matching(A.indices, A.indptr, nrows)
if np.any(perm == -1):
raise Exception('Possibly singular input matrix.')
return perm

[docs]def weighted_bipartite_matching(A, perm_type='row'):
"""
Returns an array of row permutations that attempts to maximize the product
of the ABS values of the diagonal elements in a nonsingular square CSC
sparse matrix. Such a permutation is always possible provided that the
matrix is nonsingular.

This function looks at both the structure and ABS values of the underlying
matrix.

Parameters
----------
A : csc_matrix
Input matrix

perm_type : str {'row', 'column'}
Type of permutation to generate.

Returns
-------
perm : array
Array of row or column permutations.

Notes
-----
This function uses a weighted maximum cardinality bipartite matching
algorithm based on breadth-first search (BFS).  The columns are weighted
according to the element of max ABS value in the associated rows and are
traversed in descending order by weight.  When performing the BFS
traversal, the row associated to a given column is the one with maximum
weight. Unlike other techniques[1]_, this algorithm does not guarantee the
product of the diagonal is maximized.  However, this limitation is offset
by the substantially faster runtime of this method.

References
----------
I. S. Duff and J. Koster, "The design and use of algorithms for permuting
large entries to the diagonal of sparse matrices", SIAM J.  Matrix Anal.
and Applics. 20, no. 4, 889 (1997).
"""
_deprecate()
nrows = A.shape[0]
if A.shape[0] != A.shape[1]:
raise ValueError('weighted_bfs_matching requires a square matrix.')
if sp.isspmatrix_csr(A) or sp.isspmatrix_coo(A):
A = A.tocsc()
elif not sp.isspmatrix_csc(A):
raise TypeError("matrix must be in CSC, CSR, or COO format.")

if perm_type == 'column':
A = A.transpose().tocsc()
perm = _weighted_bipartite_matching(
np.asarray(np.abs(A.data), dtype=float),
A.indices, A.indptr, nrows)
if np.any(perm == -1):
raise Exception('Possibly singular input matrix.')
return perm
```