如何使用排列数组有效地排列稀疏（Numpy）矩阵中的行？

Question

I used the Scipy Reverse Cuthill-McKee implementation ( scipy.sparse.csgraph.reverse_cuthill_mckee ) for creating a band matrix using a (high-dimensional) sparse csr_matrix. The result of this method is a permutation array whichs gives me the indices of how to permutate the rows of my matrix as I understood.

Now is there any efficient solution for doing this permutation on my sparse csr_matrix in any other sparse matrix (csr, lil_matrix, etc)? I tried a for-loop but my matrix has dimension like 200,000 x 150,000 and it takes too much time.

A = csr_matrix((data,(rowind,columnind)), shape=(200000, 150000), dtype=np.uint8)

permutation_array = csgraph.reverse_cuthill_mckee(A, false)

result_matrix = lil_matrix((200000, 150000), dtype=np.uint8)

i=0
for x in np.nditer(permutation_array):
    result_matrix[x, :]=A[i, :]
    i+=1

The result of the reverse_cuthill_mckee call is an array which is like a tupel containing the indices for my permutation. So this array is something like: [199999 54877 54873 ..., 12045 9191 0] (size = 200,000)

This means: row with index 0 has now index 199999, row with index 1 has now index 54877, row with index 2 has now index 54873, etc. see: https://en.wikipedia.org/wiki/Permutation#Definition_and_notations (As I understood the return)

Thank you

Answer 1

I wonder if you are applying the permutation array correctly.

Make a random matrix (float) and convert it to a uint8 (beware, csr calculations might not work with this dtype):

In [963]: ran=sparse.random(10,10,.3, format='csr')
In [964]: A = sparse.csr_matrix((np.ones(ran.data.shape).astype(np.uint8),ran.indices, ran.indptr))
In [965]: A.A
Out[965]: 
array([[1, 1, 0, 0, 0, 0, 1, 0, 0, 0],
       [0, 1, 1, 1, 1, 1, 1, 0, 1, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [1, 1, 0, 0, 0, 0, 0, 1, 0, 1],
       [0, 1, 0, 0, 1, 1, 0, 0, 0, 0],
       [1, 0, 1, 0, 0, 1, 0, 1, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 1, 0, 0, 0, 1],
       [0, 1, 1, 1, 0, 1, 0, 0, 0, 0],
       [0, 0, 0, 0, 1, 1, 1, 0, 0, 0]], dtype=uint8)

(oops, used the wrong matrix here):

In [994]: permutation_array = csgraph.reverse_cuthill_mckee(A, False)
In [995]: permutation_array
Out[995]: array([9, 7, 0, 4, 6, 3, 5, 1, 8, 2], dtype=int32)

My first inclination is to use such an array to simply index rows of the original matrix:

In [996]: A[permutation_array,:].A
Out[996]: 
array([[0, 0, 0, 0, 1, 1, 1, 0, 0, 0],
       [0, 0, 0, 0, 0, 1, 0, 0, 0, 1],
       [1, 1, 0, 0, 0, 0, 1, 0, 0, 0],
       [0, 1, 0, 0, 1, 1, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [1, 1, 0, 0, 0, 0, 0, 1, 0, 1],
       [1, 0, 1, 0, 0, 1, 0, 1, 0, 0],
       [0, 1, 1, 1, 1, 1, 1, 0, 1, 0],
       [0, 1, 1, 1, 0, 1, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=uint8)

I see some clustering; maybe the best we can expect from a random matrix.

You on the other hand appear to be doing:

In [997]: res = sparse.lil_matrix(A.shape,dtype=A.dtype)
In [998]: res[permutation_array,:] = A
In [999]: res.A
Out[999]: 
array([[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 0, 0, 0, 0, 1, 0, 0, 0, 1],
       [0, 0, 0, 0, 1, 1, 1, 0, 0, 0],
       [1, 0, 1, 0, 0, 1, 0, 1, 0, 0],
       [1, 1, 0, 0, 0, 0, 0, 1, 0, 1],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [0, 1, 0, 0, 1, 1, 0, 0, 0, 0],
       [0, 1, 1, 1, 1, 1, 1, 0, 1, 0],
       [0, 1, 1, 1, 0, 1, 0, 0, 0, 0],
       [1, 1, 0, 0, 0, 0, 1, 0, 0, 0]], dtype=uint8)

I don't see any improvement in clustering of 1s in res .

---

The docs for the MATLAB equivalent say

r = symrcm(S) returns the symmetric reverse Cuthill-McKee ordering of S. This is a permutation r such that S(r,r) tends to have its nonzero elements closer to the diagonal.

In numpy terms, that means:

In [1019]: I,J=np.ix_(permutation_array,permutation_array)
In [1020]: A[I,J].A
Out[1020]: 
array([[0, 0, 0, 1, 1, 0, 1, 0, 0, 0],
       [1, 0, 0, 0, 0, 0, 1, 0, 0, 0],
       [0, 0, 1, 0, 1, 0, 0, 1, 0, 0],
       [0, 0, 0, 1, 0, 0, 1, 1, 0, 0],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
       [1, 1, 1, 0, 0, 0, 0, 1, 0, 0],
       [0, 1, 1, 0, 0, 0, 1, 0, 0, 1],
       [0, 0, 0, 1, 1, 1, 1, 1, 1, 1],
       [0, 0, 0, 0, 0, 1, 1, 1, 0, 1],
       [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=uint8)

And indeed there are more 0 bands in the 2 off diagonal corners.

And using the bandwidth calculation on the MATLAB page, https://www.mathworks.com/help/matlab/ref/symrcm.html

In [1028]: i,j=A.nonzero()
In [1029]: np.max(i-j)
Out[1029]: 7
In [1030]: i,j=A[I,J].nonzero()
In [1031]: np.max(i-j)
Out[1031]: 5

The MATLAB docs say that with this permutation, the eigenvalues remain the same. Testing:

In [1032]: from scipy.sparse import linalg
In [1048]: linalg.eigs(A.astype('f'))[0]
Out[1048]: 
array([ 3.14518213+0.j        , -0.96188843+0.j        ,
       -0.58978939+0.62853903j, -0.58978939-0.62853903j,
        1.09950364+0.54544497j,  1.09950364-0.54544497j], dtype=complex64)
In [1049]: linalg.eigs(A[I,J].astype('f'))[0]
Out[1049]: 
array([ 3.14518023+0.j        ,  1.09950352+0.54544479j,
        1.09950352-0.54544479j, -0.58978981+0.62853914j,
       -0.58978981-0.62853914j, -0.96188819+0.j        ], dtype=complex64)

Eigenvalues are not the same for the row permutations we tried earlier:

In [1050]: linalg.eigs(A[permutation_array,:].astype('f'))[0]
Out[1050]: 
array([ 2.95226836+0.j        , -1.60117996+0.52467293j,
       -1.60117996-0.52467293j, -0.01723826+1.06249797j,
       -0.01723826-1.06249797j,  0.90314150+0.j        ], dtype=complex64)
In [1051]: linalg.eigs(res.astype('f'))[0]
Out[1051]: 
array([-0.05822830-0.97881651j, -0.99999994+0.j        ,
        1.17350495+0.j        , -0.91237622+0.8656373j ,
       -0.91237622-0.8656373j ,  2.26292515+0.j        ], dtype=complex64)

This [I,J] permutation works with the example matrix in http://ciprian-zavoianu.blogspot.com/2009/01/project-bandwidth-reduction.html

In [1058]: B = np.matrix('1 0 0 0 1 0 0 0;0 1 1 0 0 1 0 1;0 1 1 0 1 0 0 0;0 0 0 
      ...: 1 0 0 1 0;1 0 1 0 1 0 0 0; 0 1 0 0 0 1 0 1;0 0 0 1 0 0 1 0;0 1 0 0 0 
      ...: 1 0 1')
In [1059]: B
Out[1059]: 
matrix([[1, 0, 0, 0, 1, 0, 0, 0],
        [0, 1, 1, 0, 0, 1, 0, 1],
        [0, 1, 1, 0, 1, 0, 0, 0],
        [0, 0, 0, 1, 0, 0, 1, 0],
        [1, 0, 1, 0, 1, 0, 0, 0],
        [0, 1, 0, 0, 0, 1, 0, 1],
        [0, 0, 0, 1, 0, 0, 1, 0],
        [0, 1, 0, 0, 0, 1, 0, 1]])
In [1060]: Bm=sparse.csr_matrix(B)
In [1061]: Bm
Out[1061]: 
<8x8 sparse matrix of type '<class 'numpy.int32'>'
    with 22 stored elements in Compressed Sparse Row format>
In [1062]: permB = csgraph.reverse_cuthill_mckee(Bm, False)
In [1063]: permB
Out[1063]: array([6, 3, 7, 5, 1, 2, 4, 0], dtype=int32)
In [1064]: Bm[np.ix_(permB,permB)].A
Out[1064]: 
array([[1, 1, 0, 0, 0, 0, 0, 0],
       [1, 1, 0, 0, 0, 0, 0, 0],
       [0, 0, 1, 1, 1, 0, 0, 0],
       [0, 0, 1, 1, 1, 0, 0, 0],
       [0, 0, 1, 1, 1, 1, 0, 0],
       [0, 0, 0, 0, 1, 1, 1, 0],
       [0, 0, 0, 0, 0, 1, 1, 1],
       [0, 0, 0, 0, 0, 0, 1, 1]], dtype=int32)