List of Lists Format (LIL)

  • row-based linked list
    • each row is a Python list (sorted) of column indices of non-zero elements
    • rows stored in a NumPy array (dtype=np.object)
    • non-zero values data stored analogously
  • efficient for constructing sparse matrices incrementally
  • constructor accepts:
    • dense matrix (array)
    • sparse matrix
    • shape tuple (create empty matrix)
  • flexible slicing, changing sparsity structure is efficient
  • slow arithmetics, slow column slicing due to being row-based
  • use:
    • when sparsity pattern is not known apriori or changes
    • example: reading a sparse matrix from a text file

Examples

  • create an empty LIL matrix:

    >>> mtx = sparse.lil_matrix((4, 5))

  • prepare random data:

    >>> from numpy.random import rand

    >>> data = np.round(rand(2, 3))

    >>> data

    array([[1.,  1.,  1.],

           [1.,  0.,  1.]])

  • assign the data using fancy indexing:

    >>> mtx[:2, [1, 2, 3]] = data

    >>> mtx  

    <4x5 sparse matrix of type '<... 'numpy.float64'>'

            with 5 stored elements in LInked List format>

    >>> print(mtx)   

      (0, 1)  1.0

      (0, 2)  1.0

      (0, 3)  1.0

      (1, 1)  1.0

      (1, 3)  1.0

    >>> mtx.todense()

    matrix([[0.,  1.,  1.,  1.,  0.],

            [0.,  1.,  0.,  1.,  0.],

            [0.,  0.,  0.,  0.,  0.],

            [0.,  0.,  0.,  0.,  0.]])

    >>> mtx.toarray()

    array([[0.,  1.,  1.,  1.,  0.],

           [0.,  1.,  0.,  1.,  0.],

           [0.,  0.,  0.,  0.,  0.],

           [0.,  0.,  0.,  0.,  0.]])

  • more slicing and indexing:

    >>> mtx = sparse.lil_matrix([[0, 1, 2, 0], [3, 0, 1, 0], [1, 0, 0, 1]])

    >>> mtx.todense()    

    matrix([[0, 1, 2, 0],

            [3, 0, 1, 0],

            [1, 0, 0, 1]]...)

    >>> print(mtx) 

      (0, 1)    1

      (0, 2)    2

      (1, 0)    3

      (1, 2)    1

      (2, 0)    1

      (2, 3)    1

    >>> mtx[:2, :]  

    <2x4 sparse matrix of type '<... 'numpy.int64'>'

      with 4 stored elements in LInked List format>

    >>> mtx[:2, :].todense()    

    matrix([[0, 1, 2, 0],

            [3, 0, 1, 0]]...)

    >>> mtx[1:2, [0,2]].todense()    

    matrix([[3, 1]]...)

    >>> mtx.todense()    

    matrix([[0, 1, 2, 0],

            [3, 0, 1, 0],

            [1, 0, 0, 1]]...)