Module « scipy.sparse.csgraph »

Fonction maximum_flow - module scipy.sparse.csgraph

Signature de la fonction maximum_flow

Description

maximum_flow.__doc__

    maximum_flow(csgraph, source, sink)

    Maximize the flow between two vertices in a graph.

    .. versionadded:: 1.4.0

    Parameters
    ----------
    csgraph : csr_matrix
        The square matrix representing a directed graph whose (i, j)'th entry
        is an integer representing the capacity of the edge between
        vertices i and j.
    source : int
        The source vertex from which the flow flows.
    sink : int
        The sink vertex to which the flow flows.

    Returns
    -------
    res : MaximumFlowResult
        A maximum flow represented by a ``MaximumFlowResult``
        which includes the value of the flow in ``flow_value``,
        and the residual graph in ``residual``.

    Raises
    ------
    TypeError:
        if the input graph is not in CSR format.

    ValueError:
        if the capacity values are not integers, or the source or sink are out
        of bounds.

    Notes
    -----
    This solves the maximum flow problem on a given directed weighted graph:
    A flow associates to every edge a value, also called a flow, less than the
    capacity of the edge, so that for every vertex (apart from the source and
    the sink vertices), the total incoming flow is equal to the total outgoing
    flow. The value of a flow is the sum of the flow of all edges leaving the
    source vertex, and the maximum flow problem consists of finding a flow
    whose value is maximal.

    By the max-flow min-cut theorem, the maximal value of the flow is also the
    total weight of the edges in a minimum cut.

    To solve the problem, we use the Edmonds--Karp algorithm. [1]_ This
    particular implementation strives to exploit sparsity. Its time complexity
    is :math:`O(VE^2)` and its space complexity is :math:`O(E)`.

    The maximum flow problem is usually defined with real valued capacities,
    but we require that all capacities are integral to ensure convergence. When
    dealing with rational capacities, or capacities belonging to
    :math:`x\mathbb{Q}` for some fixed :math:`x \in \mathbb{R}`, it is possible
    to reduce the problem to the integral case by scaling all capacities
    accordingly.

    Solving a maximum-flow problem can be used for example for graph cuts
    optimization in computer vision [3]_.

    References
    ----------
    .. [1] Edmonds, J. and Karp, R. M.
           Theoretical improvements in algorithmic efficiency for network flow
           problems. 1972. Journal of the ACM. 19 (2): pp. 248-264
    .. [2] Cormen, T. H. and Leiserson, C. E. and Rivest, R. L. and Stein C.
           Introduction to Algorithms. Second Edition. 2001. MIT Press.
    .. [3] https://en.wikipedia.org/wiki/Graph_cuts_in_computer_vision

    Examples
    --------
    Perhaps the simplest flow problem is that of a graph of only two vertices
    with an edge from source (0) to sink (1)::

        (0) --5--> (1)

    Here, the maximum flow is simply the capacity of the edge:

    >>> from scipy.sparse import csr_matrix
    >>> from scipy.sparse.csgraph import maximum_flow
    >>> graph = csr_matrix([[0, 5], [0, 0]])
    >>> maximum_flow(graph, 0, 1).flow_value
    5

    If, on the other hand, there is a bottleneck between source and sink, that
    can reduce the maximum flow::

        (0) --5--> (1) --3--> (2)

    >>> graph = csr_matrix([[0, 5, 0], [0, 0, 3], [0, 0, 0]])
    >>> maximum_flow(graph, 0, 2).flow_value
    3

    A less trivial example is given in [2]_, Chapter 26.1:

    >>> graph = csr_matrix([[0, 16, 13,  0,  0,  0],
    ...                     [0, 10,  0, 12,  0,  0],
    ...                     [0,  4,  0,  0, 14,  0],
    ...                     [0,  0,  9,  0,  0, 20],
    ...                     [0,  0,  0,  7,  0,  4],
    ...                     [0,  0,  0,  0,  0,  0]])
    >>> maximum_flow(graph, 0, 5).flow_value
    23

    It is possible to reduce the problem of finding a maximum matching in a
    bipartite graph to a maximum flow problem: Let :math:`G = ((U, V), E)` be a
    bipartite graph. Then, add to the graph a source vertex with edges to every
    vertex in :math:`U` and a sink vertex with edges from every vertex in
    :math:`V`. Finally, give every edge in the resulting graph a capacity of 1.
    Then, a maximum flow in the new graph gives a maximum matching in the
    original graph consisting of the edges in :math:`E` whose flow is positive.

    Assume that the edges are represented by a
    :math:`\lvert U \rvert \times \lvert V \rvert` matrix in CSR format whose
    :math:`(i, j)`'th entry is 1 if there is an edge from :math:`i \in U` to
    :math:`j \in V` and 0 otherwise; that is, the input is of the form required
    by :func:`maximum_bipartite_matching`. Then the CSR representation of the
    graph constructed above contains this matrix as a block. Here's an example:

    >>> graph = csr_matrix([[0, 1, 0, 1], [1, 0, 1, 0], [0, 1, 1, 0]])
    >>> print(graph.toarray())
    [[0 1 0 1]
     [1 0 1 0]
     [0 1 1 0]]
    >>> i, j = graph.shape
    >>> n = graph.nnz
    >>> indptr = np.concatenate([[0],
    ...                          graph.indptr + i,
    ...                          np.arange(n + i + 1, n + i + j + 1),
    ...                          [n + i + j]])
    >>> indices = np.concatenate([np.arange(1, i + 1),
    ...                           graph.indices + i + 1,
    ...                           np.repeat(i + j + 1, j)])
    >>> data = np.ones(n + i + j, dtype=int)
    >>>
    >>> graph_flow = csr_matrix((data, indices, indptr))
    >>> print(graph_flow.toarray())
    [[0 1 1 1 0 0 0 0 0]
     [0 0 0 0 0 1 0 1 0]
     [0 0 0 0 1 0 1 0 0]
     [0 0 0 0 0 1 1 0 0]
     [0 0 0 0 0 0 0 0 1]
     [0 0 0 0 0 0 0 0 1]
     [0 0 0 0 0 0 0 0 1]
     [0 0 0 0 0 0 0 0 1]
     [0 0 0 0 0 0 0 0 0]]

    At this point, we can find the maximum flow between the added sink and the
    added source and the desired matching can be obtained by restricting the
    residual graph to the block corresponding to the original graph:

    >>> flow = maximum_flow(graph_flow, 0, i+j+1)
    >>> matching = flow.residual[1:i+1, i+1:i+j+1]
    >>> print(matching.toarray())
    [[0 1 0 0]
     [1 0 0 0]
     [0 0 1 0]]

    This tells us that the first, second, and third vertex in :math:`U` are
    matched with the second, first, and third vertex in :math:`V` respectively.

    While this solves the maximum bipartite matching problem in general, note
    that algorithms specialized to that problem, such as
    :func:`maximum_bipartite_matching`, will generally perform better.

    This approach can also be used to solve various common generalizations of
    the maximum bipartite matching problem. If, for instance, some vertices can
    be matched with more than one other vertex, this may be handled by
    modifying the capacities of the new graph appropriately.