demo + utils venv

2019-02-03 13:40:10 +01:00
parent 5fa112490b
commit cfa9c8ea23
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+r"""
+==============================================================
+Compressed Sparse Graph Routines (:mod:`scipy.sparse.csgraph`)
+==============================================================
+
+.. currentmodule:: scipy.sparse.csgraph
+
+Fast graph algorithms based on sparse matrix representations.
+
+Contents
+========
+
+.. autosummary::
+   :toctree: generated/
+
+   connected_components -- determine connected components of a graph
+   laplacian -- compute the laplacian of a graph
+   shortest_path -- compute the shortest path between points on a positive graph
+   dijkstra -- use Dijkstra's algorithm for shortest path
+   floyd_warshall -- use the Floyd-Warshall algorithm for shortest path
+   bellman_ford -- use the Bellman-Ford algorithm for shortest path
+   johnson -- use Johnson's algorithm for shortest path
+   breadth_first_order -- compute a breadth-first order of nodes
+   depth_first_order -- compute a depth-first order of nodes
+   breadth_first_tree -- construct the breadth-first tree from a given node
+   depth_first_tree -- construct a depth-first tree from a given node
+   minimum_spanning_tree -- construct the minimum spanning tree of a graph
+   reverse_cuthill_mckee -- compute permutation for reverse Cuthill-McKee ordering
+   maximum_bipartite_matching -- compute permutation to make diagonal zero free
+   structural_rank -- compute the structural rank of a graph
+   NegativeCycleError
+
+.. autosummary::
+   :toctree: generated/
+
+   construct_dist_matrix
+   csgraph_from_dense
+   csgraph_from_masked
+   csgraph_masked_from_dense
+   csgraph_to_dense
+   csgraph_to_masked
+   reconstruct_path
+
+Graph Representations
+=====================
+This module uses graphs which are stored in a matrix format.  A
+graph with N nodes can be represented by an (N x N) adjacency matrix G.
+If there is a connection from node i to node j, then G[i, j] = w, where
+w is the weight of the connection.  For nodes i and j which are
+not connected, the value depends on the representation:
+
+- for dense array representations, non-edges are represented by
+  G[i, j] = 0, infinity, or NaN.
+
+- for dense masked representations (of type np.ma.MaskedArray), non-edges
+  are represented by masked values.  This can be useful when graphs with
+  zero-weight edges are desired.
+
+- for sparse array representations, non-edges are represented by
+  non-entries in the matrix.  This sort of sparse representation also
+  allows for edges with zero weights.
+
+As a concrete example, imagine that you would like to represent the following
+undirected graph::
+
+              G
+
+             (0)
+            /   \
+           1     2
+          /       \
+        (2)       (1)
+
+This graph has three nodes, where node 0 and 1 are connected by an edge of
+weight 2, and nodes 0 and 2 are connected by an edge of weight 1.
+We can construct the dense, masked, and sparse representations as follows,
+keeping in mind that an undirected graph is represented by a symmetric matrix::
+
+    >>> G_dense = np.array([[0, 2, 1],
+    ...                     [2, 0, 0],
+    ...                     [1, 0, 0]])
+    >>> G_masked = np.ma.masked_values(G_dense, 0)
+    >>> from scipy.sparse import csr_matrix
+    >>> G_sparse = csr_matrix(G_dense)
+
+This becomes more difficult when zero edges are significant.  For example,
+consider the situation when we slightly modify the above graph::
+
+             G2
+
+             (0)
+            /   \
+           0     2
+          /       \
+        (2)       (1)
+
+This is identical to the previous graph, except nodes 0 and 2 are connected
+by an edge of zero weight.  In this case, the dense representation above
+leads to ambiguities: how can non-edges be represented if zero is a meaningful
+value?  In this case, either a masked or sparse representation must be used
+to eliminate the ambiguity::
+
+    >>> G2_data = np.array([[np.inf, 2,      0     ],
+    ...                     [2,      np.inf, np.inf],
+    ...                     [0,      np.inf, np.inf]])
+    >>> G2_masked = np.ma.masked_invalid(G2_data)
+    >>> from scipy.sparse.csgraph import csgraph_from_dense
+    >>> # G2_sparse = csr_matrix(G2_data) would give the wrong result
+    >>> G2_sparse = csgraph_from_dense(G2_data, null_value=np.inf)
+    >>> G2_sparse.data
+    array([ 2.,  0.,  2.,  0.])
+
+Here we have used a utility routine from the csgraph submodule in order to
+convert the dense representation to a sparse representation which can be
+understood by the algorithms in submodule.  By viewing the data array, we
+can see that the zero values are explicitly encoded in the graph.
+
+Directed vs. Undirected
+-----------------------
+Matrices may represent either directed or undirected graphs.  This is
+specified throughout the csgraph module by a boolean keyword.  Graphs are
+assumed to be directed by default. In a directed graph, traversal from node
+i to node j can be accomplished over the edge G[i, j], but not the edge
+G[j, i].  Consider the following dense graph::
+
+    >>> G_dense = np.array([[0, 1, 0],
+    ...                     [2, 0, 3],
+    ...                     [0, 4, 0]])
+
+When ``directed=True`` we get the graph::
+
+      ---1--> ---3-->
+    (0)     (1)     (2)
+      <--2--- <--4---
+
+In a non-directed graph, traversal from node i to node j can be
+accomplished over either G[i, j] or G[j, i].  If both edges are not null,
+and the two have unequal weights, then the smaller of the two is used.
+
+So for the same graph, when ``directed=False`` we get the graph::
+
+    (0)--1--(1)--2--(2)
+
+Note that a symmetric matrix will represent an undirected graph, regardless
+of whether the 'directed' keyword is set to True or False.  In this case,
+using ``directed=True`` generally leads to more efficient computation.
+
+The routines in this module accept as input either scipy.sparse representations
+(csr, csc, or lil format), masked representations, or dense representations
+with non-edges indicated by zeros, infinities, and NaN entries.
+"""
+
+from __future__ import division, print_function, absolute_import
+
+__docformat__ = "restructuredtext en"
+
+__all__ = ['connected_components',
+           'laplacian',
+           'shortest_path',
+           'floyd_warshall',
+           'dijkstra',
+           'bellman_ford',
+           'johnson',
+           'breadth_first_order',
+           'depth_first_order',
+           'breadth_first_tree',
+           'depth_first_tree',
+           'minimum_spanning_tree',
+           'reverse_cuthill_mckee',
+           'maximum_bipartite_matching',
+           'structural_rank',
+           'construct_dist_matrix',
+           'reconstruct_path',
+           'csgraph_masked_from_dense',
+           'csgraph_from_dense',
+           'csgraph_from_masked',
+           'csgraph_to_dense',
+           'csgraph_to_masked',
+           'NegativeCycleError']
+
+from ._laplacian import laplacian
+from ._shortest_path import shortest_path, floyd_warshall, dijkstra,\
+    bellman_ford, johnson, NegativeCycleError
+from ._traversal import breadth_first_order, depth_first_order, \
+    breadth_first_tree, depth_first_tree, connected_components
+from ._min_spanning_tree import minimum_spanning_tree
+from ._reordering import reverse_cuthill_mckee, maximum_bipartite_matching, \
+    structural_rank
+from ._tools import construct_dist_matrix, reconstruct_path,\
+    csgraph_from_dense, csgraph_to_dense, csgraph_masked_from_dense,\
+    csgraph_from_masked, csgraph_to_masked
+
+from scipy._lib._testutils import PytestTester
+test = PytestTester(__name__)
+del PytestTester
@@ -0,0 +1,128 @@
+"""
+Laplacian of a compressed-sparse graph
+"""
+
+# Authors: Aric Hagberg <hagberg@lanl.gov>
+#          Gael Varoquaux <gael.varoquaux@normalesup.org>
+#          Jake Vanderplas <vanderplas@astro.washington.edu>
+# License: BSD
+
+from __future__ import division, print_function, absolute_import
+
+import numpy as np
+from scipy.sparse import isspmatrix
+
+
+###############################################################################
+# Graph laplacian
+def laplacian(csgraph, normed=False, return_diag=False, use_out_degree=False):
+    """
+    Return the Laplacian matrix of a directed graph.
+
+    Parameters
+    ----------
+    csgraph : array_like or sparse matrix, 2 dimensions
+        compressed-sparse graph, with shape (N, N).
+    normed : bool, optional
+        If True, then compute normalized Laplacian.
+    return_diag : bool, optional
+        If True, then also return an array related to vertex degrees.
+    use_out_degree : bool, optional
+        If True, then use out-degree instead of in-degree.
+        This distinction matters only if the graph is asymmetric.
+        Default: False.
+
+    Returns
+    -------
+    lap : ndarray or sparse matrix
+        The N x N laplacian matrix of csgraph. It will be a numpy array (dense)
+        if the input was dense, or a sparse matrix otherwise.
+    diag : ndarray, optional
+        The length-N diagonal of the Laplacian matrix.
+        For the normalized Laplacian, this is the array of square roots
+        of vertex degrees or 1 if the degree is zero.
+
+    Notes
+    -----
+    The Laplacian matrix of a graph is sometimes referred to as the
+    "Kirchoff matrix" or the "admittance matrix", and is useful in many
+    parts of spectral graph theory.  In particular, the eigen-decomposition
+    of the laplacian matrix can give insight into many properties of the graph.
+
+    Examples
+    --------
+    >>> from scipy.sparse import csgraph
+    >>> G = np.arange(5) * np.arange(5)[:, np.newaxis]
+    >>> G
+    array([[ 0,  0,  0,  0,  0],
+           [ 0,  1,  2,  3,  4],
+           [ 0,  2,  4,  6,  8],
+           [ 0,  3,  6,  9, 12],
+           [ 0,  4,  8, 12, 16]])
+    >>> csgraph.laplacian(G, normed=False)
+    array([[  0,   0,   0,   0,   0],
+           [  0,   9,  -2,  -3,  -4],
+           [  0,  -2,  16,  -6,  -8],
+           [  0,  -3,  -6,  21, -12],
+           [  0,  -4,  -8, -12,  24]])
+    """
+    if csgraph.ndim != 2 or csgraph.shape[0] != csgraph.shape[1]:
+        raise ValueError('csgraph must be a square matrix or array')
+
+    if normed and (np.issubdtype(csgraph.dtype, np.signedinteger)
+                   or np.issubdtype(csgraph.dtype, np.uint)):
+        csgraph = csgraph.astype(float)
+
+    create_lap = _laplacian_sparse if isspmatrix(csgraph) else _laplacian_dense
+    degree_axis = 1 if use_out_degree else 0
+    lap, d = create_lap(csgraph, normed=normed, axis=degree_axis)
+    if return_diag:
+        return lap, d
+    return lap
+
+
+def _setdiag_dense(A, d):
+    A.flat[::len(d)+1] = d
+
+
+def _laplacian_sparse(graph, normed=False, axis=0):
+    if graph.format in ('lil', 'dok'):
+        m = graph.tocoo()
+        needs_copy = False
+    else:
+        m = graph
+        needs_copy = True
+    w = m.sum(axis=axis).getA1() - m.diagonal()
+    if normed:
+        m = m.tocoo(copy=needs_copy)
+        isolated_node_mask = (w == 0)
+        w = np.where(isolated_node_mask, 1, np.sqrt(w))
+        m.data /= w[m.row]
+        m.data /= w[m.col]
+        m.data *= -1
+        m.setdiag(1 - isolated_node_mask)
+    else:
+        if m.format == 'dia':
+            m = m.copy()
+        else:
+            m = m.tocoo(copy=needs_copy)
+        m.data *= -1
+        m.setdiag(w)
+    return m, w
+
+
+def _laplacian_dense(graph, normed=False, axis=0):
+    m = np.array(graph)
+    np.fill_diagonal(m, 0)
+    w = m.sum(axis=axis)
+    if normed:
+        isolated_node_mask = (w == 0)
+        w = np.where(isolated_node_mask, 1, np.sqrt(w))
+        m /= w
+        m /= w[:, np.newaxis]
+        m *= -1
+        _setdiag_dense(m, 1 - isolated_node_mask)
+    else:
+        m *= -1
+        _setdiag_dense(m, w)
+    return m, w
@@ -0,0 +1,58 @@
+from __future__ import division, print_function, absolute_import
+
+import numpy as np
+from scipy.sparse import csr_matrix, isspmatrix, isspmatrix_csc
+from ._tools import csgraph_to_dense, csgraph_from_dense,\
+    csgraph_masked_from_dense, csgraph_from_masked
+
+DTYPE = np.float64
+
+
+def validate_graph(csgraph, directed, dtype=DTYPE,
+                   csr_output=True, dense_output=True,
+                   copy_if_dense=False, copy_if_sparse=False,
+                   null_value_in=0, null_value_out=np.inf,
+                   infinity_null=True, nan_null=True):
+    """Routine for validation and conversion of csgraph inputs"""
+    if not (csr_output or dense_output):
+        raise ValueError("Internal: dense or csr output must be true")
+
+    # if undirected and csc storage, then transposing in-place
+    # is quicker than later converting to csr.
+    if (not directed) and isspmatrix_csc(csgraph):
+        csgraph = csgraph.T
+
+    if isspmatrix(csgraph):
+        if csr_output:
+            csgraph = csr_matrix(csgraph, dtype=DTYPE, copy=copy_if_sparse)
+        else:
+            csgraph = csgraph_to_dense(csgraph, null_value=null_value_out)
+    elif np.ma.isMaskedArray(csgraph):
+        if dense_output:
+            mask = csgraph.mask
+            csgraph = np.array(csgraph.data, dtype=DTYPE, copy=copy_if_dense)
+            csgraph[mask] = null_value_out
+        else:
+            csgraph = csgraph_from_masked(csgraph)
+    else:
+        if dense_output:
+            csgraph = csgraph_masked_from_dense(csgraph,
+                                                copy=copy_if_dense,
+                                                null_value=null_value_in,
+                                                nan_null=nan_null,
+                                                infinity_null=infinity_null)
+            mask = csgraph.mask
+            csgraph = np.asarray(csgraph.data, dtype=DTYPE)
+            csgraph[mask] = null_value_out
+        else:
+            csgraph = csgraph_from_dense(csgraph, null_value=null_value_in,
+                                         infinity_null=infinity_null,
+                                         nan_null=nan_null)
+
+    if csgraph.ndim != 2:
+        raise ValueError("compressed-sparse graph must be two dimensional")
+
+    if csgraph.shape[0] != csgraph.shape[1]:
+        raise ValueError("compressed-sparse graph must be shape (N, N)")
+
+    return csgraph
@@ -0,0 +1,32 @@
+from __future__ import division, print_function, absolute_import
+
+
+def configuration(parent_package='', top_path=None):
+    import numpy
+    from numpy.distutils.misc_util import Configuration
+
+    config = Configuration('csgraph', parent_package, top_path)
+
+    config.add_data_dir('tests')
+
+    config.add_extension('_shortest_path',
+         sources=['_shortest_path.c'],
+         include_dirs=[numpy.get_include()])
+
+    config.add_extension('_traversal',
+         sources=['_traversal.c'],
+         include_dirs=[numpy.get_include()])
+
+    config.add_extension('_min_spanning_tree',
+         sources=['_min_spanning_tree.c'],
+         include_dirs=[numpy.get_include()])
+    
+    config.add_extension('_reordering',
+         sources=['_reordering.c'],
+         include_dirs=[numpy.get_include()])
+
+    config.add_extension('_tools',
+         sources=['_tools.c'],
+         include_dirs=[numpy.get_include()])
+
+    return config
@@ -0,0 +1,101 @@
+from __future__ import division, print_function, absolute_import
+
+import numpy as np
+from numpy.testing import assert_equal, assert_array_almost_equal
+from scipy.sparse import csgraph
+
+
+def test_weak_connections():
+    Xde = np.array([[0, 1, 0],
+                    [0, 0, 0],
+                    [0, 0, 0]])
+
+    Xsp = csgraph.csgraph_from_dense(Xde, null_value=0)
+
+    for X in Xsp, Xde:
+        n_components, labels =\
+            csgraph.connected_components(X, directed=True,
+                                         connection='weak')
+
+        assert_equal(n_components, 2)
+        assert_array_almost_equal(labels, [0, 0, 1])
+
+
+def test_strong_connections():
+    X1de = np.array([[0, 1, 0],
+                     [0, 0, 0],
+                     [0, 0, 0]])
+    X2de = X1de + X1de.T
+
+    X1sp = csgraph.csgraph_from_dense(X1de, null_value=0)
+    X2sp = csgraph.csgraph_from_dense(X2de, null_value=0)
+
+    for X in X1sp, X1de:
+        n_components, labels =\
+            csgraph.connected_components(X, directed=True,
+                                         connection='strong')
+
+        assert_equal(n_components, 3)
+        labels.sort()
+        assert_array_almost_equal(labels, [0, 1, 2])
+
+    for X in X2sp, X2de:
+        n_components, labels =\
+            csgraph.connected_components(X, directed=True,
+                                         connection='strong')
+
+        assert_equal(n_components, 2)
+        labels.sort()
+        assert_array_almost_equal(labels, [0, 0, 1])
+
+
+def test_strong_connections2():
+    X = np.array([[0, 0, 0, 0, 0, 0],
+                  [1, 0, 1, 0, 0, 0],
+                  [0, 0, 0, 1, 0, 0],
+                  [0, 0, 1, 0, 1, 0],
+                  [0, 0, 0, 0, 0, 0],
+                  [0, 0, 0, 0, 1, 0]])
+    n_components, labels =\
+        csgraph.connected_components(X, directed=True,
+                                     connection='strong')
+    assert_equal(n_components, 5)
+    labels.sort()
+    assert_array_almost_equal(labels, [0, 1, 2, 2, 3, 4])
+
+
+def test_weak_connections2():
+    X = np.array([[0, 0, 0, 0, 0, 0],
+                  [1, 0, 0, 0, 0, 0],
+                  [0, 0, 0, 1, 0, 0],
+                  [0, 0, 1, 0, 1, 0],
+                  [0, 0, 0, 0, 0, 0],
+                  [0, 0, 0, 0, 1, 0]])
+    n_components, labels =\
+        csgraph.connected_components(X, directed=True,
+                                     connection='weak')
+    assert_equal(n_components, 2)
+    labels.sort()
+    assert_array_almost_equal(labels, [0, 0, 1, 1, 1, 1])
+
+
+def test_ticket1876():
+    # Regression test: this failed in the original implementation
+    # There should be two strongly-connected components; previously gave one
+    g = np.array([[0, 1, 1, 0],
+                  [1, 0, 0, 1],
+                  [0, 0, 0, 1],
+                  [0, 0, 1, 0]])
+    n_components, labels = csgraph.connected_components(g, connection='strong')
+
+    assert_equal(n_components, 2)
+    assert_equal(labels[0], labels[1])
+    assert_equal(labels[2], labels[3])
+
+
+def test_fully_connected_graph():
+    # Fully connected dense matrices raised an exception.
+    # https://github.com/scipy/scipy/issues/3818
+    g = np.ones((4, 4))
+    n_components, labels = csgraph.connected_components(g)
+    assert_equal(n_components, 1)
@@ -0,0 +1,69 @@
+from __future__ import division, print_function, absolute_import
+
+import numpy as np
+from numpy.testing import assert_array_almost_equal
+from scipy.sparse import csr_matrix
+from scipy.sparse.csgraph import csgraph_from_dense, csgraph_to_dense
+
+
+def test_csgraph_from_dense():
+    np.random.seed(1234)
+    G = np.random.random((10, 10))
+    some_nulls = (G < 0.4)
+    all_nulls = (G < 0.8)
+
+    for null_value in [0, np.nan, np.inf]:
+        G[all_nulls] = null_value
+        olderr = np.seterr(invalid="ignore")
+        try:
+            G_csr = csgraph_from_dense(G, null_value=0)
+        finally:
+            np.seterr(**olderr)
+
+        G[all_nulls] = 0
+        assert_array_almost_equal(G, G_csr.toarray())
+
+    for null_value in [np.nan, np.inf]:
+        G[all_nulls] = 0
+        G[some_nulls] = null_value
+        olderr = np.seterr(invalid="ignore")
+        try:
+            G_csr = csgraph_from_dense(G, null_value=0)
+        finally:
+            np.seterr(**olderr)
+
+        G[all_nulls] = 0
+        assert_array_almost_equal(G, G_csr.toarray())
+
+
+def test_csgraph_to_dense():
+    np.random.seed(1234)
+    G = np.random.random((10, 10))
+    nulls = (G < 0.8)
+    G[nulls] = np.inf
+
+    G_csr = csgraph_from_dense(G)
+
+    for null_value in [0, 10, -np.inf, np.inf]:
+        G[nulls] = null_value
+        assert_array_almost_equal(G, csgraph_to_dense(G_csr, null_value))
+
+
+def test_multiple_edges():
+    # create a random sqare matrix with an even number of elements
+    np.random.seed(1234)
+    X = np.random.random((10, 10))
+    Xcsr = csr_matrix(X)
+
+    # now double-up every other column
+    Xcsr.indices[::2] = Xcsr.indices[1::2]
+
+    # normal sparse toarray() will sum the duplicated edges
+    Xdense = Xcsr.toarray()
+    assert_array_almost_equal(Xdense[:, 1::2],
+                              X[:, ::2] + X[:, 1::2])
+
+    # csgraph_to_dense chooses the minimum of each duplicated edge
+    Xdense = csgraph_to_dense(Xcsr)
+    assert_array_almost_equal(Xdense[:, 1::2],
+                              np.minimum(X[:, ::2], X[:, 1::2]))
@@ -0,0 +1,136 @@
+# Author: Gael Varoquaux <gael.varoquaux@normalesup.org>
+#         Jake Vanderplas <vanderplas@astro.washington.edu>
+# License: BSD
+from __future__ import division, print_function, absolute_import
+
+import numpy as np
+from numpy.testing import assert_allclose, assert_array_almost_equal
+from pytest import raises as assert_raises
+from scipy import sparse
+
+from scipy.sparse import csgraph
+
+
+def _explicit_laplacian(x, normed=False):
+    if sparse.issparse(x):
+        x = x.todense()
+    x = np.asarray(x)
+    y = -1.0 * x
+    for j in range(y.shape[0]):
+        y[j,j] = x[j,j+1:].sum() + x[j,:j].sum()
+    if normed:
+        d = np.diag(y).copy()
+        d[d == 0] = 1.0
+        y /= d[:,None]**.5
+        y /= d[None,:]**.5
+    return y
+
+
+def _check_symmetric_graph_laplacian(mat, normed):
+    if not hasattr(mat, 'shape'):
+        mat = eval(mat, dict(np=np, sparse=sparse))
+
+    if sparse.issparse(mat):
+        sp_mat = mat
+        mat = sp_mat.todense()
+    else:
+        sp_mat = sparse.csr_matrix(mat)
+
+    laplacian = csgraph.laplacian(mat, normed=normed)
+    n_nodes = mat.shape[0]
+    if not normed:
+        assert_array_almost_equal(laplacian.sum(axis=0), np.zeros(n_nodes))
+    assert_array_almost_equal(laplacian.T, laplacian)
+    assert_array_almost_equal(laplacian,
+            csgraph.laplacian(sp_mat, normed=normed).todense())
+
+    assert_array_almost_equal(laplacian,
+            _explicit_laplacian(mat, normed=normed))
+
+
+def test_laplacian_value_error():
+    for t in int, float, complex:
+        for m in ([1, 1],
+                  [[[1]]],
+                  [[1, 2, 3], [4, 5, 6]],
+                  [[1, 2], [3, 4], [5, 5]]):
+            A = np.array(m, dtype=t)
+            assert_raises(ValueError, csgraph.laplacian, A)
+
+
+def test_symmetric_graph_laplacian():
+    symmetric_mats = ('np.arange(10) * np.arange(10)[:, np.newaxis]',
+            'np.ones((7, 7))',
+            'np.eye(19)',
+            'sparse.diags([1, 1], [-1, 1], shape=(4,4))',
+            'sparse.diags([1, 1], [-1, 1], shape=(4,4)).todense()',
+            'np.asarray(sparse.diags([1, 1], [-1, 1], shape=(4,4)).todense())',
+            'np.vander(np.arange(4)) + np.vander(np.arange(4)).T')
+    for mat_str in symmetric_mats:
+        for normed in True, False:
+            _check_symmetric_graph_laplacian(mat_str, normed)
+
+
+def _assert_allclose_sparse(a, b, **kwargs):
+    # helper function that can deal with sparse matrices
+    if sparse.issparse(a):
+        a = a.toarray()
+    if sparse.issparse(b):
+        b = a.toarray()
+    assert_allclose(a, b, **kwargs)
+
+
+def _check_laplacian(A, desired_L, desired_d, normed, use_out_degree):
+    for arr_type in np.array, sparse.csr_matrix, sparse.coo_matrix:
+        for t in int, float, complex:
+            adj = arr_type(A, dtype=t)
+            L = csgraph.laplacian(adj, normed=normed, return_diag=False,
+                                  use_out_degree=use_out_degree)
+            _assert_allclose_sparse(L, desired_L, atol=1e-12)
+            L, d = csgraph.laplacian(adj, normed=normed, return_diag=True,
+                                  use_out_degree=use_out_degree)
+            _assert_allclose_sparse(L, desired_L, atol=1e-12)
+            _assert_allclose_sparse(d, desired_d, atol=1e-12)
+
+
+def test_asymmetric_laplacian():
+    # adjacency matrix
+    A = [[0, 1, 0],
+         [4, 2, 0],
+         [0, 0, 0]]
+
+    # Laplacian matrix using out-degree
+    L = [[1, -1, 0],
+         [-4, 4, 0],
+         [0, 0, 0]]
+    d = [1, 4, 0]
+    _check_laplacian(A, L, d, normed=False, use_out_degree=True)
+
+    # normalized Laplacian matrix using out-degree
+    L = [[1, -0.5, 0],
+         [-2, 1, 0],
+         [0, 0, 0]]
+    d = [1, 2, 1]
+    _check_laplacian(A, L, d, normed=True, use_out_degree=True)
+
+    # Laplacian matrix using in-degree
+    L = [[4, -1, 0],
+         [-4, 1, 0],
+         [0, 0, 0]]
+    d = [4, 1, 0]
+    _check_laplacian(A, L, d, normed=False, use_out_degree=False)
+
+    # normalized Laplacian matrix using in-degree
+    L = [[1, -0.5, 0],
+         [-2, 1, 0],
+         [0, 0, 0]]
+    d = [2, 1, 1]
+    _check_laplacian(A, L, d, normed=True, use_out_degree=False)
+
+
+def test_sparse_formats():
+    for fmt in ('csr', 'csc', 'coo', 'lil', 'dok', 'dia', 'bsr'):
+        mat = sparse.diags([1, 1], [-1, 1], shape=(4,4), format=fmt)
+        for normed in True, False:
+            _check_symmetric_graph_laplacian(mat, normed)
+
@@ -0,0 +1,121 @@
+from __future__ import division, print_function, absolute_import
+
+import numpy as np
+from numpy.testing import assert_equal
+from scipy.sparse.csgraph import (reverse_cuthill_mckee,
+        maximum_bipartite_matching, structural_rank)
+from scipy.sparse import diags, csc_matrix, csr_matrix, coo_matrix
+
+def test_graph_reverse_cuthill_mckee():
+    A = np.array([[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]], dtype=int)
+    
+    graph = csr_matrix(A)
+    perm = reverse_cuthill_mckee(graph)
+    correct_perm = np.array([6, 3, 7, 5, 1, 2, 4, 0])
+    assert_equal(perm, correct_perm)
+    
+    # Test int64 indices input
+    graph.indices = graph.indices.astype('int64')
+    graph.indptr = graph.indptr.astype('int64')
+    perm = reverse_cuthill_mckee(graph, True)
+    assert_equal(perm, correct_perm)
+
+
+def test_graph_reverse_cuthill_mckee_ordering():
+    data = np.ones(63,dtype=int)
+    rows = np.array([0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 
+                2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5,
+                6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9,
+                9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 
+                12, 12, 12, 13, 13, 13, 13, 14, 14, 14,
+                14, 15, 15, 15, 15, 15])
+    cols = np.array([0, 2, 5, 8, 10, 1, 3, 9, 11, 0, 2,
+                7, 10, 1, 3, 11, 4, 6, 12, 14, 0, 7, 13, 
+                15, 4, 6, 14, 2, 5, 7, 15, 0, 8, 10, 13,
+                1, 9, 11, 0, 2, 8, 10, 15, 1, 3, 9, 11,
+                4, 12, 14, 5, 8, 13, 15, 4, 6, 12, 14,
+                5, 7, 10, 13, 15])
+    graph = coo_matrix((data, (rows,cols))).tocsr()
+    perm = reverse_cuthill_mckee(graph)
+    correct_perm = np.array([12, 14, 4, 6, 10, 8, 2, 15,
+                0, 13, 7, 5, 9, 11, 1, 3])
+    assert_equal(perm, correct_perm)
+
+
+def test_graph_maximum_bipartite_matching():
+    A = diags(np.ones(25), offsets=0, format='csc')
+    rand_perm = np.random.permutation(25)
+    rand_perm2 = np.random.permutation(25)
+
+    Rrow = np.arange(25)
+    Rcol = rand_perm
+    Rdata = np.ones(25,dtype=int)
+    Rmat = coo_matrix((Rdata,(Rrow,Rcol))).tocsc()
+
+    Crow = rand_perm2
+    Ccol = np.arange(25)
+    Cdata = np.ones(25,dtype=int)
+    Cmat = coo_matrix((Cdata,(Crow,Ccol))).tocsc()
+    # Randomly permute identity matrix
+    B = Rmat*A*Cmat
+    
+    # Row permute
+    perm = maximum_bipartite_matching(B,perm_type='row')
+    Rrow = np.arange(25)
+    Rcol = perm
+    Rdata = np.ones(25,dtype=int)
+    Rmat = coo_matrix((Rdata,(Rrow,Rcol))).tocsc()
+    C1 = Rmat*B
+    
+    # Column permute
+    perm2 = maximum_bipartite_matching(B,perm_type='column')
+    Crow = perm2
+    Ccol = np.arange(25)
+    Cdata = np.ones(25,dtype=int)
+    Cmat = coo_matrix((Cdata,(Crow,Ccol))).tocsc()
+    C2 = B*Cmat
+    
+    # Should get identity matrix back
+    assert_equal(any(C1.diagonal() == 0), False)
+    assert_equal(any(C2.diagonal() == 0), False)
+    
+    # Test int64 indices input
+    B.indices = B.indices.astype('int64')
+    B.indptr = B.indptr.astype('int64')
+    perm = maximum_bipartite_matching(B,perm_type='row')
+    Rrow = np.arange(25)
+    Rcol = perm
+    Rdata = np.ones(25,dtype=int)
+    Rmat = coo_matrix((Rdata,(Rrow,Rcol))).tocsc()
+    C3 = Rmat*B
+    assert_equal(any(C3.diagonal() == 0), False)
+
+
+def test_graph_structural_rank():
+    # Test square matrix #1
+    A = csc_matrix([[1, 1, 0], 
+                    [1, 0, 1],
+                    [0, 1, 0]])
+    assert_equal(structural_rank(A), 3)
+    
+    # Test square matrix #2
+    rows = np.array([0,0,0,0,0,1,1,2,2,3,3,3,3,3,3,4,4,5,5,6,6,7,7])
+    cols = np.array([0,1,2,3,4,2,5,2,6,0,1,3,5,6,7,4,5,5,6,2,6,2,4])
+    data = np.ones_like(rows)
+    B = coo_matrix((data,(rows,cols)), shape=(8,8))
+    assert_equal(structural_rank(B), 6)
+    
+    #Test non-square matrix
+    C = csc_matrix([[1, 0, 2, 0], 
+                    [2, 0, 4, 0]])
+    assert_equal(structural_rank(C), 2)
+    
+    #Test tall matrix
+    assert_equal(structural_rank(C.T), 2)
@@ -0,0 +1,202 @@
+from __future__ import division, print_function, absolute_import
+
+import numpy as np
+from numpy.testing import assert_array_almost_equal, assert_array_equal
+from pytest import raises as assert_raises
+from scipy.sparse.csgraph import (shortest_path, dijkstra, johnson,
+    bellman_ford, construct_dist_matrix, NegativeCycleError)
+
+
+directed_G = np.array([[0, 3, 3, 0, 0],
+                       [0, 0, 0, 2, 4],
+                       [0, 0, 0, 0, 0],
+                       [1, 0, 0, 0, 0],
+                       [2, 0, 0, 2, 0]], dtype=float)
+
+undirected_G = np.array([[0, 3, 3, 1, 2],
+                         [3, 0, 0, 2, 4],
+                         [3, 0, 0, 0, 0],
+                         [1, 2, 0, 0, 2],
+                         [2, 4, 0, 2, 0]], dtype=float)
+
+unweighted_G = (directed_G > 0).astype(float)
+
+directed_SP = [[0, 3, 3, 5, 7],
+               [3, 0, 6, 2, 4],
+               [np.inf, np.inf, 0, np.inf, np.inf],
+               [1, 4, 4, 0, 8],
+               [2, 5, 5, 2, 0]]
+
+directed_pred = np.array([[-9999, 0, 0, 1, 1],
+                          [3, -9999, 0, 1, 1],
+                          [-9999, -9999, -9999, -9999, -9999],
+                          [3, 0, 0, -9999, 1],
+                          [4, 0, 0, 4, -9999]], dtype=float)
+
+undirected_SP = np.array([[0, 3, 3, 1, 2],
+                          [3, 0, 6, 2, 4],
+                          [3, 6, 0, 4, 5],
+                          [1, 2, 4, 0, 2],
+                          [2, 4, 5, 2, 0]], dtype=float)
+
+undirected_SP_limit_2 = np.array([[0, np.inf, np.inf, 1, 2],
+                                  [np.inf, 0, np.inf, 2, np.inf],
+                                  [np.inf, np.inf, 0, np.inf, np.inf],
+                                  [1, 2, np.inf, 0, 2],
+                                  [2, np.inf, np.inf, 2, 0]], dtype=float)
+
+undirected_SP_limit_0 = np.ones((5, 5), dtype=float) - np.eye(5)
+undirected_SP_limit_0[undirected_SP_limit_0 > 0] = np.inf
+
+undirected_pred = np.array([[-9999, 0, 0, 0, 0],
+                            [1, -9999, 0, 1, 1],
+                            [2, 0, -9999, 0, 0],
+                            [3, 3, 0, -9999, 3],
+                            [4, 4, 0, 4, -9999]], dtype=float)
+
+methods = ['auto', 'FW', 'D', 'BF', 'J']
+
+
+def test_dijkstra_limit():
+    limits = [0, 2, np.inf]
+    results = [undirected_SP_limit_0,
+               undirected_SP_limit_2,
+               undirected_SP]
+
+    def check(limit, result):
+        SP = dijkstra(undirected_G, directed=False, limit=limit)
+        assert_array_almost_equal(SP, result)
+
+    for limit, result in zip(limits, results):
+        check(limit, result)
+
+
+def test_directed():
+    def check(method):
+        SP = shortest_path(directed_G, method=method, directed=True,
+                           overwrite=False)
+        assert_array_almost_equal(SP, directed_SP)
+
+    for method in methods:
+        check(method)
+
+
+def test_undirected():
+    def check(method, directed_in):
+        if directed_in:
+            SP1 = shortest_path(directed_G, method=method, directed=False,
+                                overwrite=False)
+            assert_array_almost_equal(SP1, undirected_SP)
+        else:
+            SP2 = shortest_path(undirected_G, method=method, directed=True,
+                                overwrite=False)
+            assert_array_almost_equal(SP2, undirected_SP)
+
+    for method in methods:
+        for directed_in in (True, False):
+            check(method, directed_in)
+
+
+def test_shortest_path_indices():
+    indices = np.arange(4)
+
+    def check(func, indshape):
+        outshape = indshape + (5,)
+        SP = func(directed_G, directed=False,
+                  indices=indices.reshape(indshape))
+        assert_array_almost_equal(SP, undirected_SP[indices].reshape(outshape))
+
+    for indshape in [(4,), (4, 1), (2, 2)]:
+        for func in (dijkstra, bellman_ford, johnson, shortest_path):
+            check(func, indshape)
+
+    assert_raises(ValueError, shortest_path, directed_G, method='FW',
+                  indices=indices)
+
+
+def test_predecessors():
+    SP_res = {True: directed_SP,
+              False: undirected_SP}
+    pred_res = {True: directed_pred,
+                False: undirected_pred}
+
+    def check(method, directed):
+        SP, pred = shortest_path(directed_G, method, directed=directed,
+                                 overwrite=False,
+                                 return_predecessors=True)
+        assert_array_almost_equal(SP, SP_res[directed])
+        assert_array_almost_equal(pred, pred_res[directed])
+
+    for method in methods:
+        for directed in (True, False):
+            check(method, directed)
+
+
+def test_construct_shortest_path():
+    def check(method, directed):
+        SP1, pred = shortest_path(directed_G,
+                                  directed=directed,
+                                  overwrite=False,
+                                  return_predecessors=True)
+        SP2 = construct_dist_matrix(directed_G, pred, directed=directed)
+        assert_array_almost_equal(SP1, SP2)
+
+    for method in methods:
+        for directed in (True, False):
+            check(method, directed)
+
+
+def test_unweighted_path():
+    def check(method, directed):
+        SP1 = shortest_path(directed_G,
+                            directed=directed,
+                            overwrite=False,
+                            unweighted=True)
+        SP2 = shortest_path(unweighted_G,
+                            directed=directed,
+                            overwrite=False,
+                            unweighted=False)
+        assert_array_almost_equal(SP1, SP2)
+
+    for method in methods:
+        for directed in (True, False):
+            check(method, directed)
+
+
+def test_negative_cycles():
+    # create a small graph with a negative cycle
+    graph = np.ones([5, 5])
+    graph.flat[::6] = 0
+    graph[1, 2] = -2
+
+    def check(method, directed):
+        assert_raises(NegativeCycleError, shortest_path, graph, method,
+                      directed)
+
+    for method in ['FW', 'J', 'BF']:
+        for directed in (True, False):
+            check(method, directed)
+
+
+def test_masked_input():
+    G = np.ma.masked_equal(directed_G, 0)
+
+    def check(method):
+        SP = shortest_path(directed_G, method=method, directed=True,
+                           overwrite=False)
+        assert_array_almost_equal(SP, directed_SP)
+
+    for method in methods:
+        check(method)
+
+
+def test_overwrite():
+    G = np.array([[0, 3, 3, 1, 2],
+                  [3, 0, 0, 2, 4],
+                  [3, 0, 0, 0, 0],
+                  [1, 2, 0, 0, 2],
+                  [2, 4, 0, 2, 0]], dtype=float)
+    foo = G.copy()
+    shortest_path(foo, overwrite=False)
+    assert_array_equal(foo, G)
+
@@ -0,0 +1,67 @@
+"""Test the minimum spanning tree function"""
+from __future__ import division, print_function, absolute_import
+
+import numpy as np
+from numpy.testing import assert_
+import numpy.testing as npt
+from scipy.sparse import csr_matrix
+from scipy.sparse.csgraph import minimum_spanning_tree
+
+
+def test_minimum_spanning_tree():
+
+    # Create a graph with two connected components.
+    graph = [[0,1,0,0,0],
+             [1,0,0,0,0],
+             [0,0,0,8,5],
+             [0,0,8,0,1],
+             [0,0,5,1,0]]
+    graph = np.asarray(graph)
+
+    # Create the expected spanning tree.
+    expected = [[0,1,0,0,0],
+                [0,0,0,0,0],
+                [0,0,0,0,5],
+                [0,0,0,0,1],
+                [0,0,0,0,0]]
+    expected = np.asarray(expected)
+
+    # Ensure minimum spanning tree code gives this expected output.
+    csgraph = csr_matrix(graph)
+    mintree = minimum_spanning_tree(csgraph)
+    npt.assert_array_equal(mintree.todense(), expected,
+        'Incorrect spanning tree found.')
+
+    # Ensure that the original graph was not modified.
+    npt.assert_array_equal(csgraph.todense(), graph,
+        'Original graph was modified.')
+
+    # Now let the algorithm modify the csgraph in place.
+    mintree = minimum_spanning_tree(csgraph, overwrite=True)
+    npt.assert_array_equal(mintree.todense(), expected,
+        'Graph was not properly modified to contain MST.')
+
+    np.random.seed(1234)
+    for N in (5, 10, 15, 20):
+
+        # Create a random graph.
+        graph = 3 + np.random.random((N, N))
+        csgraph = csr_matrix(graph)
+
+        # The spanning tree has at most N - 1 edges.
+        mintree = minimum_spanning_tree(csgraph)
+        assert_(mintree.nnz < N)
+
+        # Set the sub diagonal to 1 to create a known spanning tree.
+        idx = np.arange(N-1)
+        graph[idx,idx+1] = 1
+        csgraph = csr_matrix(graph)
+        mintree = minimum_spanning_tree(csgraph)
+
+        # We expect to see this pattern in the spanning tree and otherwise
+        # have this zero.
+        expected = np.zeros((N, N))
+        expected[idx, idx+1] = 1
+
+        npt.assert_array_equal(mintree.todense(), expected,
+            'Incorrect spanning tree found.')
@@ -0,0 +1,70 @@
+from __future__ import division, print_function, absolute_import
+
+import numpy as np
+from numpy.testing import assert_array_almost_equal
+from scipy.sparse.csgraph import (breadth_first_tree, depth_first_tree,
+    csgraph_to_dense, csgraph_from_dense)
+
+
+def test_graph_breadth_first():
+    csgraph = np.array([[0, 1, 2, 0, 0],
+                        [1, 0, 0, 0, 3],
+                        [2, 0, 0, 7, 0],
+                        [0, 0, 7, 0, 1],
+                        [0, 3, 0, 1, 0]])
+    csgraph = csgraph_from_dense(csgraph, null_value=0)
+
+    bfirst = np.array([[0, 1, 2, 0, 0],
+                       [0, 0, 0, 0, 3],
+                       [0, 0, 0, 7, 0],
+                       [0, 0, 0, 0, 0],
+                       [0, 0, 0, 0, 0]])
+
+    for directed in [True, False]:
+        bfirst_test = breadth_first_tree(csgraph, 0, directed)
+        assert_array_almost_equal(csgraph_to_dense(bfirst_test),
+                                  bfirst)
+
+
+def test_graph_depth_first():
+    csgraph = np.array([[0, 1, 2, 0, 0],
+                        [1, 0, 0, 0, 3],
+                        [2, 0, 0, 7, 0],
+                        [0, 0, 7, 0, 1],
+                        [0, 3, 0, 1, 0]])
+    csgraph = csgraph_from_dense(csgraph, null_value=0)
+
+    dfirst = np.array([[0, 1, 0, 0, 0],
+                       [0, 0, 0, 0, 3],
+                       [0, 0, 0, 0, 0],
+                       [0, 0, 7, 0, 0],
+                       [0, 0, 0, 1, 0]])
+
+    for directed in [True, False]:
+        dfirst_test = depth_first_tree(csgraph, 0, directed)
+        assert_array_almost_equal(csgraph_to_dense(dfirst_test),
+                                  dfirst)
+
+
+def test_graph_breadth_first_trivial_graph():
+    csgraph = np.array([[0]])
+    csgraph = csgraph_from_dense(csgraph, null_value=0)
+
+    bfirst = np.array([[0]])
+
+    for directed in [True, False]:
+        bfirst_test = breadth_first_tree(csgraph, 0, directed)
+        assert_array_almost_equal(csgraph_to_dense(bfirst_test),
+                                  bfirst)
+
+
+def test_graph_depth_first_trivial_graph():
+    csgraph = np.array([[0]])
+    csgraph = csgraph_from_dense(csgraph, null_value=0)
+
+    bfirst = np.array([[0]])
+
+    for directed in [True, False]:
+        bfirst_test = depth_first_tree(csgraph, 0, directed)
+        assert_array_almost_equal(csgraph_to_dense(bfirst_test),
+                                  bfirst)