Fast graph algorithms based on sparse matrix representations.

connected_components(csgraph[, directed, ...]) |
Analyze the connected components of a sparse graph |

laplacian(csgraph[, normed, return_diag]) |
Return the Laplacian matrix of a directed graph. |

shortest_path(csgraph[, method, directed, ...]) |
Perform a shortest-path graph search on a positive directed or undirected graph. |

dijkstra(csgraph[, directed, indices, ...]) |
Dijkstra algorithm using Fibonacci Heaps |

floyd_warshall(csgraph[, directed, ...]) |
Compute the shortest path lengths using the Floyd-Warshall algorithm |

bellman_ford(csgraph[, directed, indices, ...]) |
Compute the shortest path lengths using the Bellman-Ford algorithm. |

johnson(csgraph[, directed, indices, ...]) |
Compute the shortest path lengths using Johnson’s algorithm. |

breadth_first_order(csgraph, i_start[, ...]) |
Return a breadth-first ordering starting with specified node. |

depth_first_order(csgraph, i_start[, ...]) |
Return a depth-first ordering starting with specified node. |

breadth_first_tree(csgraph, i_start[, directed]) |
Return the tree generated by a breadth-first search |

depth_first_tree(csgraph, i_start[, directed]) |
Return a tree generated by a depth-first search. |

minimum_spanning_tree(csgraph[, overwrite]) |
Return a minimum spanning tree of an undirected graph |

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.

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]. 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.
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.