# Compressed Sparse Graph Routines (`scipy.sparse.csgraph`)¶

Fast graph algorithms based on sparse matrix representations.

## Contents¶

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 |

## 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]. 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.