# 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 |

`reverse_cuthill_mckee` (graph[, symmetric_mode]) |
Returns the permutation array that orders a sparse CSR or CSC matrix in Reverse-Cuthill McKee ordering. |

`maximum_bipartite_matching` (graph[, perm_type]) |
Returns an array of row or column permutations that makes the diagonal of a nonsingular square CSC sparse matrix zero free. |

`structural_rank` (graph) |
Compute the structural rank of a graph (matrix) with a given sparsity pattern. |

`NegativeCycleError` |

`construct_dist_matrix` (graph, predecessors[, …]) |
Construct distance matrix from a predecessor matrix |

`csgraph_from_dense` (graph[, null_value, …]) |
Construct a CSR-format sparse graph from a dense matrix. |

`csgraph_from_masked` (graph) |
Construct a CSR-format graph from a masked array. |

`csgraph_masked_from_dense` (graph[, …]) |
Construct a masked array graph representation from a dense matrix. |

`csgraph_to_dense` (csgraph[, null_value]) |
Convert a sparse graph representation to a dense representation |

`csgraph_to_masked` (csgraph) |
Convert a sparse graph representation to a masked array representation |

`reconstruct_path` (csgraph, predecessors[, …]) |
Construct a tree from a graph and a predecessor list. |

## 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.