# 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 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_flow`(csgraph, source, sink) Maximize the flow between two vertices in a graph. `maximum_bipartite_matching`(graph[, perm_type]) Returns a matching of a bipartite graph whose cardinality is as least that of any given matching of the graph. `min_weight_full_bipartite_matching`(...[, ...]) Returns the minimum weight full matching of a bipartite graph. `structural_rank`(graph) Compute the structural rank of a graph (matrix) with a given sparsity pattern. `NegativeCycleError`([message])
 `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. 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:

```>>> import numpy as np
>>> G_dense = np.array([[0, 2, 1],
...                     [2, 0, 0],
...                     [1, 0, 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:

```>>> import numpy as np
>>> G2_data = np.array([[np.inf, 2,      0     ],
...                     [2,      np.inf, np.inf],
...                     [0,      np.inf, np.inf]])
>>> 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:

```>>> import numpy as np
>>> 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)--3--(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.