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# 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.
`yen`(csgraph, source, sink, K, *[, directed, ...]) | Yen's K-Shortest Paths algorithm on a directed or undirected graph.
`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 at least that of any given matching of the graph.
`min_weight_full_bipartite_matching`(biadjacency) | 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.
`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:

    >>> import numpy as np
    >>> 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_array
    >>> G_sparse = csr_array(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]])
    >>> G2_masked = np.ma.masked_invalid(G2_data)
    >>> from scipy.sparse.csgraph import csgraph_from_dense
    >>> # G2_sparse = csr_array(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.