# scipy.sparse.csgraph.shortest_path

scipy.sparse.csgraph.shortest_path(csgraph, method='auto', directed=True, return_predecessors=False, unweighted=False, overwrite=False, indices=None)

  Perform a shortest-path graph search on a positive directed or undirected graph.

  New in version 0.11.0.

  Parameters:
    csgraph array, matrix, or sparse matrix, 2 dimensions

      The N x N array of distances representing the input graph.

    method string [‘auto’|’FW’|’D’], optional

      Algorithm to use for shortest paths. Options are:

      > ‘auto’ – (default) select the best among ‘FW’, ‘D’, ‘BF’, or ‘J’

      >   based on the input data.

      > ‘FW’ – Floyd-Warshall algorithm.

      >   Computational cost is approximately `O[N^3]`. The input csgraph will be converted to a dense representation.

      > ‘D’ – Dijkstra’s algorithm with Fibonacci heaps.

      >   Computational cost is approximately `O[N(N*k + N*log(N))]`, where `k` is the average number of connected edges per node. The input csgraph will be converted to a csr representation.

      > ‘BF’ – Bellman-Ford algorithm.

      >   This algorithm can be used when weights are negative. If a negative cycle is encountered, an error will be raised. Computational cost is approximately `O[N(N^2 k)]`, where `k` is the average number of connected edges per node. The input csgraph will be converted to a csr representation.

      > ‘J’ – Johnson’s algorithm.

      >   Like the Bellman-Ford algorithm, Johnson’s algorithm is designed for use when the weights are negative. It combines the Bellman-Ford algorithm with Dijkstra’s algorithm for faster computation.

    directed bool, optional

      If True (default), then find the shortest path on a directed graph: only move from point i to point j along paths csgraph[i, j]. If False, then find the shortest path on an undirected graph: the algorithm can progress from point i to j along csgraph[i, j] or csgraph[j, i]

    return_predecessors bool, optional

      If True, return the size (N, N) predecesor matrix

    unweighted bool, optional

      If True, then find unweighted distances. That is, rather than finding the path between each point such that the sum of weights is minimized, find the path such that the number of edges is minimized.

    overwrite bool, optional

      If True, overwrite csgraph with the result. This applies only if method == ‘FW’ and csgraph is a dense, c-ordered array with dtype=float64.

    indices array_like or int, optional

      If specified, only compute the paths from the points at the given indices. Incompatible with method == ‘FW’.

  Returns:
    dist_matrix ndarray

      The N x N matrix of distances between graph nodes. dist_matrix[i,j] gives the shortest distance from point i to point j along the graph.

    predecessors ndarray

      Returned only if return_predecessors == True. The N x N matrix of predecessors, which can be used to reconstruct the shortest paths. Row i of the predecessor matrix contains information on the shortest paths from point i: each entry predecessors[i, j] gives the index of the previous node in the path from point i to point j. If no path exists between point i and j, then predecessors[i, j] = -9999

  Raises:
    NegativeCycleError:

      if there are negative cycles in the graph

  Notes

  As currently implemented, Dijkstra’s algorithm and Johnson’s algorithm do not work for graphs with direction-dependent distances when directed == False. i.e., if csgraph[i,j] and csgraph[j,i] are non-equal edges, method=’D’ may yield an incorrect result.

  Examples

      >>> from scipy.sparse import csr_matrix
      >>> from scipy.sparse.csgraph import shortest_path

      >>> graph = [
      ... [0, 1, 2, 0],
      ... [0, 0, 0, 1],
      ... [2, 0, 0, 3],
      ... [0, 0, 0, 0]
      ... ]
      >>> graph = csr_matrix(graph)
      >>> print(graph)
        (0, 1)    1
        (0, 2)    2
        (1, 3)    1
        (2, 0)    2
        (2, 3)    3

      >>> dist_matrix, predecessors = shortest_path(csgraph=graph, directed=False, indices=0, return_predecessors=True)
      >>> dist_matrix
      array([0., 1., 2., 2.])
      >>> predecessors
      array([-9999,     0,     0,     1], dtype=int32)