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# bellman_ford_predecessor_and_distance

bellman_ford_predecessor_and_distance(G, source, target=None, weight='weight', heuristic=False)

  Compute shortest path lengths and predecessors on shortest paths in weighted graphs.

  The algorithm has a running time of \(O(mn)\) where \(n\) is the number of nodes and \(m\) is the number of edges. It is slower than Dijkstra but can handle negative edge weights.

  If a negative cycle is detected, you can use `find_negative_cycle()` to return the cycle and examine it. Shortest paths are not defined when a negative cycle exists because once reached, the path can cycle forever to build up arbitrarily low weights.

  Parameters:
    G NetworkX graph

      The algorithm works for all types of graphs, including directed graphs and multigraphs.

    source: node label

      Starting node for path

    target node label, optional

      Ending node for path

    weight string or function

      If this is a string, then edge weights will be accessed via the edge attribute with this key (that is, the weight of the edge joining `u` to `v` will be `G.edges[u, v][weight]`). If no such edge attribute exists, the weight of the edge is assumed to be one.

      If this is a function, the weight of an edge is the value returned by the function. The function must accept exactly three positional arguments: the two endpoints of an edge and the dictionary of edge attributes for that edge. The function must return a number.

    heuristic bool

      Determines whether to use a heuristic to early detect negative cycles at a hopefully negligible cost.

  Returns:
    pred, dist dictionaries

      Returns two dictionaries keyed by node to predecessor in the path and to the distance from the source respectively.

  Raises:
    NodeNotFound

      If `source` is not in `G`.

    NetworkXUnbounded

      If the (di)graph contains a negative (di)cycle, the algorithm raises an exception to indicate the presence of the negative (di)cycle. Note: any negative weight edge in an undirected graph is a negative cycle.

  See also

  `find_negative_cycle`

  Notes

  Edge weight attributes must be numerical. Distances are calculated as sums of weighted edges traversed.

  The dictionaries returned only have keys for nodes reachable from the source.

  In the case where the (di)graph is not connected, if a component not containing the source contains a negative (di)cycle, it will not be detected.

  In NetworkX v2.1 and prior, the source node had predecessor `[None]`. In NetworkX v2.2 this changed to the source node having predecessor `[]`

  Examples

      >>> G = nx.path_graph(5, create_using=nx.DiGraph())
      >>> pred, dist = nx.bellman_ford_predecessor_and_distance(G, 0)
      >>> sorted(pred.items())
      [(0, []), (1, [0]), (2, [1]), (3, [2]), (4, [3])]
      >>> sorted(dist.items())
      [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]

      >>> pred, dist = nx.bellman_ford_predecessor_and_distance(G, 0, 1)
      >>> sorted(pred.items())
      [(0, []), (1, [0]), (2, [1]), (3, [2]), (4, [3])]
      >>> sorted(dist.items())
      [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]

      >>> G = nx.cycle_graph(5, create_using=nx.DiGraph())
      >>> G[1][2]["weight"] = -7
      >>> nx.bellman_ford_predecessor_and_distance(G, 0)
      Traceback (most recent call last):
          ...
      networkx.exception.NetworkXUnbounded: Negative cycle detected.