dijkstra.py

# -*- coding: utf-8 -*-
"""
Shortest path algorithms for weighed graphs.
"""
__author__ = """\n""".join(['Aric Hagberg <hagberg@lanl.gov>',
                            'Loïc Séguin-C. <loicseguin@gmail.com>',
                            'Dan Schult <dschult@colgate.edu>'])
#    Copyright (C) 2004-2015 by
#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
#    All rights reserved.
#    BSD license.

__all__ = ['dijkstra_path',
           'dijkstra_path_length',
           'bidirectional_dijkstra',
           'single_source_dijkstra',
           'single_source_dijkstra_path',
           'single_source_dijkstra_path_length',
           'all_pairs_dijkstra_path',
           'all_pairs_dijkstra_path_length',
           'dijkstra_predecessor_and_distance',
           'bellman_ford',
           'negative_edge_cycle',
           'goldberg_radzik',
           'johnson']

from collections import deque
from heapq import heappush, heappop
from itertools import count
import networkx as nx
from networkx.utils import generate_unique_node


def dijkstra_path(G, source, target, weight='weight'):
    """Returns the shortest path from source to target in a weighted graph G.

    Parameters
    ----------
    G : NetworkX graph

    source : node
       Starting node

    target : node
       Ending node

    weight: string, optional (default='weight')
       Edge data key corresponding to the edge weight

    Returns
    -------
    path : list
       List of nodes in a shortest path.

    Raises
    ------
    NetworkXNoPath
       If no path exists between source and target.

    Examples
    --------
    >>> G=nx.path_graph(5)
    >>> print(nx.dijkstra_path(G,0,4))
    [0, 1, 2, 3, 4]

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

    See Also
    --------
    bidirectional_dijkstra()
    """
    (length, path) = single_source_dijkstra(G, source, target=target,
                                            weight=weight)
    try:
        return path[target]
    except KeyError:
        raise nx.NetworkXNoPath(
            "node %s not reachable from %s" % (source, target))


def dijkstra_path_length(G, source, target, weight='weight'):
    """Returns the shortest path length from source to target
    in a weighted graph.

    Parameters
    ----------
    G : NetworkX graph

    source : node label
       starting node for path

    target : node label
       ending node for path

    weight: string, optional (default='weight')
       Edge data key corresponding to the edge weight

    Returns
    -------
    length : number
        Shortest path length.

    Raises
    ------
    NetworkXNoPath
        If no path exists between source and target.

    Examples
    --------
    >>> G=nx.path_graph(5)
    >>> print(nx.dijkstra_path_length(G,0,4))
    4

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

    See Also
    --------
    bidirectional_dijkstra()
    """
    length = single_source_dijkstra_path_length(G, source, weight=weight)
    try:
        return length[target]
    except KeyError:
        raise nx.NetworkXNoPath(
            "node %s not reachable from %s" % (source, target))


def single_source_dijkstra_path(G, source, cutoff=None, weight='weight'):
    """Compute shortest path between source and all other reachable
    nodes for a weighted graph.

    Parameters
    ----------
    G : NetworkX graph

    source : node
       Starting node for path.

    weight: string, optional (default='weight')
       Edge data key corresponding to the edge weight

    cutoff : integer or float, optional
       Depth to stop the search. Only paths of length <= cutoff are returned.

    Returns
    -------
    paths : dictionary
       Dictionary of shortest path lengths keyed by target.

    Examples
    --------
    >>> G=nx.path_graph(5)
    >>> path=nx.single_source_dijkstra_path(G,0)
    >>> path[4]
    [0, 1, 2, 3, 4]

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

    See Also
    --------
    single_source_dijkstra()

    """
    (length, path) = single_source_dijkstra(
        G, source, cutoff=cutoff, weight=weight)
    return path


def single_source_dijkstra_path_length(G, source, cutoff=None,
                                       weight='weight'):
    """Compute the shortest path length between source and all other
    reachable nodes for a weighted graph.

    Parameters
    ----------
    G : NetworkX graph

    source : node label
       Starting node for path

    weight: string, optional (default='weight')
       Edge data key corresponding to the edge weight.

    cutoff : integer or float, optional
       Depth to stop the search. Only paths of length <= cutoff are returned.

    Returns
    -------
    length : dictionary
       Dictionary of shortest lengths keyed by target.

    Examples
    --------
    >>> G=nx.path_graph(5)
    >>> length=nx.single_source_dijkstra_path_length(G,0)
    >>> length[4]
    4
    >>> print(length)
    {0: 0, 1: 1, 2: 2, 3: 3, 4: 4}

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

    See Also
    --------
    single_source_dijkstra()

    """
    if G.is_multigraph():
        get_weight = lambda u, v, data: min(
            eattr.get(weight, 1) for eattr in data.values())
    else:
        get_weight = lambda u, v, data: data.get(weight, 1)

    return _dijkstra(G, source, get_weight, cutoff=cutoff)


def single_source_dijkstra(G, source, target=None, cutoff=None, weight='weight'):
    """Compute shortest paths and lengths in a weighted graph G.

    Uses Dijkstra's algorithm for shortest paths.

    Parameters
    ----------
    G : NetworkX graph

    source : node label
       Starting node for path

    target : node label, optional
       Ending node for path

    cutoff : integer or float, optional
       Depth to stop the search. Only paths of length <= cutoff are returned.

    Returns
    -------
    distance,path : dictionaries
       Returns a tuple of two dictionaries keyed by node.
       The first dictionary stores distance from the source.
       The second stores the path from the source to that node.


    Examples
    --------
    >>> G=nx.path_graph(5)
    >>> length,path=nx.single_source_dijkstra(G,0)
    >>> print(length[4])
    4
    >>> print(length)
    {0: 0, 1: 1, 2: 2, 3: 3, 4: 4}
    >>> path[4]
    [0, 1, 2, 3, 4]

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

    Based on the Python cookbook recipe (119466) at
    http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/119466

    This algorithm is not guaranteed to work if edge weights
    are negative or are floating point numbers
    (overflows and roundoff errors can cause problems).

    See Also
    --------
    single_source_dijkstra_path()
    single_source_dijkstra_path_length()
    """
    if source == target:
        return ({source: 0}, {source: [source]})

    if G.is_multigraph():
        get_weight = lambda u, v, data: min(
            eattr.get(weight, 1) for eattr in data.values())
    else:
        get_weight = lambda u, v, data: data.get(weight, 1)

    paths = {source: [source]}  # dictionary of paths
    return _dijkstra(G, source, get_weight, paths=paths, cutoff=cutoff,
                     target=target)


def _dijkstra(G, source, get_weight, pred=None, paths=None, cutoff=None,
              target=None):
    """Implementation of Dijkstra's algorithm

    Parameters
    ----------
    G : NetworkX graph

    source : node label
       Starting node for path

    get_weight: function
        Function for getting edge weight

    pred: list, optional(default=None)
        List of predecessors of a node

    paths: dict, optional (default=None)
        Path from the source to a target node.

    target : node label, optional
       Ending node for path

    cutoff : integer or float, optional
       Depth to stop the search. Only paths of length <= cutoff are returned.

    Returns
    -------
    distance,path : dictionaries
       Returns a tuple of two dictionaries keyed by node.
       The first dictionary stores distance from the source.
       The second stores the path from the source to that node.

    pred,distance : dictionaries
       Returns two dictionaries representing a list of predecessors
       of a node and the distance to each node.

    distance : dictionary
       Dictionary of shortest lengths keyed by target.
    """
    G_succ = G.succ if G.is_directed() else G.adj

    push = heappush
    pop = heappop
    dist = {}  # dictionary of final distances
    seen = {source: 0}
    c = count()
    fringe = []  # use heapq with (distance,label) tuples
    push(fringe, (0, next(c), source))
    while fringe:
        (d, _, v) = pop(fringe)
        if v in dist:
            continue  # already searched this node.
        dist[v] = d
        if v == target:
            break

        for u, e in G_succ[v].items():
            cost = get_weight(v, u, e)
            if cost is None:
                continue
            vu_dist = dist[v] + get_weight(v, u, e)
            if cutoff is not None:
                if vu_dist > cutoff:
                    continue
            if u in dist:
                if vu_dist < dist[u]:
                    raise ValueError('Contradictory paths found:',
                                     'negative weights?')
            elif u not in seen or vu_dist < seen[u]:
                seen[u] = vu_dist
                push(fringe, (vu_dist, next(c), u))
                if paths is not None:
                    paths[u] = paths[v] + [u]
                if pred is not None:
                    pred[u] = [v]
            elif vu_dist == seen[u]:
                if pred is not None:
                    pred[u].append(v)

    if paths is not None:
        return (dist, paths)
    if pred is not None:
        return (pred, dist)
    return dist


def dijkstra_predecessor_and_distance(G, source, cutoff=None, weight='weight'):
    """Compute shortest path length and predecessors on shortest paths
    in weighted graphs.

    Parameters
    ----------
    G : NetworkX graph

    source : node label
       Starting node for path

    weight: string, optional (default='weight')
       Edge data key corresponding to the edge weight

    cutoff : integer or float, optional
       Depth to stop the search. Only paths of length <= cutoff are returned.

    Returns
    -------
    pred,distance : dictionaries
       Returns two dictionaries representing a list of predecessors
       of a node and the distance to each node.

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

    The list of predecessors contains more than one element only when
    there are more than one shortest paths to the key node.
    """
    if G.is_multigraph():
        get_weight = lambda u, v, data: min(
            eattr.get(weight, 1) for eattr in data.values())
    else:
        get_weight = lambda u, v, data: data.get(weight, 1)

    pred = {source: []}  # dictionary of predecessors
    return _dijkstra(G, source, get_weight, pred=pred, cutoff=cutoff)


def all_pairs_dijkstra_path_length(G, cutoff=None, weight='weight'):
    """ Compute shortest path lengths between all nodes in a weighted graph.

    Parameters
    ----------
    G : NetworkX graph

    weight: string, optional (default='weight')
       Edge data key corresponding to the edge weight

    cutoff : integer or float, optional
       Depth to stop the search. Only paths of length <= cutoff are returned.

    Returns
    -------
    distance : dictionary
       Dictionary, keyed by source and target, of shortest path lengths.

    Examples
    --------
    >>> G=nx.path_graph(5)
    >>> length=nx.all_pairs_dijkstra_path_length(G)
    >>> print(length[1][4])
    3
    >>> length[1]
    {0: 1, 1: 0, 2: 1, 3: 2, 4: 3}

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

    The dictionary returned only has keys for reachable node pairs.
    """
    length = single_source_dijkstra_path_length
    # TODO This can be trivially parallelized.
    return {n: length(G, n, cutoff=cutoff, weight=weight) for n in G}


def all_pairs_dijkstra_path(G, cutoff=None, weight='weight'):
    """ Compute shortest paths between all nodes in a weighted graph.

    Parameters
    ----------
    G : NetworkX graph

    weight: string, optional (default='weight')
       Edge data key corresponding to the edge weight

    cutoff : integer or float, optional
       Depth to stop the search. Only paths of length <= cutoff are returned.

    Returns
    -------
    distance : dictionary
       Dictionary, keyed by source and target, of shortest paths.

    Examples
    --------
    >>> G=nx.path_graph(5)
    >>> path=nx.all_pairs_dijkstra_path(G)
    >>> print(path[0][4])
    [0, 1, 2, 3, 4]

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

    See Also
    --------
    floyd_warshall()

    """
    path = single_source_dijkstra_path
    # TODO This can be trivially parallelized.
    return {n: path(G, n, cutoff=cutoff, weight=weight) for n in G}


def bellman_ford(G, source, weight='weight'):
    """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.

    Parameters
    ----------
    G : NetworkX graph
       The algorithm works for all types of graphs, including directed
       graphs and multigraphs.

    source: node label
       Starting node for path

    weight: string, optional (default='weight')
       Edge data key corresponding to the edge weight

    Returns
    -------
    pred, dist : dictionaries
       Returns two dictionaries keyed by node to predecessor in the
       path and to the distance from the source respectively.

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

    Examples
    --------
    >>> import networkx as nx
    >>> G = nx.path_graph(5, create_using = nx.DiGraph())
    >>> pred, dist = nx.bellman_ford(G, 0)
    >>> sorted(pred.items())
    [(0, None), (1, 0), (2, 1), (3, 2), (4, 3)]
    >>> sorted(dist.items())
    [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]

    >>> from nose.tools import assert_raises
    >>> G = nx.cycle_graph(5, create_using = nx.DiGraph())
    >>> G[1][2]['weight'] = -7
    >>> assert_raises(nx.NetworkXUnbounded, nx.bellman_ford, G, 0)

    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 cost (di)cycle, it
    will not be detected.

    """
    if source not in G:
        raise KeyError("Node %s is not found in the graph" % source)

    for u, v, attr in G.selfloop_edges(data=True):
        if attr.get(weight, 1) < 0:
            raise nx.NetworkXUnbounded("Negative cost cycle detected.")

    dist = {source: 0}
    pred = {source: None}

    if len(G) == 1:
        return pred, dist

    return _bellman_ford_relaxation(G, pred, dist, [source], weight)


def _bellman_ford_relaxation(G, pred, dist, source, weight):
    """Relaxation loop for Bellman–Ford algorithm

    Parameters
    ----------
    G : NetworkX graph

    pred: dict
        Keyed by node to predecessor in the path

    dist: dict
        Keyed by node to the distance from the source

    source: list
        List of source nodes

    weight: string
       Edge data key corresponding to the edge weight

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

    Raises
    ------
    NetworkXUnbounded
       If the (di)graph contains a negative cost (di)cycle, the
       algorithm raises an exception to indicate the presence of the
       negative cost (di)cycle.  Note: any negative weight edge in an
       undirected graph is a negative cost cycle
    """
    if G.is_multigraph():
        def get_weight(edge_dict):
            return min(eattr.get(weight, 1) for eattr in edge_dict.values())
    else:
        def get_weight(edge_dict):
            return edge_dict.get(weight, 1)

    G_succ = G.succ if G.is_directed() else G.adj
    inf = float('inf')
    n = len(G)

    count = {}
    q = deque(source)
    in_q = set(source)
    while q:
        u = q.popleft()
        in_q.remove(u)
        # Skip relaxations if the predecessor of u is in the queue.
        if pred[u] not in in_q:
            dist_u = dist[u]
            for v, e in G_succ[u].items():
                dist_v = dist_u + get_weight(e)
                if dist_v < dist.get(v, inf):
                    if v not in in_q:
                        q.append(v)
                        in_q.add(v)
                        count_v = count.get(v, 0) + 1
                        if count_v == n:
                            raise nx.NetworkXUnbounded(
                                "Negative cost cycle detected.")
                        count[v] = count_v
                    dist[v] = dist_v
                    pred[v] = u

    return pred, dist


def goldberg_radzik(G, source, weight='weight'):
    """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.

    Parameters
    ----------
    G : NetworkX graph
       The algorithm works for all types of graphs, including directed
       graphs and multigraphs.

    source: node label
       Starting node for path

    weight: string, optional (default='weight')
       Edge data key corresponding to the edge weight

    Returns
    -------
    pred, dist : dictionaries
       Returns two dictionaries keyed by node to predecessor in the
       path and to the distance from the source respectively.

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

    Examples
    --------
    >>> import networkx as nx
    >>> G = nx.path_graph(5, create_using = nx.DiGraph())
    >>> pred, dist = nx.goldberg_radzik(G, 0)
    >>> sorted(pred.items())
    [(0, None), (1, 0), (2, 1), (3, 2), (4, 3)]
    >>> sorted(dist.items())
    [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]

    >>> from nose.tools import assert_raises
    >>> G = nx.cycle_graph(5, create_using = nx.DiGraph())
    >>> G[1][2]['weight'] = -7
    >>> assert_raises(nx.NetworkXUnbounded, nx.goldberg_radzik, G, 0)

    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 cost (di)cycle, it
    will not be detected.

    """
    if source not in G:
        raise KeyError("Node %s is not found in the graph" % source)

    for u, v, attr in G.selfloop_edges(data=True):
        if attr.get(weight, 1) < 0:
            raise nx.NetworkXUnbounded("Negative cost cycle detected.")

    if len(G) == 1:
        return {source: None}, {source: 0}

    if G.is_multigraph():
        def get_weight(edge_dict):
            return min(attr.get(weight, 1) for attr in edge_dict.values())
    else:
        def get_weight(edge_dict):
            return edge_dict.get(weight, 1)

    if G.is_directed():
        G_succ = G.succ
    else:
        G_succ = G.adj

    inf = float('inf')
    d = dict((u, inf) for u in G)
    d[source] = 0
    pred = {source: None}

    def topo_sort(relabeled):
        """Topologically sort nodes relabeled in the previous round and detect
        negative cycles.
        """
        # List of nodes to scan in this round. Denoted by A in Goldberg and
        # Radzik's paper.
        to_scan = []
        # In the DFS in the loop below, neg_count records for each node the
        # number of edges of negative reduced costs on the path from a DFS root
        # to the node in the DFS forest. The reduced cost of an edge (u, v) is
        # defined as d[u] + weight[u][v] - d[v].
        #
        # neg_count also doubles as the DFS visit marker array.
        neg_count = {}
        for u in relabeled:
            # Skip visited nodes.
            if u in neg_count:
                continue
            d_u = d[u]
            # Skip nodes without out-edges of negative reduced costs.
            if all(d_u + get_weight(e) >= d[v] for v, e in G_succ[u].items()):
                continue
            # Nonrecursive DFS that inserts nodes reachable from u via edges of
            # nonpositive reduced costs into to_scan in (reverse) topological
            # order.
            stack = [(u, iter(G_succ[u].items()))]
            in_stack = set([u])
            neg_count[u] = 0
            while stack:
                u, it = stack[-1]
                try:
                    v, e = next(it)
                except StopIteration:
                    to_scan.append(u)
                    stack.pop()
                    in_stack.remove(u)
                    continue
                t = d[u] + get_weight(e)
                d_v = d[v]
                if t <= d_v:
                    is_neg = t < d_v
                    d[v] = t
                    pred[v] = u
                    if v not in neg_count:
                        neg_count[v] = neg_count[u] + int(is_neg)
                        stack.append((v, iter(G_succ[v].items())))
                        in_stack.add(v)
                    elif (v in in_stack and
                          neg_count[u] + int(is_neg) > neg_count[v]):
                        # (u, v) is a back edge, and the cycle formed by the
                        # path v to u and (u, v) contains at least one edge of
                        # negative reduced cost. The cycle must be of negative
                        # cost.
                        raise nx.NetworkXUnbounded(
                            'Negative cost cycle detected.')
        to_scan.reverse()
        return to_scan

    def relax(to_scan):
        """Relax out-edges of relabeled nodes.
        """
        relabeled = set()
        # Scan nodes in to_scan in topological order and relax incident
        # out-edges. Add the relabled nodes to labeled.
        for u in to_scan:
            d_u = d[u]
            for v, e in G_succ[u].items():
                w_e = get_weight(e)
                if d_u + w_e < d[v]:
                    d[v] = d_u + w_e
                    pred[v] = u
                    relabeled.add(v)
        return relabeled

    # Set of nodes relabled in the last round of scan operations. Denoted by B
    # in Goldberg and Radzik's paper.
    relabeled = set([source])

    while relabeled:
        to_scan = topo_sort(relabeled)
        relabeled = relax(to_scan)

    d = dict((u, d[u]) for u in pred)
    return pred, d


def negative_edge_cycle(G, weight='weight'):
    """Return True if there exists a negative edge cycle anywhere in G.

    Parameters
    ----------
    G : NetworkX graph

    weight: string, optional (default='weight')
       Edge data key corresponding to the edge weight

    Returns
    -------
    negative_cycle : bool
        True if a negative edge cycle exists, otherwise False.

    Examples
    --------
    >>> import networkx as nx
    >>> G = nx.cycle_graph(5, create_using = nx.DiGraph())
    >>> print(nx.negative_edge_cycle(G))
    False
    >>> G[1][2]['weight'] = -7
    >>> print(nx.negative_edge_cycle(G))
    True

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

    This algorithm uses bellman_ford() but finds negative cycles
    on any component by first adding a new node connected to
    every node, and starting bellman_ford on that node.  It then
    removes that extra node.
    """
    newnode = generate_unique_node()
    G.add_edges_from([(newnode, n) for n in G])

    try:
        bellman_ford(G, newnode, weight)
    except nx.NetworkXUnbounded:
        return True
    finally:
        G.remove_node(newnode)
    return False


def bidirectional_dijkstra(G, source, target, weight='weight'):
    """Dijkstra's algorithm for shortest paths using bidirectional search.

    Parameters
    ----------
    G : NetworkX graph

    source : node
       Starting node.

    target : node
       Ending node.

    weight: string, optional (default='weight')
       Edge data key corresponding to the edge weight

    Returns
    -------
    length : number
        Shortest path length.

    Returns a tuple of two dictionaries keyed by node.
    The first dictionary stores distance from the source.
    The second stores the path from the source to that node.

    Raises
    ------
    NetworkXNoPath
        If no path exists between source and target.

    Examples
    --------
    >>> G=nx.path_graph(5)
    >>> length,path=nx.bidirectional_dijkstra(G,0,4)
    >>> print(length)
    4
    >>> print(path)
    [0, 1, 2, 3, 4]

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

    In practice  bidirectional Dijkstra is much more than twice as fast as
    ordinary Dijkstra.

    Ordinary Dijkstra expands nodes in a sphere-like manner from the
    source. The radius of this sphere will eventually be the length
    of the shortest path. Bidirectional Dijkstra will expand nodes
    from both the source and the target, making two spheres of half
    this radius. Volume of the first sphere is pi*r*r while the
    others are 2*pi*r/2*r/2, making up half the volume.

    This algorithm is not guaranteed to work if edge weights
    are negative or are floating point numbers
    (overflows and roundoff errors can cause problems).

    See Also
    --------
    shortest_path
    shortest_path_length
    """
    if source == target:
        return (0, [source])
    push = heappush
    pop = heappop
    # Init:   Forward             Backward
    dists  = [{},                {}]  # dictionary of final distances
    paths  = [{source: [source]}, {target: [target]}]  # dictionary of paths
    fringe = [[],                []]  # heap of (distance, node) tuples for
                                      # extracting next node to expand
    seen   = [{source: 0},        {target: 0}]  # dictionary of distances to
                                                # nodes seen
    c = count()
    # initialize fringe heap
    push(fringe[0], (0, next(c), source))
    push(fringe[1], (0, next(c), target))
    # neighs for extracting correct neighbor information
    if G.is_directed():
        neighs = [G.successors_iter, G.predecessors_iter]
    else:
        neighs = [G.neighbors_iter, G.neighbors_iter]
    # variables to hold shortest discovered path
    #finaldist = 1e30000
    finalpath = []
    dir = 1
    while fringe[0] and fringe[1]:
        # choose direction
        # dir == 0 is forward direction and dir == 1 is back
        dir = 1 - dir
        # extract closest to expand
        (dist, _, v) = pop(fringe[dir])
        if v in dists[dir]:
            # Shortest path to v has already been found
            continue
        # update distance
        dists[dir][v] = dist  # equal to seen[dir][v]
        if v in dists[1 - dir]:
            # if we have scanned v in both directions we are done
            # we have now discovered the shortest path
            return (finaldist, finalpath)

        for w in neighs[dir](v):
            if(dir == 0):  # forward
                if G.is_multigraph():
                    minweight = min((dd.get(weight, 1)
                                     for k, dd in G[v][w].items()))
                else:
                    minweight = G[v][w].get(weight, 1)
                vwLength = dists[dir][v] + minweight  # G[v][w].get(weight,1)
            else:  # back, must remember to change v,w->w,v
                if G.is_multigraph():
                    minweight = min((dd.get(weight, 1)
                                     for k, dd in G[w][v].items()))
                else:
                    minweight = G[w][v].get(weight, 1)
                vwLength = dists[dir][v] + minweight  # G[w][v].get(weight,1)

            if w in dists[dir]:
                if vwLength < dists[dir][w]:
                    raise ValueError(
                        "Contradictory paths found: negative weights?")
            elif w not in seen[dir] or vwLength < seen[dir][w]:
                # relaxing
                seen[dir][w] = vwLength
                push(fringe[dir], (vwLength, next(c), w))
                paths[dir][w] = paths[dir][v] + [w]
                if w in seen[0] and w in seen[1]:
                    # see if this path is better than than the already
                    # discovered shortest path
                    totaldist = seen[0][w] + seen[1][w]
                    if finalpath == [] or finaldist > totaldist:
                        finaldist = totaldist
                        revpath = paths[1][w][:]
                        revpath.reverse()
                        finalpath = paths[0][w] + revpath[1:]
    raise nx.NetworkXNoPath("No path between %s and %s." % (source, target))


def johnson(G, weight='weight'):
    """Compute shortest paths between all nodes in a weighted graph using
    Johnson's algorithm.

    Parameters
    ----------
    G : NetworkX graph

    weight: string, optional (default='weight')
        Edge data key corresponding to the edge weight.

    Returns
    -------
    distance : dictionary
       Dictionary, keyed by source and target, of shortest paths.

    Raises
    ------
    NetworkXError
       If given graph is not weighted.

    Examples
    --------
    >>> import networkx as nx
    >>> graph = nx.DiGraph()
    >>> graph.add_weighted_edges_from([('0', '3', 3), ('0', '1', -5),
    ... ('0', '2', 2), ('1', '2', 4), ('2', '3', 1)])
    >>> paths = nx.johnson(graph, weight='weight')
    >>> paths['0']['2']
    ['0', '1', '2']

    Notes
    ------
    Johnson's algorithm is suitable even for graphs with negative weights. It
    works by using the Bellman–Ford algorithm to compute a transformation of
    the input graph that removes all negative weights, allowing Dijkstra's
    algorithm to be used on the transformed graph.

    It may be faster than Floyd - Warshall algorithm in sparse graphs.
    Algorithm complexity: O(V^2 * logV + V * E)

    See Also
    --------
    floyd_warshall_predecessor_and_distance
    floyd_warshall_numpy
    all_pairs_shortest_path
    all_pairs_shortest_path_length
    all_pairs_dijkstra_path
    bellman_ford
    """
    if not nx.is_weighted(G, weight=weight):
        raise nx.NetworkXError('Graph is not weighted.')

    dist = {v: 0 for v in G}
    pred = {v: None for v in G}

    # Calculate distance of shortest paths
    dist_bellman = _bellman_ford_relaxation(G, pred, dist, G.nodes(),
                                            weight)[1]

    if G.is_multigraph():
        get_weight = lambda u, v, data: (
            min(eattr.get(weight, 1) for eattr in data.values()) +
            dist_bellman[u] - dist_bellman[v])
    else:
        get_weight = lambda u, v, data: (data.get(weight, 1) +
                                         dist_bellman[u] - dist_bellman[v])

    all_pairs = {v: _dijkstra(G, v, get_weight, paths={v: [v]})[1] for v in G}
    return all_pairs