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 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 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) if G.is_directed(): G_succ = G.succ else: G_succ = 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 _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 : 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. 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 """ 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 + weight(u, v, 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 bellman_ford( G: Type[nx.DiGraph], source: str, weight_index: str = 'weight' ) -> Tuple[Dict[str, Optional[int]], Dict[str, Optional[int]]]: """ Computes shortest paths from a single source vertex to all of the other vertices in a weighted digraph (allowing for negative weights). """ if source not in G: raise KeyError("Node %s is not found in the graph" % source) dist = {source: 0} pred = {source: None} if len(G) == 1: return pred, dist if G.is_multigraph(): def get_weight(edge_dict): return min( eattr.get(weight_index, 1) for eattr in list(edge_dict.values())) else: def get_weight(edge_dict): return edge_dict.get(weight_index, 1) if G.is_directed(): G_succ = G.succ else: G_succ = 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 list(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: q.remove(v) continue count[v] = count_v dist[v] = dist_v pred[v] = u return pred, dist