# -*- coding: utf-8 -*- """ Shortest path algorithms for weighed graphs. """ __author__ = """\n""".join(['Aric Hagberg ', 'Loïc Séguin-C. ', 'Dan Schult ']) # Copyright (C) 2004-2011 by # Aric Hagberg # Dan Schult # Pieter Swart # 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'] import heapq 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, 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() """ dist = {} # dictionary of final distances seen = {source:0} fringe=[] # use heapq with (distance,label) tuples heapq.heappush(fringe,(0,source)) while fringe: (d,v)=heapq.heappop(fringe) if v in dist: continue # already searched this node. dist[v] = d #for ignore,w,edgedata in G.edges_iter(v,data=True): #is about 30% slower than the following if G.is_multigraph(): edata=[] for w,keydata in G[v].items(): minweight=min((dd.get(weight,1) for k,dd in keydata.items())) edata.append((w,{weight:minweight})) else: edata=iter(G[v].items()) for w,edgedata in edata: vw_dist = dist[v] + edgedata.get(weight,1) if cutoff is not None: if vw_dist>cutoff: continue if w in dist: if vw_dist < dist[w]: raise ValueError('Contradictory paths found:', 'negative weights?') elif w not in seen or vw_dist < seen[w]: seen[w] = vw_dist heapq.heappush(fringe,(vw_dist,w)) return dist 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]}) dist = {} # dictionary of final distances paths = {source:[source]} # dictionary of paths seen = {source:0} fringe=[] # use heapq with (distance,label) tuples heapq.heappush(fringe,(0,source)) while fringe: (d,v)=heapq.heappop(fringe) if v in dist: continue # already searched this node. dist[v] = d if v == target: break #for ignore,w,edgedata in G.edges_iter(v,data=True): #is about 30% slower than the following if G.is_multigraph(): edata=[] for w,keydata in G[v].items(): minweight=min((dd.get(weight,1) for k,dd in keydata.items())) edata.append((w,{weight:minweight})) else: edata=iter(G[v].items()) for w,edgedata in edata: vw_dist = dist[v] + edgedata.get(weight,1) if cutoff is not None: if vw_dist>cutoff: continue if w in dist: if vw_dist < dist[w]: raise ValueError('Contradictory paths found:', 'negative weights?') elif w not in seen or vw_dist < seen[w]: seen[w] = vw_dist heapq.heappush(fringe,(vw_dist,w)) paths[w] = paths[v]+[w] return (dist,paths) 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. """ push=heapq.heappush pop=heapq.heappop dist = {} # dictionary of final distances pred = {source:[]} # dictionary of predecessors seen = {source:0} fringe=[] # use heapq with (distance,label) tuples push(fringe,(0,source)) while fringe: (d,v)=pop(fringe) if v in dist: continue # already searched this node. dist[v] = d if G.is_multigraph(): edata=[] for w,keydata in G[v].items(): minweight=min((dd.get(weight,1) for k,dd in keydata.items())) edata.append((w,{weight:minweight})) else: edata=iter(G[v].items()) for w,edgedata in edata: vw_dist = dist[v] + edgedata.get(weight,1) if cutoff is not None: if vw_dist>cutoff: continue if w in dist: if vw_dist < dist[w]: raise ValueError('Contradictory paths found:', 'negative weights?') elif w not in seen or vw_dist < seen[w]: seen[w] = vw_dist push(fringe,(vw_dist,w)) pred[w] = [v] elif vw_dist==seen[w]: pred[w].append(v) return (pred,dist) 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. """ paths={} for n in G: paths[n]=single_source_dijkstra_path_length(G,n, cutoff=cutoff, weight=weight) return paths 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() """ paths={} for n in G: paths[n]=single_source_dijkstra_path(G, n, cutoff=cutoff, weight=weight) return paths 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) >>> pred {0: None, 1: 0, 2: 1, 3: 2, 4: 3} >>> dist {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) numb_nodes = len(G) dist = {source: 0} pred = {source: None} if numb_nodes == 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) for i in range(numb_nodes): no_changes=True # Only need edges from nodes in dist b/c all others have dist==inf for u, dist_u in list(dist.items()): # get all edges from nodes in dist for v, edict in G[u].items(): # double loop handles undirected too dist_v = dist_u + get_weight(edict) if v not in dist or dist[v] > dist_v: dist[v] = dist_v pred[v] = u no_changes = False if no_changes: break else: raise nx.NetworkXUnbounded("Negative cost cycle detected.") return pred, dist 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: G.remove_node(newnode) return True 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]) #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 #initialize fringe heap heapq.heappush(fringe[0], (0, source)) heapq.heappush(fringe[1], (0, 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 )= heapq.heappop(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 heapq.heappush(fringe[dir], (vwLength,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))