# Dijkstra's algorithm for shortest paths # David Eppstein, UC Irvine, 4 April 2002 # http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/117228 from priodict import priorityDictionary def Dijkstra(G,start,end=None): """ Find shortest paths from the start vertex to all vertices nearer than or equal to the end. The input graph G is assumed to have the following representation: A vertex can be any object that can be used as an index into a dictionary. G is a dictionary, indexed by vertices. For any vertex v, G[v] is itself a dictionary, indexed by the neighbors of v. For any edge v->w, G[v][w] is the length of the edge. This is related to the representation in <http://www.python.org/doc/essays/graphs.html> where Guido van Rossum suggests representing graphs as dictionaries mapping vertices to lists of neighbors, however dictionaries of edges have many advantages over lists: they can store extra information (here, the lengths), they support fast existence tests, and they allow easy modification of the graph by edge insertion and removal. Such modifications are not needed here but are important in other graph algorithms. Since dictionaries obey iterator protocol, a graph represented as described here could be handed without modification to an algorithm using Guido's representation. Of course, G and G[v] need not be Python dict objects; they can be any other object that obeys dict protocol, for instance a wrapper in which vertices are URLs and a call to G[v] loads the web page and finds its links. The output is a pair (D,P) where D[v] is the distance from start to v and P[v] is the predecessor of v along the shortest path from s to v. Dijkstra's algorithm is only guaranteed to work correctly when all edge lengths are positive. This code does not verify this property for all edges (only the edges seen before the end vertex is reached), but will correctly compute shortest paths even for some graphs with negative edges, and will raise an exception if it discovers that a negative edge has caused it to make a mistake. """ D = {} # dictionary of final distances P = {} # dictionary of predecessors Q = priorityDictionary() # est.dist. of non-final vert. Q[start] = 0 for v in Q: D[v] = Q[v] if v == end: break for w in G[v]: vwLength = D[v] + G[v][w] if w in D: if vwLength < D[w]: raise ValueError, \ "Dijkstra: found better path to already-final vertex" elif w not in Q or vwLength < Q[w]: Q[w] = vwLength P[w] = v return (D,P) def shortestPath(G,start,end): """ Find a single shortest path from the given start vertex to the given end vertex. The input has the same conventions as Dijkstra(). The output is a list of the vertices in order along the shortest path. """ D,P = Dijkstra(G,start,end) Path = [] while 1: Path.append(end) if end == start: break end = P[end] Path.reverse() return Path