# Copyright (c) 2007-2009 Pedro Matiello # Peter Sagerson # Johannes Reinhardt # Rhys Ulerich # Roy Smith # Salim Fadhley # Tomaz Kovacic # # Permission is hereby granted, free of charge, to any person # obtaining a copy of this software and associated documentation # files (the "Software"), to deal in the Software without # restriction, including without limitation the rights to use, # copy, modify, merge, publish, distribute, sublicense, and/or sell # copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following # conditions: # The above copyright notice and this permission notice shall be # included in all copies or substantial portions of the Software. # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, # EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES # OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND # NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT # HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, # WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING # FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE. """ Minimization and maximization algorithms. @sort: heuristic_search, minimal_spanning_tree, shortest_path, shortest_path_bellman_ford """ from pygraph.algorithms.utils import heappush, heappop from pygraph.classes.exceptions import NodeUnreachable from pygraph.classes.exceptions import NegativeWeightCycleError from pygraph.classes.digraph import digraph import bisect # Minimal spanning tree def minimal_spanning_tree(graph, root=None): """ Minimal spanning tree. @attention: Minimal spanning tree is meaningful only for weighted graphs. @type graph: graph @param graph: Graph. @type root: node @param root: Optional root node (will explore only root's connected component) @rtype: dictionary @return: Generated spanning tree. """ visited = [] # List for marking visited and non-visited nodes spanning_tree = {} # MInimal Spanning tree # Initialization if (root is not None): visited.append(root) nroot = root spanning_tree[root] = None else: nroot = 1 # Algorithm loop while (nroot is not None): ledge = _lightest_edge(graph, visited) if (ledge == None): if (root is not None): break nroot = _first_unvisited(graph, visited) if (nroot is not None): spanning_tree[nroot] = None visited.append(nroot) else: spanning_tree[ledge[1]] = ledge[0] visited.append(ledge[1]) return spanning_tree def _first_unvisited(graph, visited): """ Return first unvisited node. @type graph: graph @param graph: Graph. @type visited: list @param visited: List of nodes. @rtype: node @return: First unvisited node. """ for each in graph: if (each not in visited): return each return None def _lightest_edge(graph, visited): """ Return the lightest edge in graph going from a visited node to an unvisited one. @type graph: graph @param graph: Graph. @type visited: list @param visited: List of nodes. @rtype: tuple @return: Lightest edge in graph going from a visited node to an unvisited one. """ lightest_edge = None weight = None for each in visited: for other in graph[each]: if (other not in visited): w = graph.edge_weight((each, other)) if (weight is None or w < weight or weight < 0): lightest_edge = (each, other) weight = w return lightest_edge # Shortest Path def shortest_path(graph, source): """ Return the shortest path distance between source and all other nodes using Dijkstra's algorithm. @attention: All weights must be nonnegative. @see: shortest_path_bellman_ford @type graph: graph, digraph @param graph: Graph. @type source: node @param source: Node from which to start the search. @rtype: tuple @return: A tuple containing two dictionaries, each keyed by target nodes. 1. Shortest path spanning tree 2. Shortest distance from given source to each target node Inaccessible target nodes do not appear in either dictionary. """ # Initialization dist = {source: 0} previous = {source: None} # This is a sorted queue of (dist, node) 2-tuples. The first item in the # queue is always either a finalized node that we can ignore or the node # with the smallest estimated distance from the source. Note that we will # not remove nodes from this list as they are finalized; we just ignore them # when they come up. q = [(0, source)] # The set of nodes for which we have final distances. finished = set() # Algorithm loop while len(q) > 0: du, u = q.pop(0) # Process reachable, remaining nodes from u if u not in finished: finished.add(u) for v in graph[u]: if v not in finished: alt = du + graph.edge_weight((u, v)) if (v not in dist) or (alt < dist[v]): dist[v] = alt previous[v] = u bisect.insort(q, (alt, v)) return previous, dist def shortest_path_bellman_ford(graph, source): """ Return the shortest path distance between the source node and all other nodes in the graph using Bellman-Ford's algorithm. This algorithm is useful when you have a weighted (and obviously a directed) graph with negative weights. @attention: The algorithm can detect negative weight cycles and will raise an exception. It's meaningful only for directed weighted graphs. @see: shortest_path @type graph: digraph @param graph: Digraph @type source: node @param source: Source node of the graph @raise NegativeWeightCycleError: raises if it finds a negative weight cycle. If this condition is met d(v) > d(u) + W(u, v) then raise the error. @rtype: tuple @return: A tuple containing two dictionaries, each keyed by target nodes. (same as shortest_path function that implements Dijkstra's algorithm) 1. Shortest path spanning tree 2. Shortest distance from given source to each target node """ # initialize the required data structures distance = {source : 0} predecessor = {source : None} # iterate and relax edges for i in range(1,graph.order()-1): for src,dst in graph.edges(): if (src in distance) and (dst not in distance): distance[dst] = distance[src] + graph.edge_weight((src,dst)) predecessor[dst] = src elif (src in distance) and (dst in distance) and \ distance[src] + graph.edge_weight((src,dst)) < distance[dst]: distance[dst] = distance[src] + graph.edge_weight((src,dst)) predecessor[dst] = src # detect negative weight cycles for src,dst in graph.edges(): if src in distance and \ dst in distance and \ distance[src] + graph.edge_weight((src,dst)) < distance[dst]: raise NegativeWeightCycleError("Detected a negative weight cycle on edge (%s, %s)" % (src,dst)) return predecessor, distance #Heuristics search def heuristic_search(graph, start, goal, heuristic): """ A* search algorithm. A set of heuristics is available under C{graph.algorithms.heuristics}. User-created heuristics are allowed too. @type graph: graph, digraph @param graph: Graph @type start: node @param start: Start node @type goal: node @param goal: Goal node @type heuristic: function @param heuristic: Heuristic function @rtype: list @return: Optimized path from start to goal node """ # The queue stores priority, node, cost to reach, and parent. queue = [ (0, start, 0, None) ] # This dictionary maps queued nodes to distance of discovered paths # and the computed heuristics to goal. We avoid to compute the heuristics # more than once and to insert too many times the node in the queue. g = {} # This maps explored nodes to parent closest to the start explored = {} while queue: _, current, dist, parent = heappop(queue) if current == goal: path = [current] + [ n for n in _reconstruct_path( parent, explored ) ] path.reverse() return path if current in explored: continue explored[current] = parent for neighbor in graph[current]: if neighbor in explored: continue ncost = dist + graph.edge_weight((current, neighbor)) if neighbor in g: qcost, h = g[neighbor] if qcost <= ncost: continue # if ncost < qcost, a longer path to neighbor remains # g. Removing it would need to filter the whole # queue, it's better just to leave it there and ignore # it when we visit the node a second time. else: h = heuristic(neighbor, goal) g[neighbor] = ncost, h heappush(queue, (ncost + h, neighbor, ncost, current)) raise NodeUnreachable( start, goal ) def _reconstruct_path(node, parents): while node is not None: yield node node = parents[node] #maximum flow/minimum cut def maximum_flow(graph, source, sink, caps = None): """ Find a maximum flow and minimum cut of a directed graph by the Edmonds-Karp algorithm. @type graph: digraph @param graph: Graph @type source: node @param source: Source of the flow @type sink: node @param sink: Sink of the flow @type caps: dictionary @param caps: Dictionary specifying a maximum capacity for each edge. If not given, the weight of the edge will be used as its capacity. Otherwise, for each edge (a,b), caps[(a,b)] should be given. @rtype: tuple @return: A tuple containing two dictionaries 1. contains the flow through each edge for a maximal flow through the graph 2. contains to which component of a minimum cut each node belongs """ #handle optional argument, if weights are available, use them, if not, assume one if caps == None: caps = {} for edge in graph.edges(): caps[edge] = graph.edge_weight((edge[0],edge[1])) #data structures to maintain f = {}.fromkeys(graph.edges(),0) label = {}.fromkeys(graph.nodes(),[]) label[source] = ['-',float('Inf')] u = {}.fromkeys(graph.nodes(),False) d = {}.fromkeys(graph.nodes(),float('Inf')) #queue for labelling q = [source] finished = False while not finished: #choose first labelled vertex with u == false for i in range(len(q)): if not u[q[i]]: v = q.pop(i) break #find augmenting path for w in graph.neighbors(v): if label[w] == [] and f[(v,w)] < caps[(v,w)]: d[w] = min(caps[(v,w)] - f[(v,w)],d[v]) label[w] = [v,'+',d[w]] q.append(w) for w in graph.incidents(v): if label[w] == [] and f[(w,v)] > 0: d[w] = min(f[(w,v)],d[v]) label[w] = [v,'-',d[w]] q.append(w) u[v] = True #extend flow by augmenting path if label[sink] != []: delta = label[sink][-1] w = sink while w != source: v = label[w][0] if label[w][1] == '-': f[(w,v)] = f[(w,v)] - delta else: f[(v,w)] = f[(v,w)] + delta w = v #reset labels label = {}.fromkeys(graph.nodes(),[]) label[source] = ['-',float('Inf')] q = [source] u = {}.fromkeys(graph.nodes(),False) d = {}.fromkeys(graph.nodes(),float('Inf')) #check whether finished finished = True for node in graph.nodes(): if label[node] != [] and u[node] == False: finished = False #find the two components of the cut cut = {} for node in graph.nodes(): if label[node] == []: cut[node] = 1 else: cut[node] = 0 return (f,cut) def cut_value(graph, flow, cut): """ Calculate the value of a cut. @type graph: digraph @param graph: Graph @type flow: dictionary @param flow: Dictionary containing a flow for each edge. @type cut: dictionary @param cut: Dictionary mapping each node to a subset index. The function only considers the flow between nodes with 0 and 1. @rtype: float @return: The value of the flow between the subsets 0 and 1 """ #max flow/min cut value calculation S = [] T = [] for node in cut.keys(): if cut[node] == 0: S.append(node) elif cut[node] == 1: T.append(node) value = 0 for node in S: for neigh in graph.neighbors(node): if neigh in T: value = value + flow[(node,neigh)] for inc in graph.incidents(node): if inc in T: value = value - flow[(inc,node)] return value def cut_tree(igraph, caps = None): """ Construct a Gomory-Hu cut tree by applying the algorithm of Gusfield. @type igraph: graph @param igraph: Graph @type caps: dictionary @param caps: Dictionary specifying a maximum capacity for each edge. If not given, the weight of the edge will be used as its capacity. Otherwise, for each edge (a,b), caps[(a,b)] should be given. @rtype: dictionary @return: Gomory-Hu cut tree as a dictionary, where each edge is associated with its weight """ #maximum flow needs a digraph, we get a graph #I think this conversion relies on implementation details outside the api and may break in the future graph = digraph() graph.add_graph(igraph) #handle optional argument if not caps: caps = {} for edge in graph.edges(): caps[edge] = igraph.edge_weight(edge) #temporary flow variable f = {} #we use a numbering of the nodes for easier handling n = {} N = 0 for node in graph.nodes(): n[N] = node N = N + 1 #predecessor function p = {}.fromkeys(range(N),0) p[0] = None for s in range(1,N): t = p[s] S = [] #max flow calculation (flow,cut) = maximum_flow(graph,n[s],n[t],caps) for i in range(N): if cut[n[i]] == 0: S.append(i) value = cut_value(graph,flow,cut) f[s] = value for i in range(N): if i == s: continue if i in S and p[i] == t: p[i] = s if p[t] in S: p[s] = p[t] p[t] = s f[s] = f[t] f[t] = value #cut tree is a dictionary, where each edge is associated with its weight b = {} for i in range(1,N): b[(n[i],n[p[i]])] = f[i] return b