def all_pairs_node_connectivity(G, nbunch=None, flow_func=None): """Compute node connectivity between all pairs of nodes of G. Parameters ---------- G : NetworkX graph Undirected graph nbunch: container Container of nodes. If provided node connectivity will be computed only over pairs of nodes in nbunch. flow_func : function A function for computing the maximum flow among a pair of nodes. The function has to accept at least three parameters: a Digraph, a source node, and a target node. And return a residual network that follows NetworkX conventions (see :meth:`maximum_flow` for details). If flow_func is None, the default maximum flow function (:meth:`edmonds_karp`) is used. See below for details. The choice of the default function may change from version to version and should not be relied on. Default value: None. Returns ------- all_pairs : dict A dictionary with node connectivity between all pairs of nodes in G, or in nbunch if provided. See also -------- :meth:`local_node_connectivity` :meth:`edge_connectivity` :meth:`local_edge_connectivity` :meth:`maximum_flow` :meth:`edmonds_karp` :meth:`preflow_push` :meth:`shortest_augmenting_path` """ if nbunch is None: nbunch = G else: nbunch = set(nbunch) if G.is_directed(): iter_func = itertools.permutations else: iter_func = itertools.combinations all_pairs = dict.fromkeys(nbunch, dict()) # Reuse auxiliary digraph and residual network H = build_auxiliary_node_connectivity(G) mapping = H.graph['mapping'] R = build_residual_network(H, 'capacity') kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R) for u, v in iter_func(nbunch, 2): K = local_node_connectivity(G, u, v, **kwargs) all_pairs[u][v] = K return all_pairs
def edge_connectivity(G, s=None, t=None, flow_func=None): r"""Returns the edge connectivity of the graph or digraph G. The edge connectivity is equal to the minimum number of edges that must be removed to disconnect G or render it trivial. If source and target nodes are provided, this function returns the local edge connectivity: the minimum number of edges that must be removed to break all paths from source to target in G. Parameters ---------- G : NetworkX graph Undirected or directed graph s : node Source node. Optional. Default value: None. t : node Target node. Optional. Default value: None. flow_func : function A function for computing the maximum flow among a pair of nodes. The function has to accept at least three parameters: a Digraph, a source node, and a target node. And return a residual network that follows NetworkX conventions (see :meth:`maximum_flow` for details). If flow_func is None, the default maximum flow function (:meth:`edmonds_karp`) is used. See below for details. The choice of the default function may change from version to version and should not be relied on. Default value: None. Returns ------- K : integer Edge connectivity for G, or local edge connectivity if source and target were provided Examples -------- >>> # Platonic icosahedral graph is 5-edge-connected >>> G = nx.icosahedral_graph() >>> nx.edge_connectivity(G) 5 You can use alternative flow algorithms for the underlying maximum flow computation. In dense networks the algorithm :meth:`shortest_augmenting_path` will usually perform better than the default :meth:`edmonds_karp`, which is faster for sparse networks with highly skewed degree distributions. >>> nx.edge_connectivity(G, flow_func=nx.shortest_augmenting_path) 5 If you specify a pair of nodes (source and target) as parameters, this function returns the value of local edge connectivity. >>> nx.edge_connectivity(G, 3, 7) 5 If you need to perform several local computations among different pairs of nodes on the same graph, it is recommended that you reuse the data structures used in the maximum flow computations. See :meth:`local_edge_connectivity` for details. Notes ----- This is a flow based implementation of global edge connectivity. For undirected graphs the algorithm works by finding a 'small' dominating set of nodes of G (see algorithm 7 in [1]_ ) and computing local maximum flow (see :meth:`local_edge_connectivity`) between an arbitrary node in the dominating set and the rest of nodes in it. This is an implementation of algorithm 6 in [1]_ . For directed graphs, the algorithm does n calls to the maximum flow function. This is an implementation of algorithm 8 in [1]_ . See also -------- :meth:`local_edge_connectivity` :meth:`local_node_connectivity` :meth:`node_connectivity` :meth:`maximum_flow` :meth:`edmonds_karp` :meth:`preflow_push` :meth:`shortest_augmenting_path` References ---------- .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf """ if (s is not None and t is None) or (s is None and t is not None): raise nx.NetworkXError('Both source and target must be specified.') # Local edge connectivity if s is not None and t is not None: if s not in G: raise nx.NetworkXError('node %s not in graph' % s) if t not in G: raise nx.NetworkXError('node %s not in graph' % t) return local_edge_connectivity(G, s, t, flow_func=flow_func) # Global edge connectivity # reuse auxiliary digraph and residual network H = build_auxiliary_edge_connectivity(G) R = build_residual_network(H, 'capacity') kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R) if G.is_directed(): # Algorithm 8 in [1] if not nx.is_weakly_connected(G): return 0 # initial value for \lambda is minimum degree L = min(G.degree().values()) nodes = G.nodes() n = len(nodes) for i in range(n): kwargs['cutoff'] = L try: L = min(L, local_edge_connectivity(G, nodes[i], nodes[i+1], **kwargs)) except IndexError: # last node! L = min(L, local_edge_connectivity(G, nodes[i], nodes[0], **kwargs)) return L else: # undirected # Algorithm 6 in [1] if not nx.is_connected(G): return 0 # initial value for \lambda is minimum degree L = min(G.degree().values()) # A dominating set is \lambda-covering # We need a dominating set with at least two nodes for node in G: D = nx.dominating_set(G, start_with=node) v = D.pop() if D: break else: # in complete graphs the dominating sets will always be of one node # thus we return min degree return L for w in D: kwargs['cutoff'] = L L = min(L, local_edge_connectivity(G, v, w, **kwargs)) return L
def node_connectivity(G, s=None, t=None, flow_func=None): r"""Returns node connectivity for a graph or digraph G. Node connectivity is equal to the minimum number of nodes that must be removed to disconnect G or render it trivial. If source and target nodes are provided, this function returns the local node connectivity: the minimum number of nodes that must be removed to break all paths from source to target in G. Parameters ---------- G : NetworkX graph Undirected graph s : node Source node. Optional. Default value: None. t : node Target node. Optional. Default value: None. flow_func : function A function for computing the maximum flow among a pair of nodes. The function has to accept at least three parameters: a Digraph, a source node, and a target node. And return a residual network that follows NetworkX conventions (see :meth:`maximum_flow` for details). If flow_func is None, the default maximum flow function (:meth:`edmonds_karp`) is used. See below for details. The choice of the default function may change from version to version and should not be relied on. Default value: None. Returns ------- K : integer Node connectivity of G, or local node connectivity if source and target are provided. Examples -------- >>> # Platonic icosahedral graph is 5-node-connected >>> G = nx.icosahedral_graph() >>> nx.node_connectivity(G) 5 You can use alternative flow algorithms for the underlying maximum flow computation. In dense networks the algorithm :meth:`shortest_augmenting_path` will usually perform better than the default :meth:`edmonds_karp`, which is faster for sparse networks with highly skewed degree distributions. >>> nx.node_connectivity(G, flow_func=nx.shortest_augmenting_path) 5 If you specify a pair of nodes (source and target) as parameters, this function returns the value of local node connectivity. >>> nx.node_connectivity(G, 3, 7) 5 If you need to perform several local computations among different pairs of nodes on the same graph, it is recommended that you reuse the data structures used in the maximum flow computations. See :meth:`local_node_connectivity` for details. Notes ----- This is a flow based implementation of node connectivity. The algorithm works by solving `O((n-\delta-1+\delta(\delta-1)/2)` maximum flow problems on an auxiliary digraph. Where `\delta` is the minimum degree of G. For details about the auxiliary digraph and the computation of local node connectivity see :meth:`local_node_connectivity`. This implementation is based on algorithm 11 in [1]_. See also -------- :meth:`local_node_connectivity` :meth:`edge_connectivity` :meth:`maximum_flow` :meth:`edmonds_karp` :meth:`preflow_push` :meth:`shortest_augmenting_path` References ---------- .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf """ if (s is not None and t is None) or (s is None and t is not None): raise nx.NetworkXError('Both source and target must be specified.') # Local node connectivity if s is not None and t is not None: if s not in G: raise nx.NetworkXError('node %s not in graph' % s) if t not in G: raise nx.NetworkXError('node %s not in graph' % t) return local_node_connectivity(G, s, t, flow_func=flow_func) # Global node connectivity if G.is_directed(): if not nx.is_weakly_connected(G): return 0 iter_func = itertools.permutations # It is necessary to consider both predecessors # and successors for directed graphs def neighbors(v): return itertools.chain.from_iterable([G.predecessors_iter(v), G.successors_iter(v)]) else: if not nx.is_connected(G): return 0 iter_func = itertools.combinations neighbors = G.neighbors_iter # Reuse the auxiliary digraph and the residual network H = build_auxiliary_node_connectivity(G) R = build_residual_network(H, 'capacity') kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R) # Pick a node with minimum degree degree = G.degree() minimum_degree = min(degree.values()) v = next(n for n, d in degree.items() if d == minimum_degree) # Node connectivity is bounded by degree. K = minimum_degree # compute local node connectivity with all its non-neighbors nodes for w in set(G) - set(neighbors(v)) - set([v]): kwargs['cutoff'] = K K = min(K, local_node_connectivity(G, v, w, **kwargs)) # Also for non adjacent pairs of neighbors of v for x, y in iter_func(neighbors(v), 2): if y in G[x]: continue kwargs['cutoff'] = K K = min(K, local_node_connectivity(G, x, y, **kwargs)) return K
def average_node_connectivity(G, flow_func=None): r"""Returns the average connectivity of a graph G. The average connectivity `\bar{\kappa}` of a graph G is the average of local node connectivity over all pairs of nodes of G [1]_ . .. math:: \bar{\kappa}(G) = \frac{\sum_{u,v} \kappa_{G}(u,v)}{{n \choose 2}} Parameters ---------- G : NetworkX graph Undirected graph flow_func : function A function for computing the maximum flow among a pair of nodes. The function has to accept at least three parameters: a Digraph, a source node, and a target node. And return a residual network that follows NetworkX conventions (see :meth:`maximum_flow` for details). If flow_func is None, the default maximum flow function (:meth:`edmonds_karp`) is used. See :meth:`local_node_connectivity` for details. The choice of the default function may change from version to version and should not be relied on. Default value: None. Returns ------- K : float Average node connectivity See also -------- :meth:`local_node_connectivity` :meth:`node_connectivity` :meth:`edge_connectivity` :meth:`maximum_flow` :meth:`edmonds_karp` :meth:`preflow_push` :meth:`shortest_augmenting_path` References ---------- .. [1] Beineke, L., O. Oellermann, and R. Pippert (2002). The average connectivity of a graph. Discrete mathematics 252(1-3), 31-45. http://www.sciencedirect.com/science/article/pii/S0012365X01001807 """ if G.is_directed(): iter_func = itertools.permutations else: iter_func = itertools.combinations # Reuse the auxiliary digraph and the residual network H = build_auxiliary_node_connectivity(G) R = build_residual_network(H, 'capacity') kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R) num, den = 0, 0 for u, v in iter_func(G, 2): num += local_node_connectivity(G, u, v, **kwargs) den += 1 if den == 0: # Null Graph return 0 return num / den
def minimum_edge_cut(G, s=None, t=None, flow_func=None): r"""Returns a set of edges of minimum cardinality that disconnects G. If source and target nodes are provided, this function returns the set of edges of minimum cardinality that, if removed, would break all paths among source and target in G. If not, it returns a set of edges of minimum cardinality that disconnects G. Parameters ---------- G : NetworkX graph s : node Source node. Optional. Default value: None. t : node Target node. Optional. Default value: None. flow_func : function A function for computing the maximum flow among a pair of nodes. The function has to accept at least three parameters: a Digraph, a source node, and a target node. And return a residual network that follows NetworkX conventions (see :meth:`maximum_flow` for details). If flow_func is None, the default maximum flow function (:meth:`edmonds_karp`) is used. See below for details. The choice of the default function may change from version to version and should not be relied on. Default value: None. Returns ------- cutset : set Set of edges that, if removed, would disconnect G. If source and target nodes are provided, the set contians the edges that if removed, would destroy all paths between source and target. Examples -------- >>> # Platonic icosahedral graph has edge connectivity 5 >>> G = nx.icosahedral_graph() >>> len(nx.minimum_edge_cut(G)) 5 You can use alternative flow algorithms for the underlying maximum flow computation. In dense networks the algorithm :meth:`shortest_augmenting_path` will usually perform better than the default :meth:`edmonds_karp`, which is faster for sparse networks with highly skewed degree distributions. >>> len(nx.minimum_edge_cut(G, flow_func=nx.shortest_augmenting_path)) 5 If you specify a pair of nodes (source and target) as parameters, this function returns the value of local edge connectivity. >>> nx.edge_connectivity(G, 3, 7) 5 If you need to perform several local computations among different pairs of nodes on the same graph, it is recommended that you reuse the data structures used in the maximum flow computations. See :meth:`local_edge_connectivity` for details. Notes ----- This is a flow based implementation of minimum edge cut. For undirected graphs the algorithm works by finding a 'small' dominating set of nodes of G (see algorithm 7 in [1]_) and computing the maximum flow between an arbitrary node in the dominating set and the rest of nodes in it. This is an implementation of algorithm 6 in [1]_. For directed graphs, the algorithm does n calls to the max flow function. It is an implementation of algorithm 8 in [1]_. See also -------- :meth:`minimum_st_edge_cut` :meth:`minimum_node_cut` :meth:`stoer_wagner` :meth:`node_connectivity` :meth:`edge_connectivity` :meth:`maximum_flow` :meth:`edmonds_karp` :meth:`preflow_push` :meth:`shortest_augmenting_path` References ---------- .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf """ if (s is not None and t is None) or (s is None and t is not None): raise nx.NetworkXError('Both source and target must be specified.') # reuse auxiliary digraph and residual network H = build_auxiliary_edge_connectivity(G) R = build_residual_network(H, 'capacity') kwargs = dict(flow_func=flow_func, residual=R, auxiliary=H) # Local minimum edge cut if s and t are not None if s is not None and t is not None: if s not in G: raise nx.NetworkXError('node %s not in graph' % s) if t not in G: raise nx.NetworkXError('node %s not in graph' % t) return minimum_st_edge_cut(H, s, t, **kwargs) # Global minimum edge cut # Analog to the algoritm for global edge connectivity if G.is_directed(): # Based on algorithm 8 in [1] if not nx.is_weakly_connected(G): raise nx.NetworkXError('Input graph is not connected') # Initial cutset is all edges of a node with minimum degree node = min(G, key=G.degree) min_cut = G.edges(node) nodes = G.nodes() n = len(nodes) for i in range(n): try: this_cut = minimum_st_edge_cut(H, nodes[i], nodes[i+1], **kwargs) if len(this_cut) <= len(min_cut): min_cut = this_cut except IndexError: # Last node! this_cut = minimum_st_edge_cut(H, nodes[i], nodes[0], **kwargs) if len(this_cut) <= len(min_cut): min_cut = this_cut return min_cut else: # undirected # Based on algorithm 6 in [1] if not nx.is_connected(G): raise nx.NetworkXError('Input graph is not connected') # Initial cutset is all edges of a node with minimum degree node = min(G, key=G.degree) min_cut = G.edges(node) # A dominating set is \lambda-covering # We need a dominating set with at least two nodes for node in G: D = nx.dominating_set(G, start_with=node) v = D.pop() if D: break else: # in complete graphs the dominating set will always be of one node # thus we return min_cut, which now contains the edges of a node # with minimum degree return min_cut for w in D: this_cut = minimum_st_edge_cut(H, v, w, **kwargs) if len(this_cut) <= len(min_cut): min_cut = this_cut return min_cut
def dinitz_impl(G, s, t, capacity, residual, cutoff): if s not in G: raise nx.NetworkXError('node %s not in graph' % str(s)) if t not in G: raise nx.NetworkXError('node %s not in graph' % str(t)) if s == t: raise nx.NetworkXError('source and sink are the same node') if residual is None: R = build_residual_network(G, capacity) else: R = residual # Initialize/reset the residual network. for u in R: for e in R[u].values(): e['flow'] = 0 # Use an arbitrary high value as infinite. It is computed # when building the residual network. Useful when checking # for infinite capacity paths. INF = R.graph['inf'] if cutoff is None: cutoff = INF R_succ = R.succ def breath_first_search(G, R, s, t): rank = {} rank[s] = 0 queue = deque([s]) while queue: if t in rank: break u = queue.popleft() for v in R_succ[u]: attr = R_succ[u][v] if v not in rank and attr['capacity'] - attr['flow'] > 0: rank[v] = rank[u] + 1 queue.append(v) return rank def depth_first_search(G, R, u, t, flow, rank): if u == t: return flow for v in (n for n in R_succ[u] if n in rank and rank[n] == rank[u] + 1): attr = R_succ[u][v] if attr['capacity'] > attr['flow']: min_flow = min(flow, attr['capacity'] - attr['flow']) this_flow = depth_first_search(G, R, v, t, min_flow, rank) if this_flow > 0: R_succ[u][v]['flow'] += this_flow R_succ[v][u]['flow'] -= this_flow return this_flow return 0 flow_value = 0 while flow_value < cutoff: rank = breath_first_search(G, R, s, t) if t not in rank: break while flow_value < cutoff: blocking_flow = depth_first_search(G, R, s, t, INF, rank) if blocking_flow * 2 > INF: raise nx.NetworkXUnbounded( 'Infinite capacity path, flow unbounded above.') elif blocking_flow == 0: break flow_value += blocking_flow R.graph['flow_value'] = flow_value return R
def minimum_node_cut(G, s=None, t=None, flow_func=None): r"""Returns a set of nodes of minimum cardinality that disconnects G. If source and target nodes are provided, this function returns the set of nodes of minimum cardinality that, if removed, would destroy all paths among source and target in G. If not, it returns a set of nodes of minimum cardinality that disconnects G. Parameters ---------- G : NetworkX graph s : node Source node. Optional. Default value: None. t : node Target node. Optional. Default value: None. flow_func : function A function for computing the maximum flow among a pair of nodes. The function has to accept at least three parameters: a Digraph, a source node, and a target node. And return a residual network that follows NetworkX conventions (see :meth:`maximum_flow` for details). If flow_func is None, the default maximum flow function (:meth:`edmonds_karp`) is used. See below for details. The choice of the default function may change from version to version and should not be relied on. Default value: None. Returns ------- cutset : set Set of nodes that, if removed, would disconnect G. If source and target nodes are provided, the set contians the nodes that if removed, would destroy all paths between source and target. Examples -------- >>> # Platonic icosahedral graph has node connectivity 5 >>> G = nx.icosahedral_graph() >>> node_cut = nx.minimum_node_cut(G) >>> len(node_cut) 5 You can use alternative flow algorithms for the underlying maximum flow computation. In dense networks the algorithm :meth:`shortest_augmenting_path` will usually perform better than the default :meth:`edmonds_karp`, which is faster for sparse networks with highly skewed degree distributions. >>> node_cut == nx.minimum_node_cut(G, flow_func=nx.shortest_augmenting_path) True If you specify a pair of nodes (source and target) as parameters, this function returns a local st node cut. >>> len(nx.minimum_node_cut(G, 3, 7)) 5 If you need to perform several local st cuts among different pairs of nodes on the same graph, it is recommended that you reuse the data structures used in the maximum flow computations. See :meth:`minimum_st_node_cut` for details. Notes ----- This is a flow based implementation of minimum node cut. The algorithm is based in solving a number of maximum flow computations to determine the capacity of the minimum cut on an auxiliary directed network that corresponds to the minimum node cut of G. It handles both directed and undirected graphs. This implementation is based on algorithm 11 in [1]_. See also -------- :meth:`minimum_st_node_cut` :meth:`minimum_cut` :meth:`minimum_edge_cut` :meth:`stoer_wagner` :meth:`node_connectivity` :meth:`edge_connectivity` :meth:`maximum_flow` :meth:`edmonds_karp` :meth:`preflow_push` :meth:`shortest_augmenting_path` References ---------- .. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms. http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf """ if (s is not None and t is None) or (s is None and t is not None): raise nx.NetworkXError('Both source and target must be specified.') # Local minimum node cut. if s is not None and t is not None: if s not in G: raise nx.NetworkXError('node %s not in graph' % s) if t not in G: raise nx.NetworkXError('node %s not in graph' % t) return minimum_st_node_cut(G, s, t, flow_func=flow_func) # Global minimum node cut. # Analog to the algoritm 11 for global node connectivity in [1]. if G.is_directed(): if not nx.is_weakly_connected(G): raise nx.NetworkXError('Input graph is not connected') iter_func = itertools.permutations def neighbors(v): return itertools.chain.from_iterable([G.predecessors_iter(v), G.successors_iter(v)]) else: if not nx.is_connected(G): raise nx.NetworkXError('Input graph is not connected') iter_func = itertools.combinations neighbors = G.neighbors_iter # Reuse the auxiliary digraph and the residual network. H = build_auxiliary_node_connectivity(G) R = build_residual_network(H, 'capacity') kwargs = dict(flow_func=flow_func, auxiliary=H, residual=R) # Choose a node with minimum degree. v = min(G, key=G.degree) # Initial node cutset is all neighbors of the node with minimum degree. min_cut = set(G[v]) # Compute st node cuts between v and all its non-neighbors nodes in G. for w in set(G) - set(neighbors(v)) - set([v]): this_cut = minimum_st_node_cut(G, v, w, **kwargs) if len(min_cut) >= len(this_cut): min_cut = this_cut # Also for non adjacent pairs of neighbors of v. for x, y in iter_func(neighbors(v), 2): if y in G[x]: continue this_cut = minimum_st_node_cut(G, x, y, **kwargs) if len(min_cut) >= len(this_cut): min_cut = this_cut return min_cut
def boykov_kolmogorov_impl(G, s, t, capacity, residual, cutoff): if s not in G: raise nx.NetworkXError('node %s not in graph' % str(s)) if t not in G: raise nx.NetworkXError('node %s not in graph' % str(t)) if s == t: raise nx.NetworkXError('source and sink are the same node') if residual is None: R = build_residual_network(G, capacity) else: R = residual # Initialize/reset the residual network. # This is way too slow #nx.set_edge_attributes(R, 0, 'flow') for u in R: for e in R[u].values(): e['flow'] = 0 # Use an arbitrary high value as infinite. It is computed # when building the residual network. INF = R.graph['inf'] if cutoff is None: cutoff = INF R_succ = R.succ R_pred = R.pred def grow(): """Bidirectional breadth-first search for the growth stage. Returns a connecting edge, that is and edge that connects a node from the source search tree with a node from the target search tree. The first node in the connecting edge is always from the source tree and the last node from the target tree. """ while active: u = active[0] if u in source_tree: this_tree = source_tree other_tree = target_tree neighbors = R_succ else: this_tree = target_tree other_tree = source_tree neighbors = R_pred for v, attr in neighbors[u].items(): if attr['capacity'] - attr['flow'] > 0: if v not in this_tree: if v in other_tree: return (u, v) if this_tree is source_tree else (v, u) this_tree[v] = u dist[v] = dist[u] + 1 timestamp[v] = timestamp[u] active.append(v) elif v in this_tree and _is_closer(u, v): this_tree[v] = u dist[v] = dist[u] + 1 timestamp[v] = timestamp[u] _ = active.popleft() return None, None def augment(u, v): """Augmentation stage. Reconstruct path and determine its residual capacity. We start from a connecting edge, which links a node from the source tree to a node from the target tree. The connecting edge is the output of the grow function and the input of this function. """ attr = R_succ[u][v] flow = min(INF, attr['capacity'] - attr['flow']) path = [u] # Trace a path from u to s in source_tree. w = u while w != s: n = w w = source_tree[n] attr = R_pred[n][w] flow = min(flow, attr['capacity'] - attr['flow']) path.append(w) path.reverse() # Trace a path from v to t in target_tree. path.append(v) w = v while w != t: n = w w = target_tree[n] attr = R_succ[n][w] flow = min(flow, attr['capacity'] - attr['flow']) path.append(w) # Augment flow along the path and check for saturated edges. it = iter(path) u = next(it) these_orphans = [] for v in it: R_succ[u][v]['flow'] += flow R_succ[v][u]['flow'] -= flow if R_succ[u][v]['flow'] == R_succ[u][v]['capacity']: if v in source_tree: source_tree[v] = None these_orphans.append(v) if u in target_tree: target_tree[u] = None these_orphans.append(u) u = v orphans.extend(sorted(these_orphans, key=dist.get)) return flow def adopt(): """Adoption stage. Reconstruct search trees by adopting or discarding orphans. During augmentation stage some edges got saturated and thus the source and target search trees broke down to forests, with orphans as roots of some of its trees. We have to reconstruct the search trees rooted to source and target before we can grow them again. """ while orphans: u = orphans.popleft() if u in source_tree: tree = source_tree neighbors = R_pred else: tree = target_tree neighbors = R_succ nbrs = ((n, attr, dist[n]) for n, attr in neighbors[u].items() if n in tree) for v, attr, d in sorted(nbrs, key=itemgetter(2)): if attr['capacity'] - attr['flow'] > 0: if _has_valid_root(v, tree): tree[u] = v dist[u] = dist[v] + 1 timestamp[u] = time break else: nbrs = ((n, attr, dist[n]) for n, attr in neighbors[u].items() if n in tree) for v, attr, d in sorted(nbrs, key=itemgetter(2)): if attr['capacity'] - attr['flow'] > 0: if v not in active: active.append(v) if tree[v] == u: tree[v] = None orphans.appendleft(v) if u in active: active.remove(u) del tree[u] def _has_valid_root(n, tree): path = [] v = n while v is not None: path.append(v) if v == s or v == t: base_dist = 0 break elif timestamp[v] == time: base_dist = dist[v] break v = tree[v] else: return False length = len(path) for i, u in enumerate(path, 1): dist[u] = base_dist + length - i timestamp[u] = time return True def _is_closer(u, v): return timestamp[v] <= timestamp[u] and dist[v] > dist[u] + 1 source_tree = {s: None} target_tree = {t: None} active = deque([s, t]) orphans = deque() flow_value = 0 # data structures for the marking heuristic time = 1 timestamp = {s: time, t: time} dist = {s: 0, t: 0} while flow_value < cutoff: # Growth stage u, v = grow() if u is None: break time += 1 # Augmentation stage flow_value += augment(u, v) # Adoption stage adopt() if flow_value * 2 > INF: raise nx.NetworkXUnbounded('Infinite capacity path, flow unbounded above.') # Add source and target tree in a graph attribute. # A partition that defines a minimum cut can be directly # computed from the search trees as explained in the docstrings. R.graph['trees'] = (source_tree, target_tree) # Add the standard flow_value graph attribute. R.graph['flow_value'] = flow_value return R
def dinitz_impl(G, s, t, capacity, residual, cutoff): if s not in G: raise nx.NetworkXError('node %s not in graph' % str(s)) if t not in G: raise nx.NetworkXError('node %s not in graph' % str(t)) if s == t: raise nx.NetworkXError('source and sink are the same node') if residual is None: R = build_residual_network(G, capacity) else: R = residual # Initialize/reset the residual network. for u in R: for e in R[u].values(): e['flow'] = 0 # Use an arbitrary high value as infinite. It is computed # when building the residual network. INF = R.graph['inf'] if cutoff is None: cutoff = INF R_succ = R.succ R_pred = R.pred def breath_first_search(): parents = {} queue = deque([s]) while queue: if t in parents: break u = queue.popleft() for v in R_succ[u]: attr = R_succ[u][v] if v not in parents and attr['capacity'] - attr['flow'] > 0: parents[v] = u queue.append(v) return parents def depth_first_search(parents): """Build a path using DFS starting from the sink""" path = [] u = t flow = INF while u != s: path.append(u) v = parents[u] flow = min(flow, R_pred[u][v]['capacity'] - R_pred[u][v]['flow']) u = v path.append(s) # Augment the flow along the path found if flow > 0: for u, v in pairwise(path): R_pred[u][v]['flow'] += flow R_pred[v][u]['flow'] -= flow return flow flow_value = 0 while flow_value < cutoff: parents = breath_first_search() if t not in parents: break this_flow = depth_first_search(parents) if this_flow * 2 > INF: raise nx.NetworkXUnbounded( 'Infinite capacity path, flow unbounded above.') flow_value += this_flow R.graph['flow_value'] = flow_value return R
def boykov_kolmogorov_impl(G, s, t, capacity, residual, cutoff): if s not in G: raise nx.NetworkXError(f"node {str(s)} not in graph") if t not in G: raise nx.NetworkXError(f"node {str(t)} not in graph") if s == t: raise nx.NetworkXError("source and sink are the same node") if residual is None: R = build_residual_network(G, capacity) else: R = residual # Initialize/reset the residual network. # This is way too slow # nx.set_edge_attributes(R, 0, 'flow') for u in R: for e in R[u].values(): e["flow"] = 0 # Use an arbitrary high value as infinite. It is computed # when building the residual network. INF = R.graph["inf"] if cutoff is None: cutoff = INF R_succ = R.succ R_pred = R.pred def grow(): """Bidirectional breadth-first search for the growth stage. Returns a connecting edge, that is and edge that connects a node from the source search tree with a node from the target search tree. The first node in the connecting edge is always from the source tree and the last node from the target tree. """ while active: u = active[0] if u in source_tree: this_tree = source_tree other_tree = target_tree neighbors = R_succ else: this_tree = target_tree other_tree = source_tree neighbors = R_pred for v, attr in neighbors[u].items(): if attr["capacity"] - attr["flow"] > 0: if v not in this_tree: if v in other_tree: return (u, v) if this_tree is source_tree else (v, u) this_tree[v] = u dist[v] = dist[u] + 1 timestamp[v] = timestamp[u] active.append(v) elif v in this_tree and _is_closer(u, v): this_tree[v] = u dist[v] = dist[u] + 1 timestamp[v] = timestamp[u] _ = active.popleft() return None, None def augment(u, v): """Augmentation stage. Reconstruct path and determine its residual capacity. We start from a connecting edge, which links a node from the source tree to a node from the target tree. The connecting edge is the output of the grow function and the input of this function. """ attr = R_succ[u][v] flow = min(INF, attr["capacity"] - attr["flow"]) path = [u] # Trace a path from u to s in source_tree. w = u while w != s: n = w w = source_tree[n] attr = R_pred[n][w] flow = min(flow, attr["capacity"] - attr["flow"]) path.append(w) path.reverse() # Trace a path from v to t in target_tree. path.append(v) w = v while w != t: n = w w = target_tree[n] attr = R_succ[n][w] flow = min(flow, attr["capacity"] - attr["flow"]) path.append(w) # Augment flow along the path and check for saturated edges. it = iter(path) u = next(it) these_orphans = [] for v in it: R_succ[u][v]["flow"] += flow R_succ[v][u]["flow"] -= flow if R_succ[u][v]["flow"] == R_succ[u][v]["capacity"]: if v in source_tree: source_tree[v] = None these_orphans.append(v) if u in target_tree: target_tree[u] = None these_orphans.append(u) u = v orphans.extend(sorted(these_orphans, key=dist.get)) return flow def adopt(): """Adoption stage. Reconstruct search trees by adopting or discarding orphans. During augmentation stage some edges got saturated and thus the source and target search trees broke down to forests, with orphans as roots of some of its trees. We have to reconstruct the search trees rooted to source and target before we can grow them again. """ while orphans: u = orphans.popleft() if u in source_tree: tree = source_tree neighbors = R_pred else: tree = target_tree neighbors = R_succ nbrs = ((n, attr, dist[n]) for n, attr in neighbors[u].items() if n in tree) for v, attr, d in sorted(nbrs, key=itemgetter(2)): if attr["capacity"] - attr["flow"] > 0: if _has_valid_root(v, tree): tree[u] = v dist[u] = dist[v] + 1 timestamp[u] = time break else: nbrs = ((n, attr, dist[n]) for n, attr in neighbors[u].items() if n in tree) for v, attr, d in sorted(nbrs, key=itemgetter(2)): if attr["capacity"] - attr["flow"] > 0: if v not in active: active.append(v) if tree[v] == u: tree[v] = None orphans.appendleft(v) if u in active: active.remove(u) del tree[u] def _has_valid_root(n, tree): path = [] v = n while v is not None: path.append(v) if v == s or v == t: base_dist = 0 break elif timestamp[v] == time: base_dist = dist[v] break v = tree[v] else: return False length = len(path) for i, u in enumerate(path, 1): dist[u] = base_dist + length - i timestamp[u] = time return True def _is_closer(u, v): return timestamp[v] <= timestamp[u] and dist[v] > dist[u] + 1 source_tree = {s: None} target_tree = {t: None} active = deque([s, t]) orphans = deque() flow_value = 0 # data structures for the marking heuristic time = 1 timestamp = {s: time, t: time} dist = {s: 0, t: 0} while flow_value < cutoff: # Growth stage u, v = grow() if u is None: break time += 1 # Augmentation stage flow_value += augment(u, v) # Adoption stage adopt() if flow_value * 2 > INF: raise nx.NetworkXUnbounded( "Infinite capacity path, flow unbounded above.") # Add source and target tree in a graph attribute. # A partition that defines a minimum cut can be directly # computed from the search trees as explained in the docstrings. R.graph["trees"] = (source_tree, target_tree) # Add the standard flow_value graph attribute. R.graph["flow_value"] = flow_value return R