Source code for networkx.algorithms.flow.dinitz_alg

"""
Dinitz' algorithm for maximum flow problems.
"""
from collections import deque

import networkx as nx
from networkx.algorithms.flow.utils import build_residual_network
from networkx.utils import pairwise

__all__ = ['dinitz']


[docs]def dinitz(G, s, t, capacity='capacity', residual=None, value_only=False, cutoff=None): """Find a maximum single-commodity flow using Dinitz' algorithm. This function returns the residual network resulting after computing the maximum flow. See below for details about the conventions NetworkX uses for defining residual networks. This algorithm has a running time of $O(n^2 m)$ for $n$ nodes and $m$ edges [1]_. Parameters ---------- G : NetworkX graph Edges of the graph are expected to have an attribute called 'capacity'. If this attribute is not present, the edge is considered to have infinite capacity. s : node Source node for the flow. t : node Sink node for the flow. capacity : string Edges of the graph G are expected to have an attribute capacity that indicates how much flow the edge can support. If this attribute is not present, the edge is considered to have infinite capacity. Default value: 'capacity'. residual : NetworkX graph Residual network on which the algorithm is to be executed. If None, a new residual network is created. Default value: None. value_only : bool If True compute only the value of the maximum flow. This parameter will be ignored by this algorithm because it is not applicable. cutoff : integer, float If specified, the algorithm will terminate when the flow value reaches or exceeds the cutoff. In this case, it may be unable to immediately determine a minimum cut. Default value: None. Returns ------- R : NetworkX DiGraph Residual network after computing the maximum flow. Raises ------ NetworkXError The algorithm does not support MultiGraph and MultiDiGraph. If the input graph is an instance of one of these two classes, a NetworkXError is raised. NetworkXUnbounded If the graph has a path of infinite capacity, the value of a feasible flow on the graph is unbounded above and the function raises a NetworkXUnbounded. See also -------- :meth:`maximum_flow` :meth:`minimum_cut` :meth:`preflow_push` :meth:`shortest_augmenting_path` Notes ----- The residual network :samp:`R` from an input graph :samp:`G` has the same nodes as :samp:`G`. :samp:`R` is a DiGraph that contains a pair of edges :samp:`(u, v)` and :samp:`(v, u)` iff :samp:`(u, v)` is not a self-loop, and at least one of :samp:`(u, v)` and :samp:`(v, u)` exists in :samp:`G`. For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['capacity']` is equal to the capacity of :samp:`(u, v)` in :samp:`G` if it exists in :samp:`G` or zero otherwise. If the capacity is infinite, :samp:`R[u][v]['capacity']` will have a high arbitrary finite value that does not affect the solution of the problem. This value is stored in :samp:`R.graph['inf']`. For each edge :samp:`(u, v)` in :samp:`R`, :samp:`R[u][v]['flow']` represents the flow function of :samp:`(u, v)` and satisfies :samp:`R[u][v]['flow'] == -R[v][u]['flow']`. The flow value, defined as the total flow into :samp:`t`, the sink, is stored in :samp:`R.graph['flow_value']`. If :samp:`cutoff` is not specified, reachability to :samp:`t` using only edges :samp:`(u, v)` such that :samp:`R[u][v]['flow'] < R[u][v]['capacity']` induces a minimum :samp:`s`-:samp:`t` cut. Examples -------- >>> import networkx as nx >>> from networkx.algorithms.flow import dinitz The functions that implement flow algorithms and output a residual network, such as this one, are not imported to the base NetworkX namespace, so you have to explicitly import them from the flow package. >>> G = nx.DiGraph() >>> G.add_edge('x','a', capacity=3.0) >>> G.add_edge('x','b', capacity=1.0) >>> G.add_edge('a','c', capacity=3.0) >>> G.add_edge('b','c', capacity=5.0) >>> G.add_edge('b','d', capacity=4.0) >>> G.add_edge('d','e', capacity=2.0) >>> G.add_edge('c','y', capacity=2.0) >>> G.add_edge('e','y', capacity=3.0) >>> R = dinitz(G, 'x', 'y') >>> flow_value = nx.maximum_flow_value(G, 'x', 'y') >>> flow_value 3.0 >>> flow_value == R.graph['flow_value'] True References ---------- .. [1] Dinitz' Algorithm: The Original Version and Even's Version. 2006. Yefim Dinitz. In Theoretical Computer Science. Lecture Notes in Computer Science. Volume 3895. pp 218-240. http://www.cs.bgu.ac.il/~dinitz/Papers/Dinitz_alg.pdf """ R = dinitz_impl(G, s, t, capacity, residual, cutoff) R.graph['algorithm'] = 'dinitz' return R
def dinitz_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. 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