ó žÃÒYc@spdZdZddlmZddlZddlTddlmZdgZ d „Z d de e dd „Z dS( s? Shortest augmenting path algorithm for maximum flow problems. s&ysitu iÿÿÿÿ(tdequeNi(t*(tedmonds_karp_coretshortest_augmenting_pathcsÚ||kr(tjdt|ƒƒ‚n||krPtjdt|ƒƒ‚n||krntjdƒ‚n|dkrŒt||ƒ}n|}|j‰|j}|j‰x3|D]+} x"ˆ| jƒD]} d| dImplementation of the shortest augmenting path algorithm. snode %s not in graphs!source and sink are the same nodeitflowitcapacityt flow_valueitheightt curr_edgetinfcsãˆ}t|ƒ}t|ƒ}x=|D]5}ˆ||}t||d|dƒ}|}q%W|dˆkr€tjdƒ‚nt|ƒ}t|ƒ}xD|D]<}ˆ||dc|7<ˆ||dc|8<|}qŸW|S(s/Augment flow along a path from s to t. RRis-Infinite capacity path, flow unbounded above.(titertnexttmintnxtNetworkXUnbounded(tpathRtittutvtattr(tR_succR (s…/private/var/folders/w6/vb91730s7bb1k90y_rnhql1dhvdd44/T/pip-build-w4MwvS/networkx/networkx/algorithms/flow/shortestaugmentingpath.pytaugmentLs"         csaˆd}xLˆ|jƒD]:\}}|d|dkrt|ˆ|dƒ}qqW|dS(s5Relabel a node to create an admissible edge. iRRR(titemsR (RRRR(tR_nodesRtn(s…/private/var/folders/w6/vb91730s7bb1k90y_rnhql1dhvdd44/T/pip-build-w4MwvS/networkx/networkx/algorithms/flow/shortestaugmentingpath.pytrelabelcs  gà?g@iiÿÿÿÿN(R t NetworkXErrortstrtNonetbuild_residual_networktnodestpredtsucctvaluesRtpopleftRtappendtgraphtlentsizet CurrentEdgetfloattintR tTruetgett move_to_nextt StopIterationtpopR(tGtsttRtresidualt two_phasetcutofftRtR_predRtetheightstqRRRtmtcountsRRRRtdtdoneR((RRR Rs…/private/var/folders/w6/vb91730s7bb1k90y_rnhql1dhvdd44/T/pip-build-w4MwvS/networkx/networkx/algorithms/flow/shortestaugmentingpath.pytshortest_augmenting_path_implsª                 $    1                 Rc Cs/t|||||||ƒ}d|jd<|S(sÑFind a maximum single-commodity flow using the shortest augmenting path 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. 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. two_phase : bool If True, a two-phase variant is used. The two-phase variant improves the running time on unit-capacity networks from $O(nm)$ to $O(\min(n^{2/3}, m^{1/2}) m)$. Default value: False. 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:`edmonds_karp` :meth:`preflow_push` 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 shortest_augmenting_path 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 = shortest_augmenting_path(G, 'x', 'y') >>> flow_value = nx.maximum_flow_value(G, 'x', 'y') >>> flow_value 3.0 >>> flow_value == R.graph['flow_value'] True Rt algorithm(R>R$( R/R0R1RR2t value_onlyR3R4R5((s…/private/var/folders/w6/vb91730s7bb1k90y_rnhql1dhvdd44/T/pip-build-w4MwvS/networkx/networkx/algorithms/flow/shortestaugmentingpath.pyR°s{  (t__doc__t __author__t collectionsRtnetworkxR tutilst edmondskarpRt__all__R>RtFalseR(((s…/private/var/folders/w6/vb91730s7bb1k90y_rnhql1dhvdd44/T/pip-build-w4MwvS/networkx/networkx/algorithms/flow/shortestaugmentingpath.pyts