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<h1>Source code for networkx.algorithms.shortest_paths.weighted</h1><div class="highlight"><pre>
<span></span><span class="sd">"""</span>
<span class="sd">Shortest path algorithms for weighted graphs.</span>
<span class="sd">"""</span>
<span class="kn">from</span> <span class="nn">collections</span> <span class="kn">import</span> <span class="n">deque</span>
<span class="kn">from</span> <span class="nn">heapq</span> <span class="kn">import</span> <span class="n">heappop</span><span class="p">,</span> <span class="n">heappush</span>
<span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">count</span>
<span class="kn">import</span> <span class="nn">networkx</span> <span class="k">as</span> <span class="nn">nx</span>
<span class="kn">from</span> <span class="nn">networkx.algorithms.shortest_paths.generic</span> <span class="kn">import</span> <span class="n">_build_paths_from_predecessors</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
<span class="s2">"dijkstra_path"</span><span class="p">,</span>
<span class="s2">"dijkstra_path_length"</span><span class="p">,</span>
<span class="s2">"bidirectional_dijkstra"</span><span class="p">,</span>
<span class="s2">"single_source_dijkstra"</span><span class="p">,</span>
<span class="s2">"single_source_dijkstra_path"</span><span class="p">,</span>
<span class="s2">"single_source_dijkstra_path_length"</span><span class="p">,</span>
<span class="s2">"multi_source_dijkstra"</span><span class="p">,</span>
<span class="s2">"multi_source_dijkstra_path"</span><span class="p">,</span>
<span class="s2">"multi_source_dijkstra_path_length"</span><span class="p">,</span>
<span class="s2">"all_pairs_dijkstra"</span><span class="p">,</span>
<span class="s2">"all_pairs_dijkstra_path"</span><span class="p">,</span>
<span class="s2">"all_pairs_dijkstra_path_length"</span><span class="p">,</span>
<span class="s2">"dijkstra_predecessor_and_distance"</span><span class="p">,</span>
<span class="s2">"bellman_ford_path"</span><span class="p">,</span>
<span class="s2">"bellman_ford_path_length"</span><span class="p">,</span>
<span class="s2">"single_source_bellman_ford"</span><span class="p">,</span>
<span class="s2">"single_source_bellman_ford_path"</span><span class="p">,</span>
<span class="s2">"single_source_bellman_ford_path_length"</span><span class="p">,</span>
<span class="s2">"all_pairs_bellman_ford_path"</span><span class="p">,</span>
<span class="s2">"all_pairs_bellman_ford_path_length"</span><span class="p">,</span>
<span class="s2">"bellman_ford_predecessor_and_distance"</span><span class="p">,</span>
<span class="s2">"negative_edge_cycle"</span><span class="p">,</span>
<span class="s2">"find_negative_cycle"</span><span class="p">,</span>
<span class="s2">"goldberg_radzik"</span><span class="p">,</span>
<span class="s2">"johnson"</span><span class="p">,</span>
<span class="p">]</span>
<span class="k">def</span> <span class="nf">_weight_function</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="p">):</span>
<span class="sd">"""Returns a function that returns the weight of an edge.</span>
<span class="sd"> The returned function is specifically suitable for input to</span>
<span class="sd"> functions :func:`_dijkstra` and :func:`_bellman_ford_relaxation`.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph.</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If it is callable, `weight` itself is returned. If it is a string,</span>
<span class="sd"> it is assumed to be the name of the edge attribute that represents</span>
<span class="sd"> the weight of an edge. In that case, a function is returned that</span>
<span class="sd"> gets the edge weight according to the specified edge attribute.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> function</span>
<span class="sd"> This function returns a callable that accepts exactly three inputs:</span>
<span class="sd"> a node, an node adjacent to the first one, and the edge attribute</span>
<span class="sd"> dictionary for the eedge joining those nodes. That function returns</span>
<span class="sd"> a number representing the weight of an edge.</span>
<span class="sd"> If `G` is a multigraph, and `weight` is not callable, the</span>
<span class="sd"> minimum edge weight over all parallel edges is returned. If any edge</span>
<span class="sd"> does not have an attribute with key `weight`, it is assumed to</span>
<span class="sd"> have weight one.</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">callable</span><span class="p">(</span><span class="n">weight</span><span class="p">):</span>
<span class="k">return</span> <span class="n">weight</span>
<span class="c1"># If the weight keyword argument is not callable, we assume it is a</span>
<span class="c1"># string representing the edge attribute containing the weight of</span>
<span class="c1"># the edge.</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_multigraph</span><span class="p">():</span>
<span class="k">return</span> <span class="k">lambda</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span><span class="p">:</span> <span class="nb">min</span><span class="p">(</span><span class="n">attr</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">attr</span> <span class="ow">in</span> <span class="n">d</span><span class="o">.</span><span class="n">values</span><span class="p">())</span>
<span class="k">return</span> <span class="k">lambda</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">data</span><span class="p">:</span> <span class="n">data</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">weight</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<div class="viewcode-block" id="dijkstra_path"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.dijkstra_path.html#networkx.algorithms.shortest_paths.weighted.dijkstra_path">[docs]</a><span class="k">def</span> <span class="nf">dijkstra_path</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Returns the shortest weighted path from source to target in G.</span>
<span class="sd"> Uses Dijkstra's Method to compute the shortest weighted path</span>
<span class="sd"> between two nodes in a graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> source : node</span>
<span class="sd"> Starting node</span>
<span class="sd"> target : node</span>
<span class="sd"> Ending node</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number or None to indicate a hidden edge.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> path : list</span>
<span class="sd"> List of nodes in a shortest path.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If `source` is not in `G`.</span>
<span class="sd"> NetworkXNoPath</span>
<span class="sd"> If no path exists between source and target.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> print(nx.dijkstra_path(G, 0, 4))</span>
<span class="sd"> [0, 1, 2, 3, 4]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> The weight function can be used to hide edges by returning None.</span>
<span class="sd"> So ``weight = lambda u, v, d: 1 if d['color']=="red" else None``</span>
<span class="sd"> will find the shortest red path.</span>
<span class="sd"> The weight function can be used to include node weights.</span>
<span class="sd"> >>> def func(u, v, d):</span>
<span class="sd"> ... node_u_wt = G.nodes[u].get("node_weight", 1)</span>
<span class="sd"> ... node_v_wt = G.nodes[v].get("node_weight", 1)</span>
<span class="sd"> ... edge_wt = d.get("weight", 1)</span>
<span class="sd"> ... return node_u_wt / 2 + node_v_wt / 2 + edge_wt</span>
<span class="sd"> In this example we take the average of start and end node</span>
<span class="sd"> weights of an edge and add it to the weight of the edge.</span>
<span class="sd"> The function :func:`single_source_dijkstra` computes both</span>
<span class="sd"> path and length-of-path if you need both, use that.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> bidirectional_dijkstra</span>
<span class="sd"> bellman_ford_path</span>
<span class="sd"> single_source_dijkstra</span>
<span class="sd"> """</span>
<span class="p">(</span><span class="n">length</span><span class="p">,</span> <span class="n">path</span><span class="p">)</span> <span class="o">=</span> <span class="n">single_source_dijkstra</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="n">target</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)</span>
<span class="k">return</span> <span class="n">path</span></div>
<div class="viewcode-block" id="dijkstra_path_length"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.dijkstra_path_length.html#networkx.algorithms.shortest_paths.weighted.dijkstra_path_length">[docs]</a><span class="k">def</span> <span class="nf">dijkstra_path_length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Returns the shortest weighted path length in G from source to target.</span>
<span class="sd"> Uses Dijkstra's Method to compute the shortest weighted path length</span>
<span class="sd"> between two nodes in a graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> source : node label</span>
<span class="sd"> starting node for path</span>
<span class="sd"> target : node label</span>
<span class="sd"> ending node for path</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number or None to indicate a hidden edge.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> length : number</span>
<span class="sd"> Shortest path length.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If `source` is not in `G`.</span>
<span class="sd"> NetworkXNoPath</span>
<span class="sd"> If no path exists between source and target.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> nx.dijkstra_path_length(G, 0, 4)</span>
<span class="sd"> 4</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> The weight function can be used to hide edges by returning None.</span>
<span class="sd"> So ``weight = lambda u, v, d: 1 if d['color']=="red" else None``</span>
<span class="sd"> will find the shortest red path.</span>
<span class="sd"> The function :func:`single_source_dijkstra` computes both</span>
<span class="sd"> path and length-of-path if you need both, use that.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> bidirectional_dijkstra</span>
<span class="sd"> bellman_ford_path_length</span>
<span class="sd"> single_source_dijkstra</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">source</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NodeNotFound</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Node </span><span class="si">{</span><span class="n">source</span><span class="si">}</span><span class="s2"> not found in graph"</span><span class="p">)</span>
<span class="k">if</span> <span class="n">source</span> <span class="o">==</span> <span class="n">target</span><span class="p">:</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="n">weight</span> <span class="o">=</span> <span class="n">_weight_function</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="n">length</span> <span class="o">=</span> <span class="n">_dijkstra</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="n">target</span><span class="p">)</span>
<span class="k">try</span><span class="p">:</span>
<span class="k">return</span> <span class="n">length</span><span class="p">[</span><span class="n">target</span><span class="p">]</span>
<span class="k">except</span> <span class="ne">KeyError</span> <span class="k">as</span> <span class="n">err</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXNoPath</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Node </span><span class="si">{</span><span class="n">target</span><span class="si">}</span><span class="s2"> not reachable from </span><span class="si">{</span><span class="n">source</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span> <span class="kn">from</span> <span class="nn">err</span></div>
<div class="viewcode-block" id="single_source_dijkstra_path"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.single_source_dijkstra_path.html#networkx.algorithms.shortest_paths.weighted.single_source_dijkstra_path">[docs]</a><span class="k">def</span> <span class="nf">single_source_dijkstra_path</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Find shortest weighted paths in G from a source node.</span>
<span class="sd"> Compute shortest path between source and all other reachable</span>
<span class="sd"> nodes for a weighted graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> source : node</span>
<span class="sd"> Starting node for path.</span>
<span class="sd"> cutoff : integer or float, optional</span>
<span class="sd"> Length (sum of edge weights) at which the search is stopped.</span>
<span class="sd"> If cutoff is provided, only return paths with summed weight <= cutoff.</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number or None to indicate a hidden edge.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> paths : dictionary</span>
<span class="sd"> Dictionary of shortest path lengths keyed by target.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If `source` is not in `G`.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> path = nx.single_source_dijkstra_path(G, 0)</span>
<span class="sd"> >>> path[4]</span>
<span class="sd"> [0, 1, 2, 3, 4]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> The weight function can be used to hide edges by returning None.</span>
<span class="sd"> So ``weight = lambda u, v, d: 1 if d['color']=="red" else None``</span>
<span class="sd"> will find the shortest red path.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> single_source_dijkstra, single_source_bellman_ford</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="n">multi_source_dijkstra_path</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="p">{</span><span class="n">source</span><span class="p">},</span> <span class="n">cutoff</span><span class="o">=</span><span class="n">cutoff</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)</span></div>
<div class="viewcode-block" id="single_source_dijkstra_path_length"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.single_source_dijkstra_path_length.html#networkx.algorithms.shortest_paths.weighted.single_source_dijkstra_path_length">[docs]</a><span class="k">def</span> <span class="nf">single_source_dijkstra_path_length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Find shortest weighted path lengths in G from a source node.</span>
<span class="sd"> Compute the shortest path length between source and all other</span>
<span class="sd"> reachable nodes for a weighted graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> source : node label</span>
<span class="sd"> Starting node for path</span>
<span class="sd"> cutoff : integer or float, optional</span>
<span class="sd"> Length (sum of edge weights) at which the search is stopped.</span>
<span class="sd"> If cutoff is provided, only return paths with summed weight <= cutoff.</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number or None to indicate a hidden edge.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> length : dict</span>
<span class="sd"> Dict keyed by node to shortest path length from source.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If `source` is not in `G`.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> length = nx.single_source_dijkstra_path_length(G, 0)</span>
<span class="sd"> >>> length[4]</span>
<span class="sd"> 4</span>
<span class="sd"> >>> for node in [0, 1, 2, 3, 4]:</span>
<span class="sd"> ... print(f"{node}: {length[node]}")</span>
<span class="sd"> 0: 0</span>
<span class="sd"> 1: 1</span>
<span class="sd"> 2: 2</span>
<span class="sd"> 3: 3</span>
<span class="sd"> 4: 4</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> The weight function can be used to hide edges by returning None.</span>
<span class="sd"> So ``weight = lambda u, v, d: 1 if d['color']=="red" else None``</span>
<span class="sd"> will find the shortest red path.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> single_source_dijkstra, single_source_bellman_ford_path_length</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="n">multi_source_dijkstra_path_length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="p">{</span><span class="n">source</span><span class="p">},</span> <span class="n">cutoff</span><span class="o">=</span><span class="n">cutoff</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)</span></div>
<div class="viewcode-block" id="single_source_dijkstra"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.single_source_dijkstra.html#networkx.algorithms.shortest_paths.weighted.single_source_dijkstra">[docs]</a><span class="k">def</span> <span class="nf">single_source_dijkstra</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Find shortest weighted paths and lengths from a source node.</span>
<span class="sd"> Compute the shortest path length between source and all other</span>
<span class="sd"> reachable nodes for a weighted graph.</span>
<span class="sd"> Uses Dijkstra's algorithm to compute shortest paths and lengths</span>
<span class="sd"> between a source and all other reachable nodes in a weighted graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> source : node label</span>
<span class="sd"> Starting node for path</span>
<span class="sd"> target : node label, optional</span>
<span class="sd"> Ending node for path</span>
<span class="sd"> cutoff : integer or float, optional</span>
<span class="sd"> Length (sum of edge weights) at which the search is stopped.</span>
<span class="sd"> If cutoff is provided, only return paths with summed weight <= cutoff.</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number or None to indicate a hidden edge.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> distance, path : pair of dictionaries, or numeric and list.</span>
<span class="sd"> If target is None, paths and lengths to all nodes are computed.</span>
<span class="sd"> The return value is a tuple of two dictionaries keyed by target nodes.</span>
<span class="sd"> The first dictionary stores distance to each target node.</span>
<span class="sd"> The second stores the path to each target node.</span>
<span class="sd"> If target is not None, returns a tuple (distance, path), where</span>
<span class="sd"> distance is the distance from source to target and path is a list</span>
<span class="sd"> representing the path from source to target.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If `source` is not in `G`.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> length, path = nx.single_source_dijkstra(G, 0)</span>
<span class="sd"> >>> length[4]</span>
<span class="sd"> 4</span>
<span class="sd"> >>> for node in [0, 1, 2, 3, 4]:</span>
<span class="sd"> ... print(f"{node}: {length[node]}")</span>
<span class="sd"> 0: 0</span>
<span class="sd"> 1: 1</span>
<span class="sd"> 2: 2</span>
<span class="sd"> 3: 3</span>
<span class="sd"> 4: 4</span>
<span class="sd"> >>> path[4]</span>
<span class="sd"> [0, 1, 2, 3, 4]</span>
<span class="sd"> >>> length, path = nx.single_source_dijkstra(G, 0, 1)</span>
<span class="sd"> >>> length</span>
<span class="sd"> 1</span>
<span class="sd"> >>> path</span>
<span class="sd"> [0, 1]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> The weight function can be used to hide edges by returning None.</span>
<span class="sd"> So ``weight = lambda u, v, d: 1 if d['color']=="red" else None``</span>
<span class="sd"> will find the shortest red path.</span>
<span class="sd"> Based on the Python cookbook recipe (119466) at</span>
<span class="sd"> https://code.activestate.com/recipes/119466/</span>
<span class="sd"> This algorithm is not guaranteed to work if edge weights</span>
<span class="sd"> are negative or are floating point numbers</span>
<span class="sd"> (overflows and roundoff errors can cause problems).</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> single_source_dijkstra_path</span>
<span class="sd"> single_source_dijkstra_path_length</span>
<span class="sd"> single_source_bellman_ford</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="n">multi_source_dijkstra</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="p">{</span><span class="n">source</span><span class="p">},</span> <span class="n">cutoff</span><span class="o">=</span><span class="n">cutoff</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="n">target</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span>
<span class="p">)</span></div>
<div class="viewcode-block" id="multi_source_dijkstra_path"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.multi_source_dijkstra_path.html#networkx.algorithms.shortest_paths.weighted.multi_source_dijkstra_path">[docs]</a><span class="k">def</span> <span class="nf">multi_source_dijkstra_path</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">sources</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Find shortest weighted paths in G from a given set of source</span>
<span class="sd"> nodes.</span>
<span class="sd"> Compute shortest path between any of the source nodes and all other</span>
<span class="sd"> reachable nodes for a weighted graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> sources : non-empty set of nodes</span>
<span class="sd"> Starting nodes for paths. If this is just a set containing a</span>
<span class="sd"> single node, then all paths computed by this function will start</span>
<span class="sd"> from that node. If there are two or more nodes in the set, the</span>
<span class="sd"> computed paths may begin from any one of the start nodes.</span>
<span class="sd"> cutoff : integer or float, optional</span>
<span class="sd"> Length (sum of edge weights) at which the search is stopped.</span>
<span class="sd"> If cutoff is provided, only return paths with summed weight <= cutoff.</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number or None to indicate a hidden edge.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> paths : dictionary</span>
<span class="sd"> Dictionary of shortest paths keyed by target.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> path = nx.multi_source_dijkstra_path(G, {0, 4})</span>
<span class="sd"> >>> path[1]</span>
<span class="sd"> [0, 1]</span>
<span class="sd"> >>> path[3]</span>
<span class="sd"> [4, 3]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> The weight function can be used to hide edges by returning None.</span>
<span class="sd"> So ``weight = lambda u, v, d: 1 if d['color']=="red" else None``</span>
<span class="sd"> will find the shortest red path.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> ValueError</span>
<span class="sd"> If `sources` is empty.</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If any of `sources` is not in `G`.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> multi_source_dijkstra, multi_source_bellman_ford</span>
<span class="sd"> """</span>
<span class="n">length</span><span class="p">,</span> <span class="n">path</span> <span class="o">=</span> <span class="n">multi_source_dijkstra</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">sources</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="n">cutoff</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)</span>
<span class="k">return</span> <span class="n">path</span></div>
<div class="viewcode-block" id="multi_source_dijkstra_path_length"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.multi_source_dijkstra_path_length.html#networkx.algorithms.shortest_paths.weighted.multi_source_dijkstra_path_length">[docs]</a><span class="k">def</span> <span class="nf">multi_source_dijkstra_path_length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">sources</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Find shortest weighted path lengths in G from a given set of</span>
<span class="sd"> source nodes.</span>
<span class="sd"> Compute the shortest path length between any of the source nodes and</span>
<span class="sd"> all other reachable nodes for a weighted graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> sources : non-empty set of nodes</span>
<span class="sd"> Starting nodes for paths. If this is just a set containing a</span>
<span class="sd"> single node, then all paths computed by this function will start</span>
<span class="sd"> from that node. If there are two or more nodes in the set, the</span>
<span class="sd"> computed paths may begin from any one of the start nodes.</span>
<span class="sd"> cutoff : integer or float, optional</span>
<span class="sd"> Length (sum of edge weights) at which the search is stopped.</span>
<span class="sd"> If cutoff is provided, only return paths with summed weight <= cutoff.</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number or None to indicate a hidden edge.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> length : dict</span>
<span class="sd"> Dict keyed by node to shortest path length to nearest source.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> length = nx.multi_source_dijkstra_path_length(G, {0, 4})</span>
<span class="sd"> >>> for node in [0, 1, 2, 3, 4]:</span>
<span class="sd"> ... print(f"{node}: {length[node]}")</span>
<span class="sd"> 0: 0</span>
<span class="sd"> 1: 1</span>
<span class="sd"> 2: 2</span>
<span class="sd"> 3: 1</span>
<span class="sd"> 4: 0</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> The weight function can be used to hide edges by returning None.</span>
<span class="sd"> So ``weight = lambda u, v, d: 1 if d['color']=="red" else None``</span>
<span class="sd"> will find the shortest red path.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> ValueError</span>
<span class="sd"> If `sources` is empty.</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If any of `sources` is not in `G`.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> multi_source_dijkstra</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">sources</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">"sources must not be empty"</span><span class="p">)</span>
<span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">sources</span><span class="p">:</span>
<span class="k">if</span> <span class="n">s</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NodeNotFound</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Node </span><span class="si">{</span><span class="n">s</span><span class="si">}</span><span class="s2"> not found in graph"</span><span class="p">)</span>
<span class="n">weight</span> <span class="o">=</span> <span class="n">_weight_function</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="k">return</span> <span class="n">_dijkstra_multisource</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">sources</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="n">cutoff</span><span class="p">)</span></div>
<div class="viewcode-block" id="multi_source_dijkstra"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.multi_source_dijkstra.html#networkx.algorithms.shortest_paths.weighted.multi_source_dijkstra">[docs]</a><span class="k">def</span> <span class="nf">multi_source_dijkstra</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">sources</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Find shortest weighted paths and lengths from a given set of</span>
<span class="sd"> source nodes.</span>
<span class="sd"> Uses Dijkstra's algorithm to compute the shortest paths and lengths</span>
<span class="sd"> between one of the source nodes and the given `target`, or all other</span>
<span class="sd"> reachable nodes if not specified, for a weighted graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> sources : non-empty set of nodes</span>
<span class="sd"> Starting nodes for paths. If this is just a set containing a</span>
<span class="sd"> single node, then all paths computed by this function will start</span>
<span class="sd"> from that node. If there are two or more nodes in the set, the</span>
<span class="sd"> computed paths may begin from any one of the start nodes.</span>
<span class="sd"> target : node label, optional</span>
<span class="sd"> Ending node for path</span>
<span class="sd"> cutoff : integer or float, optional</span>
<span class="sd"> Length (sum of edge weights) at which the search is stopped.</span>
<span class="sd"> If cutoff is provided, only return paths with summed weight <= cutoff.</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number or None to indicate a hidden edge.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> distance, path : pair of dictionaries, or numeric and list</span>
<span class="sd"> If target is None, returns a tuple of two dictionaries keyed by node.</span>
<span class="sd"> The first dictionary stores distance from one of the source nodes.</span>
<span class="sd"> The second stores the path from one of the sources to that node.</span>
<span class="sd"> If target is not None, returns a tuple of (distance, path) where</span>
<span class="sd"> distance is the distance from source to target and path is a list</span>
<span class="sd"> representing the path from source to target.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> length, path = nx.multi_source_dijkstra(G, {0, 4})</span>
<span class="sd"> >>> for node in [0, 1, 2, 3, 4]:</span>
<span class="sd"> ... print(f"{node}: {length[node]}")</span>
<span class="sd"> 0: 0</span>
<span class="sd"> 1: 1</span>
<span class="sd"> 2: 2</span>
<span class="sd"> 3: 1</span>
<span class="sd"> 4: 0</span>
<span class="sd"> >>> path[1]</span>
<span class="sd"> [0, 1]</span>
<span class="sd"> >>> path[3]</span>
<span class="sd"> [4, 3]</span>
<span class="sd"> >>> length, path = nx.multi_source_dijkstra(G, {0, 4}, 1)</span>
<span class="sd"> >>> length</span>
<span class="sd"> 1</span>
<span class="sd"> >>> path</span>
<span class="sd"> [0, 1]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> The weight function can be used to hide edges by returning None.</span>
<span class="sd"> So ``weight = lambda u, v, d: 1 if d['color']=="red" else None``</span>
<span class="sd"> will find the shortest red path.</span>
<span class="sd"> Based on the Python cookbook recipe (119466) at</span>
<span class="sd"> https://code.activestate.com/recipes/119466/</span>
<span class="sd"> This algorithm is not guaranteed to work if edge weights</span>
<span class="sd"> are negative or are floating point numbers</span>
<span class="sd"> (overflows and roundoff errors can cause problems).</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> ValueError</span>
<span class="sd"> If `sources` is empty.</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If any of `sources` is not in `G`.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> multi_source_dijkstra_path</span>
<span class="sd"> multi_source_dijkstra_path_length</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">sources</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">"sources must not be empty"</span><span class="p">)</span>
<span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">sources</span><span class="p">:</span>
<span class="k">if</span> <span class="n">s</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NodeNotFound</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Node </span><span class="si">{</span><span class="n">s</span><span class="si">}</span><span class="s2"> not found in graph"</span><span class="p">)</span>
<span class="k">if</span> <span class="n">target</span> <span class="ow">in</span> <span class="n">sources</span><span class="p">:</span>
<span class="k">return</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">[</span><span class="n">target</span><span class="p">])</span>
<span class="n">weight</span> <span class="o">=</span> <span class="n">_weight_function</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="n">paths</span> <span class="o">=</span> <span class="p">{</span><span class="n">source</span><span class="p">:</span> <span class="p">[</span><span class="n">source</span><span class="p">]</span> <span class="k">for</span> <span class="n">source</span> <span class="ow">in</span> <span class="n">sources</span><span class="p">}</span> <span class="c1"># dictionary of paths</span>
<span class="n">dist</span> <span class="o">=</span> <span class="n">_dijkstra_multisource</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">sources</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">paths</span><span class="o">=</span><span class="n">paths</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="n">cutoff</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="n">target</span>
<span class="p">)</span>
<span class="k">if</span> <span class="n">target</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">return</span> <span class="p">(</span><span class="n">dist</span><span class="p">,</span> <span class="n">paths</span><span class="p">)</span>
<span class="k">try</span><span class="p">:</span>
<span class="k">return</span> <span class="p">(</span><span class="n">dist</span><span class="p">[</span><span class="n">target</span><span class="p">],</span> <span class="n">paths</span><span class="p">[</span><span class="n">target</span><span class="p">])</span>
<span class="k">except</span> <span class="ne">KeyError</span> <span class="k">as</span> <span class="n">err</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXNoPath</span><span class="p">(</span><span class="sa">f</span><span class="s2">"No path to </span><span class="si">{</span><span class="n">target</span><span class="si">}</span><span class="s2">."</span><span class="p">)</span> <span class="kn">from</span> <span class="nn">err</span></div>
<span class="k">def</span> <span class="nf">_dijkstra</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">pred</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">paths</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="sd">"""Uses Dijkstra's algorithm to find shortest weighted paths from a</span>
<span class="sd"> single source.</span>
<span class="sd"> This is a convenience function for :func:`_dijkstra_multisource`</span>
<span class="sd"> with all the arguments the same, except the keyword argument</span>
<span class="sd"> `sources` set to ``[source]``.</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="n">_dijkstra_multisource</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="p">[</span><span class="n">source</span><span class="p">],</span> <span class="n">weight</span><span class="p">,</span> <span class="n">pred</span><span class="o">=</span><span class="n">pred</span><span class="p">,</span> <span class="n">paths</span><span class="o">=</span><span class="n">paths</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="n">cutoff</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="n">target</span>
<span class="p">)</span>
<span class="k">def</span> <span class="nf">_dijkstra_multisource</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">sources</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">pred</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">paths</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="kc">None</span>
<span class="p">):</span>
<span class="sd">"""Uses Dijkstra's algorithm to find shortest weighted paths</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> sources : non-empty iterable of nodes</span>
<span class="sd"> Starting nodes for paths. If this is just an iterable containing</span>
<span class="sd"> a single node, then all paths computed by this function will</span>
<span class="sd"> start from that node. If there are two or more nodes in this</span>
<span class="sd"> iterable, the computed paths may begin from any one of the start</span>
<span class="sd"> nodes.</span>
<span class="sd"> weight: function</span>
<span class="sd"> Function with (u, v, data) input that returns that edge's weight</span>
<span class="sd"> or None to indicate a hidden edge</span>
<span class="sd"> pred: dict of lists, optional(default=None)</span>
<span class="sd"> dict to store a list of predecessors keyed by that node</span>
<span class="sd"> If None, predecessors are not stored.</span>
<span class="sd"> paths: dict, optional (default=None)</span>
<span class="sd"> dict to store the path list from source to each node, keyed by node.</span>
<span class="sd"> If None, paths are not stored.</span>
<span class="sd"> target : node label, optional</span>
<span class="sd"> Ending node for path. Search is halted when target is found.</span>
<span class="sd"> cutoff : integer or float, optional</span>
<span class="sd"> Length (sum of edge weights) at which the search is stopped.</span>
<span class="sd"> If cutoff is provided, only return paths with summed weight <= cutoff.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> distance : dictionary</span>
<span class="sd"> A mapping from node to shortest distance to that node from one</span>
<span class="sd"> of the source nodes.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If any of `sources` is not in `G`.</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> The optional predecessor and path dictionaries can be accessed by</span>
<span class="sd"> the caller through the original pred and paths objects passed</span>
<span class="sd"> as arguments. No need to explicitly return pred or paths.</span>
<span class="sd"> """</span>
<span class="n">G_succ</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">_adj</span> <span class="c1"># For speed-up (and works for both directed and undirected graphs)</span>
<span class="n">push</span> <span class="o">=</span> <span class="n">heappush</span>
<span class="n">pop</span> <span class="o">=</span> <span class="n">heappop</span>
<span class="n">dist</span> <span class="o">=</span> <span class="p">{}</span> <span class="c1"># dictionary of final distances</span>
<span class="n">seen</span> <span class="o">=</span> <span class="p">{}</span>
<span class="c1"># fringe is heapq with 3-tuples (distance,c,node)</span>
<span class="c1"># use the count c to avoid comparing nodes (may not be able to)</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">count</span><span class="p">()</span>
<span class="n">fringe</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">for</span> <span class="n">source</span> <span class="ow">in</span> <span class="n">sources</span><span class="p">:</span>
<span class="n">seen</span><span class="p">[</span><span class="n">source</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">push</span><span class="p">(</span><span class="n">fringe</span><span class="p">,</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">next</span><span class="p">(</span><span class="n">c</span><span class="p">),</span> <span class="n">source</span><span class="p">))</span>
<span class="k">while</span> <span class="n">fringe</span><span class="p">:</span>
<span class="p">(</span><span class="n">d</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="o">=</span> <span class="n">pop</span><span class="p">(</span><span class="n">fringe</span><span class="p">)</span>
<span class="k">if</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">dist</span><span class="p">:</span>
<span class="k">continue</span> <span class="c1"># already searched this node.</span>
<span class="n">dist</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">d</span>
<span class="k">if</span> <span class="n">v</span> <span class="o">==</span> <span class="n">target</span><span class="p">:</span>
<span class="k">break</span>
<span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">G_succ</span><span class="p">[</span><span class="n">v</span><span class="p">]</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
<span class="n">cost</span> <span class="o">=</span> <span class="n">weight</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">u</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
<span class="k">if</span> <span class="n">cost</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">continue</span>
<span class="n">vu_dist</span> <span class="o">=</span> <span class="n">dist</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">+</span> <span class="n">cost</span>
<span class="k">if</span> <span class="n">cutoff</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">if</span> <span class="n">vu_dist</span> <span class="o">></span> <span class="n">cutoff</span><span class="p">:</span>
<span class="k">continue</span>
<span class="k">if</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">dist</span><span class="p">:</span>
<span class="n">u_dist</span> <span class="o">=</span> <span class="n">dist</span><span class="p">[</span><span class="n">u</span><span class="p">]</span>
<span class="k">if</span> <span class="n">vu_dist</span> <span class="o"><</span> <span class="n">u_dist</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">"Contradictory paths found:"</span><span class="p">,</span> <span class="s2">"negative weights?"</span><span class="p">)</span>
<span class="k">elif</span> <span class="n">pred</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="ow">and</span> <span class="n">vu_dist</span> <span class="o">==</span> <span class="n">u_dist</span><span class="p">:</span>
<span class="n">pred</span><span class="p">[</span><span class="n">u</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="k">elif</span> <span class="n">u</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">seen</span> <span class="ow">or</span> <span class="n">vu_dist</span> <span class="o"><</span> <span class="n">seen</span><span class="p">[</span><span class="n">u</span><span class="p">]:</span>
<span class="n">seen</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="o">=</span> <span class="n">vu_dist</span>
<span class="n">push</span><span class="p">(</span><span class="n">fringe</span><span class="p">,</span> <span class="p">(</span><span class="n">vu_dist</span><span class="p">,</span> <span class="nb">next</span><span class="p">(</span><span class="n">c</span><span class="p">),</span> <span class="n">u</span><span class="p">))</span>
<span class="k">if</span> <span class="n">paths</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">paths</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="o">=</span> <span class="n">paths</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">u</span><span class="p">]</span>
<span class="k">if</span> <span class="n">pred</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">pred</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">v</span><span class="p">]</span>
<span class="k">elif</span> <span class="n">vu_dist</span> <span class="o">==</span> <span class="n">seen</span><span class="p">[</span><span class="n">u</span><span class="p">]:</span>
<span class="k">if</span> <span class="n">pred</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">pred</span><span class="p">[</span><span class="n">u</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="c1"># The optional predecessor and path dictionaries can be accessed</span>
<span class="c1"># by the caller via the pred and paths objects passed as arguments.</span>
<span class="k">return</span> <span class="n">dist</span>
<div class="viewcode-block" id="dijkstra_predecessor_and_distance"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.dijkstra_predecessor_and_distance.html#networkx.algorithms.shortest_paths.weighted.dijkstra_predecessor_and_distance">[docs]</a><span class="k">def</span> <span class="nf">dijkstra_predecessor_and_distance</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Compute weighted shortest path length and predecessors.</span>
<span class="sd"> Uses Dijkstra's Method to obtain the shortest weighted paths</span>
<span class="sd"> and return dictionaries of predecessors for each node and</span>
<span class="sd"> distance for each node from the `source`.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> source : node label</span>
<span class="sd"> Starting node for path</span>
<span class="sd"> cutoff : integer or float, optional</span>
<span class="sd"> Length (sum of edge weights) at which the search is stopped.</span>
<span class="sd"> If cutoff is provided, only return paths with summed weight <= cutoff.</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number or None to indicate a hidden edge.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> pred, distance : dictionaries</span>
<span class="sd"> Returns two dictionaries representing a list of predecessors</span>
<span class="sd"> of a node and the distance to each node.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If `source` is not in `G`.</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> The list of predecessors contains more than one element only when</span>
<span class="sd"> there are more than one shortest paths to the key node.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5, create_using=nx.DiGraph())</span>
<span class="sd"> >>> pred, dist = nx.dijkstra_predecessor_and_distance(G, 0)</span>
<span class="sd"> >>> sorted(pred.items())</span>
<span class="sd"> [(0, []), (1, [0]), (2, [1]), (3, [2]), (4, [3])]</span>
<span class="sd"> >>> sorted(dist.items())</span>
<span class="sd"> [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]</span>
<span class="sd"> >>> pred, dist = nx.dijkstra_predecessor_and_distance(G, 0, 1)</span>
<span class="sd"> >>> sorted(pred.items())</span>
<span class="sd"> [(0, []), (1, [0])]</span>
<span class="sd"> >>> sorted(dist.items())</span>
<span class="sd"> [(0, 0), (1, 1)]</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">source</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NodeNotFound</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Node </span><span class="si">{</span><span class="n">source</span><span class="si">}</span><span class="s2"> is not found in the graph"</span><span class="p">)</span>
<span class="n">weight</span> <span class="o">=</span> <span class="n">_weight_function</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="n">pred</span> <span class="o">=</span> <span class="p">{</span><span class="n">source</span><span class="p">:</span> <span class="p">[]}</span> <span class="c1"># dictionary of predecessors</span>
<span class="k">return</span> <span class="p">(</span><span class="n">pred</span><span class="p">,</span> <span class="n">_dijkstra</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">weight</span><span class="p">,</span> <span class="n">pred</span><span class="o">=</span><span class="n">pred</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="n">cutoff</span><span class="p">))</span></div>
<div class="viewcode-block" id="all_pairs_dijkstra"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.all_pairs_dijkstra.html#networkx.algorithms.shortest_paths.weighted.all_pairs_dijkstra">[docs]</a><span class="k">def</span> <span class="nf">all_pairs_dijkstra</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Find shortest weighted paths and lengths between all nodes.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> cutoff : integer or float, optional</span>
<span class="sd"> Length (sum of edge weights) at which the search is stopped.</span>
<span class="sd"> If cutoff is provided, only return paths with summed weight <= cutoff.</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edge[u][v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number or None to indicate a hidden edge.</span>
<span class="sd"> Yields</span>
<span class="sd"> ------</span>
<span class="sd"> (node, (distance, path)) : (node obj, (dict, dict))</span>
<span class="sd"> Each source node has two associated dicts. The first holds distance</span>
<span class="sd"> keyed by target and the second holds paths keyed by target.</span>
<span class="sd"> (See single_source_dijkstra for the source/target node terminology.)</span>
<span class="sd"> If desired you can apply `dict()` to this function to create a dict</span>
<span class="sd"> keyed by source node to the two dicts.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> len_path = dict(nx.all_pairs_dijkstra(G))</span>
<span class="sd"> >>> len_path[3][0][1]</span>
<span class="sd"> 2</span>
<span class="sd"> >>> for node in [0, 1, 2, 3, 4]:</span>
<span class="sd"> ... print(f"3 - {node}: {len_path[3][0][node]}")</span>
<span class="sd"> 3 - 0: 3</span>
<span class="sd"> 3 - 1: 2</span>
<span class="sd"> 3 - 2: 1</span>
<span class="sd"> 3 - 3: 0</span>
<span class="sd"> 3 - 4: 1</span>
<span class="sd"> >>> len_path[3][1][1]</span>
<span class="sd"> [3, 2, 1]</span>
<span class="sd"> >>> for n, (dist, path) in nx.all_pairs_dijkstra(G):</span>
<span class="sd"> ... print(path[1])</span>
<span class="sd"> [0, 1]</span>
<span class="sd"> [1]</span>
<span class="sd"> [2, 1]</span>
<span class="sd"> [3, 2, 1]</span>
<span class="sd"> [4, 3, 2, 1]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> The yielded dicts only have keys for reachable nodes.</span>
<span class="sd"> """</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="n">dist</span><span class="p">,</span> <span class="n">path</span> <span class="o">=</span> <span class="n">single_source_dijkstra</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="n">cutoff</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)</span>
<span class="k">yield</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">dist</span><span class="p">,</span> <span class="n">path</span><span class="p">))</span></div>
<div class="viewcode-block" id="all_pairs_dijkstra_path_length"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.all_pairs_dijkstra_path_length.html#networkx.algorithms.shortest_paths.weighted.all_pairs_dijkstra_path_length">[docs]</a><span class="k">def</span> <span class="nf">all_pairs_dijkstra_path_length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Compute shortest path lengths between all nodes in a weighted graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> cutoff : integer or float, optional</span>
<span class="sd"> Length (sum of edge weights) at which the search is stopped.</span>
<span class="sd"> If cutoff is provided, only return paths with summed weight <= cutoff.</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number or None to indicate a hidden edge.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> distance : iterator</span>
<span class="sd"> (source, dictionary) iterator with dictionary keyed by target and</span>
<span class="sd"> shortest path length as the key value.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> length = dict(nx.all_pairs_dijkstra_path_length(G))</span>
<span class="sd"> >>> for node in [0, 1, 2, 3, 4]:</span>
<span class="sd"> ... print(f"1 - {node}: {length[1][node]}")</span>
<span class="sd"> 1 - 0: 1</span>
<span class="sd"> 1 - 1: 0</span>
<span class="sd"> 1 - 2: 1</span>
<span class="sd"> 1 - 3: 2</span>
<span class="sd"> 1 - 4: 3</span>
<span class="sd"> >>> length[3][2]</span>
<span class="sd"> 1</span>
<span class="sd"> >>> length[2][2]</span>
<span class="sd"> 0</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> The dictionary returned only has keys for reachable node pairs.</span>
<span class="sd"> """</span>
<span class="n">length</span> <span class="o">=</span> <span class="n">single_source_dijkstra_path_length</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">yield</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="n">cutoff</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">))</span></div>
<div class="viewcode-block" id="all_pairs_dijkstra_path"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.all_pairs_dijkstra_path.html#networkx.algorithms.shortest_paths.weighted.all_pairs_dijkstra_path">[docs]</a><span class="k">def</span> <span class="nf">all_pairs_dijkstra_path</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Compute shortest paths between all nodes in a weighted graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> cutoff : integer or float, optional</span>
<span class="sd"> Length (sum of edge weights) at which the search is stopped.</span>
<span class="sd"> If cutoff is provided, only return paths with summed weight <= cutoff.</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number or None to indicate a hidden edge.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> distance : dictionary</span>
<span class="sd"> Dictionary, keyed by source and target, of shortest paths.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> path = dict(nx.all_pairs_dijkstra_path(G))</span>
<span class="sd"> >>> path[0][4]</span>
<span class="sd"> [0, 1, 2, 3, 4]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> floyd_warshall, all_pairs_bellman_ford_path</span>
<span class="sd"> """</span>
<span class="n">path</span> <span class="o">=</span> <span class="n">single_source_dijkstra_path</span>
<span class="c1"># TODO This can be trivially parallelized.</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">yield</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">path</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">cutoff</span><span class="o">=</span><span class="n">cutoff</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">))</span></div>
<div class="viewcode-block" id="bellman_ford_predecessor_and_distance"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.bellman_ford_predecessor_and_distance.html#networkx.algorithms.shortest_paths.weighted.bellman_ford_predecessor_and_distance">[docs]</a><span class="k">def</span> <span class="nf">bellman_ford_predecessor_and_distance</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">,</span> <span class="n">heuristic</span><span class="o">=</span><span class="kc">False</span>
<span class="p">):</span>
<span class="sd">"""Compute shortest path lengths and predecessors on shortest paths</span>
<span class="sd"> in weighted graphs.</span>
<span class="sd"> The algorithm has a running time of $O(mn)$ where $n$ is the number of</span>
<span class="sd"> nodes and $m$ is the number of edges. It is slower than Dijkstra but</span>
<span class="sd"> can handle negative edge weights.</span>
<span class="sd"> If a negative cycle is detected, you can use :func:`find_negative_cycle`</span>
<span class="sd"> to return the cycle and examine it. Shortest paths are not defined when</span>
<span class="sd"> a negative cycle exists because once reached, the path can cycle forever</span>
<span class="sd"> to build up arbitrarily low weights.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> The algorithm works for all types of graphs, including directed</span>
<span class="sd"> graphs and multigraphs.</span>
<span class="sd"> source: node label</span>
<span class="sd"> Starting node for path</span>
<span class="sd"> target : node label, optional</span>
<span class="sd"> Ending node for path</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number.</span>
<span class="sd"> heuristic : bool</span>
<span class="sd"> Determines whether to use a heuristic to early detect negative</span>
<span class="sd"> cycles at a hopefully negligible cost.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> pred, dist : dictionaries</span>
<span class="sd"> Returns two dictionaries keyed by node to predecessor in the</span>
<span class="sd"> path and to the distance from the source respectively.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If `source` is not in `G`.</span>
<span class="sd"> NetworkXUnbounded</span>
<span class="sd"> If the (di)graph contains a negative (di)cycle, the</span>
<span class="sd"> algorithm raises an exception to indicate the presence of the</span>
<span class="sd"> negative (di)cycle. Note: any negative weight edge in an</span>
<span class="sd"> undirected graph is a negative cycle.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5, create_using=nx.DiGraph())</span>
<span class="sd"> >>> pred, dist = nx.bellman_ford_predecessor_and_distance(G, 0)</span>
<span class="sd"> >>> sorted(pred.items())</span>
<span class="sd"> [(0, []), (1, [0]), (2, [1]), (3, [2]), (4, [3])]</span>
<span class="sd"> >>> sorted(dist.items())</span>
<span class="sd"> [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]</span>
<span class="sd"> >>> pred, dist = nx.bellman_ford_predecessor_and_distance(G, 0, 1)</span>
<span class="sd"> >>> sorted(pred.items())</span>
<span class="sd"> [(0, []), (1, [0]), (2, [1]), (3, [2]), (4, [3])]</span>
<span class="sd"> >>> sorted(dist.items())</span>
<span class="sd"> [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]</span>
<span class="sd"> >>> G = nx.cycle_graph(5, create_using=nx.DiGraph())</span>
<span class="sd"> >>> G[1][2]["weight"] = -7</span>
<span class="sd"> >>> nx.bellman_ford_predecessor_and_distance(G, 0)</span>
<span class="sd"> Traceback (most recent call last):</span>
<span class="sd"> ...</span>
<span class="sd"> networkx.exception.NetworkXUnbounded: Negative cycle detected.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> find_negative_cycle</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> The dictionaries returned only have keys for nodes reachable from</span>
<span class="sd"> the source.</span>
<span class="sd"> In the case where the (di)graph is not connected, if a component</span>
<span class="sd"> not containing the source contains a negative (di)cycle, it</span>
<span class="sd"> will not be detected.</span>
<span class="sd"> In NetworkX v2.1 and prior, the source node had predecessor `[None]`.</span>
<span class="sd"> In NetworkX v2.2 this changed to the source node having predecessor `[]`</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">source</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NodeNotFound</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Node </span><span class="si">{</span><span class="n">source</span><span class="si">}</span><span class="s2"> is not found in the graph"</span><span class="p">)</span>
<span class="n">weight</span> <span class="o">=</span> <span class="n">_weight_function</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">weight</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span> <span class="o"><</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">nx</span><span class="o">.</span><span class="n">selfloop_edges</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="kc">True</span><span class="p">)):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXUnbounded</span><span class="p">(</span><span class="s2">"Negative cycle detected."</span><span class="p">)</span>
<span class="n">dist</span> <span class="o">=</span> <span class="p">{</span><span class="n">source</span><span class="p">:</span> <span class="mi">0</span><span class="p">}</span>
<span class="n">pred</span> <span class="o">=</span> <span class="p">{</span><span class="n">source</span><span class="p">:</span> <span class="p">[]}</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<span class="k">return</span> <span class="n">pred</span><span class="p">,</span> <span class="n">dist</span>
<span class="n">weight</span> <span class="o">=</span> <span class="n">_weight_function</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="n">dist</span> <span class="o">=</span> <span class="n">_bellman_ford</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="p">[</span><span class="n">source</span><span class="p">],</span> <span class="n">weight</span><span class="p">,</span> <span class="n">pred</span><span class="o">=</span><span class="n">pred</span><span class="p">,</span> <span class="n">dist</span><span class="o">=</span><span class="n">dist</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="n">target</span><span class="p">,</span> <span class="n">heuristic</span><span class="o">=</span><span class="n">heuristic</span>
<span class="p">)</span>
<span class="k">return</span> <span class="p">(</span><span class="n">pred</span><span class="p">,</span> <span class="n">dist</span><span class="p">)</span></div>
<span class="k">def</span> <span class="nf">_bellman_ford</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span>
<span class="n">source</span><span class="p">,</span>
<span class="n">weight</span><span class="p">,</span>
<span class="n">pred</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="n">paths</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="n">dist</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="n">target</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="n">heuristic</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
<span class="p">):</span>
<span class="sd">"""Calls relaxation loop for Bellman–Ford algorithm and builds paths</span>
<span class="sd"> This is an implementation of the SPFA variant.</span>
<span class="sd"> See https://en.wikipedia.org/wiki/Shortest_Path_Faster_Algorithm</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> source: list</span>
<span class="sd"> List of source nodes. The shortest path from any of the source</span>
<span class="sd"> nodes will be found if multiple sources are provided.</span>
<span class="sd"> weight : function</span>
<span class="sd"> The weight of an edge is the value returned by the function. The</span>
<span class="sd"> function must accept exactly three positional arguments: the two</span>
<span class="sd"> endpoints of an edge and the dictionary of edge attributes for</span>
<span class="sd"> that edge. The function must return a number.</span>
<span class="sd"> pred: dict of lists, optional (default=None)</span>
<span class="sd"> dict to store a list of predecessors keyed by that node</span>
<span class="sd"> If None, predecessors are not stored</span>
<span class="sd"> paths: dict, optional (default=None)</span>
<span class="sd"> dict to store the path list from source to each node, keyed by node</span>
<span class="sd"> If None, paths are not stored</span>
<span class="sd"> dist: dict, optional (default=None)</span>
<span class="sd"> dict to store distance from source to the keyed node</span>
<span class="sd"> If None, returned dist dict contents default to 0 for every node in the</span>
<span class="sd"> source list</span>
<span class="sd"> target: node label, optional</span>
<span class="sd"> Ending node for path. Path lengths to other destinations may (and</span>
<span class="sd"> probably will) be incorrect.</span>
<span class="sd"> heuristic : bool</span>
<span class="sd"> Determines whether to use a heuristic to early detect negative</span>
<span class="sd"> cycles at a hopefully negligible cost.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> dist : dict</span>
<span class="sd"> Returns a dict keyed by node to the distance from the source.</span>
<span class="sd"> Dicts for paths and pred are in the mutated input dicts by those names.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If any of `source` is not in `G`.</span>
<span class="sd"> NetworkXUnbounded</span>
<span class="sd"> If the (di)graph contains a negative (di)cycle, the</span>
<span class="sd"> algorithm raises an exception to indicate the presence of the</span>
<span class="sd"> negative (di)cycle. Note: any negative weight edge in an</span>
<span class="sd"> undirected graph is a negative cycle</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">pred</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">pred</span> <span class="o">=</span> <span class="p">{</span><span class="n">v</span><span class="p">:</span> <span class="p">[]</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">source</span><span class="p">}</span>
<span class="k">if</span> <span class="n">dist</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">dist</span> <span class="o">=</span> <span class="p">{</span><span class="n">v</span><span class="p">:</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">source</span><span class="p">}</span>
<span class="n">negative_cycle_found</span> <span class="o">=</span> <span class="n">_inner_bellman_ford</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span>
<span class="n">source</span><span class="p">,</span>
<span class="n">weight</span><span class="p">,</span>
<span class="n">pred</span><span class="p">,</span>
<span class="n">dist</span><span class="p">,</span>
<span class="n">heuristic</span><span class="p">,</span>
<span class="p">)</span>
<span class="k">if</span> <span class="n">negative_cycle_found</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXUnbounded</span><span class="p">(</span><span class="s2">"Negative cycle detected."</span><span class="p">)</span>
<span class="k">if</span> <span class="n">paths</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">sources</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">source</span><span class="p">)</span>
<span class="n">dsts</span> <span class="o">=</span> <span class="p">[</span><span class="n">target</span><span class="p">]</span> <span class="k">if</span> <span class="n">target</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">pred</span>
<span class="k">for</span> <span class="n">dst</span> <span class="ow">in</span> <span class="n">dsts</span><span class="p">:</span>
<span class="n">gen</span> <span class="o">=</span> <span class="n">_build_paths_from_predecessors</span><span class="p">(</span><span class="n">sources</span><span class="p">,</span> <span class="n">dst</span><span class="p">,</span> <span class="n">pred</span><span class="p">)</span>
<span class="n">paths</span><span class="p">[</span><span class="n">dst</span><span class="p">]</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">gen</span><span class="p">)</span>
<span class="k">return</span> <span class="n">dist</span>
<span class="k">def</span> <span class="nf">_inner_bellman_ford</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span>
<span class="n">sources</span><span class="p">,</span>
<span class="n">weight</span><span class="p">,</span>
<span class="n">pred</span><span class="p">,</span>
<span class="n">dist</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span>
<span class="n">heuristic</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
<span class="p">):</span>
<span class="sd">"""Inner Relaxation loop for Bellman–Ford algorithm.</span>
<span class="sd"> This is an implementation of the SPFA variant.</span>
<span class="sd"> See https://en.wikipedia.org/wiki/Shortest_Path_Faster_Algorithm</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> source: list</span>
<span class="sd"> List of source nodes. The shortest path from any of the source</span>
<span class="sd"> nodes will be found if multiple sources are provided.</span>
<span class="sd"> weight : function</span>
<span class="sd"> The weight of an edge is the value returned by the function. The</span>
<span class="sd"> function must accept exactly three positional arguments: the two</span>
<span class="sd"> endpoints of an edge and the dictionary of edge attributes for</span>
<span class="sd"> that edge. The function must return a number.</span>
<span class="sd"> pred: dict of lists</span>
<span class="sd"> dict to store a list of predecessors keyed by that node</span>
<span class="sd"> dist: dict, optional (default=None)</span>
<span class="sd"> dict to store distance from source to the keyed node</span>
<span class="sd"> If None, returned dist dict contents default to 0 for every node in the</span>
<span class="sd"> source list</span>
<span class="sd"> heuristic : bool</span>
<span class="sd"> Determines whether to use a heuristic to early detect negative</span>
<span class="sd"> cycles at a hopefully negligible cost.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> node or None</span>
<span class="sd"> Return a node `v` where processing discovered a negative cycle.</span>
<span class="sd"> If no negative cycle found, return None.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If any of `source` is not in `G`.</span>
<span class="sd"> """</span>
<span class="k">for</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">sources</span><span class="p">:</span>
<span class="k">if</span> <span class="n">s</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NodeNotFound</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Source </span><span class="si">{</span><span class="n">s</span><span class="si">}</span><span class="s2"> not in G"</span><span class="p">)</span>
<span class="k">if</span> <span class="n">pred</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">pred</span> <span class="o">=</span> <span class="p">{</span><span class="n">v</span><span class="p">:</span> <span class="p">[]</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">sources</span><span class="p">}</span>
<span class="k">if</span> <span class="n">dist</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">dist</span> <span class="o">=</span> <span class="p">{</span><span class="n">v</span><span class="p">:</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">sources</span><span class="p">}</span>
<span class="c1"># Heuristic Storage setup. Note: use None because nodes cannot be None</span>
<span class="n">nonexistent_edge</span> <span class="o">=</span> <span class="p">(</span><span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">)</span>
<span class="n">pred_edge</span> <span class="o">=</span> <span class="p">{</span><span class="n">v</span><span class="p">:</span> <span class="kc">None</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">sources</span><span class="p">}</span>
<span class="n">recent_update</span> <span class="o">=</span> <span class="p">{</span><span class="n">v</span><span class="p">:</span> <span class="n">nonexistent_edge</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">sources</span><span class="p">}</span>
<span class="n">G_succ</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">_adj</span> <span class="c1"># For speed-up (and works for both directed and undirected graphs)</span>
<span class="n">inf</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="s2">"inf"</span><span class="p">)</span>
<span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">count</span> <span class="o">=</span> <span class="p">{}</span>
<span class="n">q</span> <span class="o">=</span> <span class="n">deque</span><span class="p">(</span><span class="n">sources</span><span class="p">)</span>
<span class="n">in_q</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">sources</span><span class="p">)</span>
<span class="k">while</span> <span class="n">q</span><span class="p">:</span>
<span class="n">u</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">popleft</span><span class="p">()</span>
<span class="n">in_q</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
<span class="c1"># Skip relaxations if any of the predecessors of u is in the queue.</span>
<span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">pred_u</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">in_q</span> <span class="k">for</span> <span class="n">pred_u</span> <span class="ow">in</span> <span class="n">pred</span><span class="p">[</span><span class="n">u</span><span class="p">]):</span>
<span class="n">dist_u</span> <span class="o">=</span> <span class="n">dist</span><span class="p">[</span><span class="n">u</span><span class="p">]</span>
<span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">G_succ</span><span class="p">[</span><span class="n">u</span><span class="p">]</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
<span class="n">dist_v</span> <span class="o">=</span> <span class="n">dist_u</span> <span class="o">+</span> <span class="n">weight</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
<span class="k">if</span> <span class="n">dist_v</span> <span class="o"><</span> <span class="n">dist</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">inf</span><span class="p">):</span>
<span class="c1"># In this conditional branch we are updating the path with v.</span>
<span class="c1"># If it happens that some earlier update also added node v</span>
<span class="c1"># that implies the existence of a negative cycle since</span>
<span class="c1"># after the update node v would lie on the update path twice.</span>
<span class="c1"># The update path is stored up to one of the source nodes,</span>
<span class="c1"># therefore u is always in the dict recent_update</span>
<span class="k">if</span> <span class="n">heuristic</span><span class="p">:</span>
<span class="k">if</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">recent_update</span><span class="p">[</span><span class="n">u</span><span class="p">]:</span>
<span class="c1"># Negative cycle found!</span>
<span class="n">pred</span><span class="p">[</span><span class="n">v</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
<span class="k">return</span> <span class="n">v</span>
<span class="c1"># Transfer the recent update info from u to v if the</span>
<span class="c1"># same source node is the head of the update path.</span>
<span class="c1"># If the source node is responsible for the cost update,</span>
<span class="c1"># then clear the history and use it instead.</span>
<span class="k">if</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">pred_edge</span> <span class="ow">and</span> <span class="n">pred_edge</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">==</span> <span class="n">u</span><span class="p">:</span>
<span class="n">recent_update</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">recent_update</span><span class="p">[</span><span class="n">u</span><span class="p">]</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">recent_update</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="k">if</span> <span class="n">v</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">in_q</span><span class="p">:</span>
<span class="n">q</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="n">in_q</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="n">count_v</span> <span class="o">=</span> <span class="n">count</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
<span class="k">if</span> <span class="n">count_v</span> <span class="o">==</span> <span class="n">n</span><span class="p">:</span>
<span class="c1"># Negative cycle found!</span>
<span class="k">return</span> <span class="n">v</span>
<span class="n">count</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">count_v</span>
<span class="n">dist</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">dist_v</span>
<span class="n">pred</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="n">u</span><span class="p">]</span>
<span class="n">pred_edge</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">u</span>
<span class="k">elif</span> <span class="n">dist</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="ow">and</span> <span class="n">dist_v</span> <span class="o">==</span> <span class="n">dist</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">v</span><span class="p">):</span>
<span class="n">pred</span><span class="p">[</span><span class="n">v</span><span class="p">]</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
<span class="c1"># successfully found shortest_path. No negative cycles found.</span>
<span class="k">return</span> <span class="kc">None</span>
<div class="viewcode-block" id="bellman_ford_path"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.bellman_ford_path.html#networkx.algorithms.shortest_paths.weighted.bellman_ford_path">[docs]</a><span class="k">def</span> <span class="nf">bellman_ford_path</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Returns the shortest path from source to target in a weighted graph G.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> source : node</span>
<span class="sd"> Starting node</span>
<span class="sd"> target : node</span>
<span class="sd"> Ending node</span>
<span class="sd"> weight : string or function (default="weight")</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> path : list</span>
<span class="sd"> List of nodes in a shortest path.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If `source` is not in `G`.</span>
<span class="sd"> NetworkXNoPath</span>
<span class="sd"> If no path exists between source and target.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> nx.bellman_ford_path(G, 0, 4)</span>
<span class="sd"> [0, 1, 2, 3, 4]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> dijkstra_path, bellman_ford_path_length</span>
<span class="sd"> """</span>
<span class="n">length</span><span class="p">,</span> <span class="n">path</span> <span class="o">=</span> <span class="n">single_source_bellman_ford</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="n">target</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)</span>
<span class="k">return</span> <span class="n">path</span></div>
<div class="viewcode-block" id="bellman_ford_path_length"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.bellman_ford_path_length.html#networkx.algorithms.shortest_paths.weighted.bellman_ford_path_length">[docs]</a><span class="k">def</span> <span class="nf">bellman_ford_path_length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Returns the shortest path length from source to target</span>
<span class="sd"> in a weighted graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> source : node label</span>
<span class="sd"> starting node for path</span>
<span class="sd"> target : node label</span>
<span class="sd"> ending node for path</span>
<span class="sd"> weight : string or function (default="weight")</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> length : number</span>
<span class="sd"> Shortest path length.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If `source` is not in `G`.</span>
<span class="sd"> NetworkXNoPath</span>
<span class="sd"> If no path exists between source and target.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> nx.bellman_ford_path_length(G, 0, 4)</span>
<span class="sd"> 4</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> dijkstra_path_length, bellman_ford_path</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">source</span> <span class="o">==</span> <span class="n">target</span><span class="p">:</span>
<span class="k">if</span> <span class="n">source</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NodeNotFound</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Node </span><span class="si">{</span><span class="n">source</span><span class="si">}</span><span class="s2"> not found in graph"</span><span class="p">)</span>
<span class="k">return</span> <span class="mi">0</span>
<span class="n">weight</span> <span class="o">=</span> <span class="n">_weight_function</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="n">length</span> <span class="o">=</span> <span class="n">_bellman_ford</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="p">[</span><span class="n">source</span><span class="p">],</span> <span class="n">weight</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="n">target</span><span class="p">)</span>
<span class="k">try</span><span class="p">:</span>
<span class="k">return</span> <span class="n">length</span><span class="p">[</span><span class="n">target</span><span class="p">]</span>
<span class="k">except</span> <span class="ne">KeyError</span> <span class="k">as</span> <span class="n">err</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXNoPath</span><span class="p">(</span><span class="sa">f</span><span class="s2">"node </span><span class="si">{</span><span class="n">target</span><span class="si">}</span><span class="s2"> not reachable from </span><span class="si">{</span><span class="n">source</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span> <span class="kn">from</span> <span class="nn">err</span></div>
<div class="viewcode-block" id="single_source_bellman_ford_path"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.single_source_bellman_ford_path.html#networkx.algorithms.shortest_paths.weighted.single_source_bellman_ford_path">[docs]</a><span class="k">def</span> <span class="nf">single_source_bellman_ford_path</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Compute shortest path between source and all other reachable</span>
<span class="sd"> nodes for a weighted graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> source : node</span>
<span class="sd"> Starting node for path.</span>
<span class="sd"> weight : string or function (default="weight")</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> paths : dictionary</span>
<span class="sd"> Dictionary of shortest path lengths keyed by target.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If `source` is not in `G`.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> path = nx.single_source_bellman_ford_path(G, 0)</span>
<span class="sd"> >>> path[4]</span>
<span class="sd"> [0, 1, 2, 3, 4]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> single_source_dijkstra, single_source_bellman_ford</span>
<span class="sd"> """</span>
<span class="p">(</span><span class="n">length</span><span class="p">,</span> <span class="n">path</span><span class="p">)</span> <span class="o">=</span> <span class="n">single_source_bellman_ford</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)</span>
<span class="k">return</span> <span class="n">path</span></div>
<div class="viewcode-block" id="single_source_bellman_ford_path_length"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.single_source_bellman_ford_path_length.html#networkx.algorithms.shortest_paths.weighted.single_source_bellman_ford_path_length">[docs]</a><span class="k">def</span> <span class="nf">single_source_bellman_ford_path_length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Compute the shortest path length between source and all other</span>
<span class="sd"> reachable nodes for a weighted graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> source : node label</span>
<span class="sd"> Starting node for path</span>
<span class="sd"> weight : string or function (default="weight")</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> length : iterator</span>
<span class="sd"> (target, shortest path length) iterator</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If `source` is not in `G`.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> length = dict(nx.single_source_bellman_ford_path_length(G, 0))</span>
<span class="sd"> >>> length[4]</span>
<span class="sd"> 4</span>
<span class="sd"> >>> for node in [0, 1, 2, 3, 4]:</span>
<span class="sd"> ... print(f"{node}: {length[node]}")</span>
<span class="sd"> 0: 0</span>
<span class="sd"> 1: 1</span>
<span class="sd"> 2: 2</span>
<span class="sd"> 3: 3</span>
<span class="sd"> 4: 4</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> single_source_dijkstra, single_source_bellman_ford</span>
<span class="sd"> """</span>
<span class="n">weight</span> <span class="o">=</span> <span class="n">_weight_function</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="k">return</span> <span class="n">_bellman_ford</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="p">[</span><span class="n">source</span><span class="p">],</span> <span class="n">weight</span><span class="p">)</span></div>
<div class="viewcode-block" id="single_source_bellman_ford"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.single_source_bellman_ford.html#networkx.algorithms.shortest_paths.weighted.single_source_bellman_ford">[docs]</a><span class="k">def</span> <span class="nf">single_source_bellman_ford</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Compute shortest paths and lengths in a weighted graph G.</span>
<span class="sd"> Uses Bellman-Ford algorithm for shortest paths.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> source : node label</span>
<span class="sd"> Starting node for path</span>
<span class="sd"> target : node label, optional</span>
<span class="sd"> Ending node for path</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> distance, path : pair of dictionaries, or numeric and list</span>
<span class="sd"> If target is None, returns a tuple of two dictionaries keyed by node.</span>
<span class="sd"> The first dictionary stores distance from one of the source nodes.</span>
<span class="sd"> The second stores the path from one of the sources to that node.</span>
<span class="sd"> If target is not None, returns a tuple of (distance, path) where</span>
<span class="sd"> distance is the distance from source to target and path is a list</span>
<span class="sd"> representing the path from source to target.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If `source` is not in `G`.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> length, path = nx.single_source_bellman_ford(G, 0)</span>
<span class="sd"> >>> length[4]</span>
<span class="sd"> 4</span>
<span class="sd"> >>> for node in [0, 1, 2, 3, 4]:</span>
<span class="sd"> ... print(f"{node}: {length[node]}")</span>
<span class="sd"> 0: 0</span>
<span class="sd"> 1: 1</span>
<span class="sd"> 2: 2</span>
<span class="sd"> 3: 3</span>
<span class="sd"> 4: 4</span>
<span class="sd"> >>> path[4]</span>
<span class="sd"> [0, 1, 2, 3, 4]</span>
<span class="sd"> >>> length, path = nx.single_source_bellman_ford(G, 0, 1)</span>
<span class="sd"> >>> length</span>
<span class="sd"> 1</span>
<span class="sd"> >>> path</span>
<span class="sd"> [0, 1]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> single_source_dijkstra</span>
<span class="sd"> single_source_bellman_ford_path</span>
<span class="sd"> single_source_bellman_ford_path_length</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">source</span> <span class="o">==</span> <span class="n">target</span><span class="p">:</span>
<span class="k">if</span> <span class="n">source</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NodeNotFound</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Node </span><span class="si">{</span><span class="n">source</span><span class="si">}</span><span class="s2"> is not found in the graph"</span><span class="p">)</span>
<span class="k">return</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">[</span><span class="n">source</span><span class="p">])</span>
<span class="n">weight</span> <span class="o">=</span> <span class="n">_weight_function</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="n">paths</span> <span class="o">=</span> <span class="p">{</span><span class="n">source</span><span class="p">:</span> <span class="p">[</span><span class="n">source</span><span class="p">]}</span> <span class="c1"># dictionary of paths</span>
<span class="n">dist</span> <span class="o">=</span> <span class="n">_bellman_ford</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="p">[</span><span class="n">source</span><span class="p">],</span> <span class="n">weight</span><span class="p">,</span> <span class="n">paths</span><span class="o">=</span><span class="n">paths</span><span class="p">,</span> <span class="n">target</span><span class="o">=</span><span class="n">target</span><span class="p">)</span>
<span class="k">if</span> <span class="n">target</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">return</span> <span class="p">(</span><span class="n">dist</span><span class="p">,</span> <span class="n">paths</span><span class="p">)</span>
<span class="k">try</span><span class="p">:</span>
<span class="k">return</span> <span class="p">(</span><span class="n">dist</span><span class="p">[</span><span class="n">target</span><span class="p">],</span> <span class="n">paths</span><span class="p">[</span><span class="n">target</span><span class="p">])</span>
<span class="k">except</span> <span class="ne">KeyError</span> <span class="k">as</span> <span class="n">err</span><span class="p">:</span>
<span class="n">msg</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"Node </span><span class="si">{</span><span class="n">target</span><span class="si">}</span><span class="s2"> not reachable from </span><span class="si">{</span><span class="n">source</span><span class="si">}</span><span class="s2">"</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXNoPath</span><span class="p">(</span><span class="n">msg</span><span class="p">)</span> <span class="kn">from</span> <span class="nn">err</span></div>
<div class="viewcode-block" id="all_pairs_bellman_ford_path_length"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.all_pairs_bellman_ford_path_length.html#networkx.algorithms.shortest_paths.weighted.all_pairs_bellman_ford_path_length">[docs]</a><span class="k">def</span> <span class="nf">all_pairs_bellman_ford_path_length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Compute shortest path lengths between all nodes in a weighted graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> weight : string or function (default="weight")</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> distance : iterator</span>
<span class="sd"> (source, dictionary) iterator with dictionary keyed by target and</span>
<span class="sd"> shortest path length as the key value.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> length = dict(nx.all_pairs_bellman_ford_path_length(G))</span>
<span class="sd"> >>> for node in [0, 1, 2, 3, 4]:</span>
<span class="sd"> ... print(f"1 - {node}: {length[1][node]}")</span>
<span class="sd"> 1 - 0: 1</span>
<span class="sd"> 1 - 1: 0</span>
<span class="sd"> 1 - 2: 1</span>
<span class="sd"> 1 - 3: 2</span>
<span class="sd"> 1 - 4: 3</span>
<span class="sd"> >>> length[3][2]</span>
<span class="sd"> 1</span>
<span class="sd"> >>> length[2][2]</span>
<span class="sd"> 0</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> The dictionary returned only has keys for reachable node pairs.</span>
<span class="sd"> """</span>
<span class="n">length</span> <span class="o">=</span> <span class="n">single_source_bellman_ford_path_length</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">yield</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="nb">dict</span><span class="p">(</span><span class="n">length</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">)))</span></div>
<div class="viewcode-block" id="all_pairs_bellman_ford_path"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.all_pairs_bellman_ford_path.html#networkx.algorithms.shortest_paths.weighted.all_pairs_bellman_ford_path">[docs]</a><span class="k">def</span> <span class="nf">all_pairs_bellman_ford_path</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Compute shortest paths between all nodes in a weighted graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> weight : string or function (default="weight")</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> distance : dictionary</span>
<span class="sd"> Dictionary, keyed by source and target, of shortest paths.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> path = dict(nx.all_pairs_bellman_ford_path(G))</span>
<span class="sd"> >>> path[0][4]</span>
<span class="sd"> [0, 1, 2, 3, 4]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> floyd_warshall, all_pairs_dijkstra_path</span>
<span class="sd"> """</span>
<span class="n">path</span> <span class="o">=</span> <span class="n">single_source_bellman_ford_path</span>
<span class="c1"># TODO This can be trivially parallelized.</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">yield</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">path</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">))</span></div>
<div class="viewcode-block" id="goldberg_radzik"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.goldberg_radzik.html#networkx.algorithms.shortest_paths.weighted.goldberg_radzik">[docs]</a><span class="k">def</span> <span class="nf">goldberg_radzik</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Compute shortest path lengths and predecessors on shortest paths</span>
<span class="sd"> in weighted graphs.</span>
<span class="sd"> The algorithm has a running time of $O(mn)$ where $n$ is the number of</span>
<span class="sd"> nodes and $m$ is the number of edges. It is slower than Dijkstra but</span>
<span class="sd"> can handle negative edge weights.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> The algorithm works for all types of graphs, including directed</span>
<span class="sd"> graphs and multigraphs.</span>
<span class="sd"> source: node label</span>
<span class="sd"> Starting node for path</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> pred, dist : dictionaries</span>
<span class="sd"> Returns two dictionaries keyed by node to predecessor in the</span>
<span class="sd"> path and to the distance from the source respectively.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If `source` is not in `G`.</span>
<span class="sd"> NetworkXUnbounded</span>
<span class="sd"> If the (di)graph contains a negative (di)cycle, the</span>
<span class="sd"> algorithm raises an exception to indicate the presence of the</span>
<span class="sd"> negative (di)cycle. Note: any negative weight edge in an</span>
<span class="sd"> undirected graph is a negative cycle.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5, create_using=nx.DiGraph())</span>
<span class="sd"> >>> pred, dist = nx.goldberg_radzik(G, 0)</span>
<span class="sd"> >>> sorted(pred.items())</span>
<span class="sd"> [(0, None), (1, 0), (2, 1), (3, 2), (4, 3)]</span>
<span class="sd"> >>> sorted(dist.items())</span>
<span class="sd"> [(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)]</span>
<span class="sd"> >>> G = nx.cycle_graph(5, create_using=nx.DiGraph())</span>
<span class="sd"> >>> G[1][2]["weight"] = -7</span>
<span class="sd"> >>> nx.goldberg_radzik(G, 0)</span>
<span class="sd"> Traceback (most recent call last):</span>
<span class="sd"> ...</span>
<span class="sd"> networkx.exception.NetworkXUnbounded: Negative cycle detected.</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> The dictionaries returned only have keys for nodes reachable from</span>
<span class="sd"> the source.</span>
<span class="sd"> In the case where the (di)graph is not connected, if a component</span>
<span class="sd"> not containing the source contains a negative (di)cycle, it</span>
<span class="sd"> will not be detected.</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">source</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NodeNotFound</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Node </span><span class="si">{</span><span class="n">source</span><span class="si">}</span><span class="s2"> is not found in the graph"</span><span class="p">)</span>
<span class="n">weight</span> <span class="o">=</span> <span class="n">_weight_function</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">any</span><span class="p">(</span><span class="n">weight</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span> <span class="o"><</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">nx</span><span class="o">.</span><span class="n">selfloop_edges</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="kc">True</span><span class="p">)):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXUnbounded</span><span class="p">(</span><span class="s2">"Negative cycle detected."</span><span class="p">)</span>
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<span class="k">return</span> <span class="p">{</span><span class="n">source</span><span class="p">:</span> <span class="kc">None</span><span class="p">},</span> <span class="p">{</span><span class="n">source</span><span class="p">:</span> <span class="mi">0</span><span class="p">}</span>
<span class="n">G_succ</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">_adj</span> <span class="c1"># For speed-up (and works for both directed and undirected graphs)</span>
<span class="n">inf</span> <span class="o">=</span> <span class="nb">float</span><span class="p">(</span><span class="s2">"inf"</span><span class="p">)</span>
<span class="n">d</span> <span class="o">=</span> <span class="p">{</span><span class="n">u</span><span class="p">:</span> <span class="n">inf</span> <span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">G</span><span class="p">}</span>
<span class="n">d</span><span class="p">[</span><span class="n">source</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">pred</span> <span class="o">=</span> <span class="p">{</span><span class="n">source</span><span class="p">:</span> <span class="kc">None</span><span class="p">}</span>
<span class="k">def</span> <span class="nf">topo_sort</span><span class="p">(</span><span class="n">relabeled</span><span class="p">):</span>
<span class="sd">"""Topologically sort nodes relabeled in the previous round and detect</span>
<span class="sd"> negative cycles.</span>
<span class="sd"> """</span>
<span class="c1"># List of nodes to scan in this round. Denoted by A in Goldberg and</span>
<span class="c1"># Radzik's paper.</span>
<span class="n">to_scan</span> <span class="o">=</span> <span class="p">[]</span>
<span class="c1"># In the DFS in the loop below, neg_count records for each node the</span>
<span class="c1"># number of edges of negative reduced costs on the path from a DFS root</span>
<span class="c1"># to the node in the DFS forest. The reduced cost of an edge (u, v) is</span>
<span class="c1"># defined as d[u] + weight[u][v] - d[v].</span>
<span class="c1">#</span>
<span class="c1"># neg_count also doubles as the DFS visit marker array.</span>
<span class="n">neg_count</span> <span class="o">=</span> <span class="p">{}</span>
<span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">relabeled</span><span class="p">:</span>
<span class="c1"># Skip visited nodes.</span>
<span class="k">if</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">neg_count</span><span class="p">:</span>
<span class="k">continue</span>
<span class="n">d_u</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="n">u</span><span class="p">]</span>
<span class="c1"># Skip nodes without out-edges of negative reduced costs.</span>
<span class="k">if</span> <span class="nb">all</span><span class="p">(</span><span class="n">d_u</span> <span class="o">+</span> <span class="n">weight</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span> <span class="o">>=</span> <span class="n">d</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">G_succ</span><span class="p">[</span><span class="n">u</span><span class="p">]</span><span class="o">.</span><span class="n">items</span><span class="p">()):</span>
<span class="k">continue</span>
<span class="c1"># Nonrecursive DFS that inserts nodes reachable from u via edges of</span>
<span class="c1"># nonpositive reduced costs into to_scan in (reverse) topological</span>
<span class="c1"># order.</span>
<span class="n">stack</span> <span class="o">=</span> <span class="p">[(</span><span class="n">u</span><span class="p">,</span> <span class="nb">iter</span><span class="p">(</span><span class="n">G_succ</span><span class="p">[</span><span class="n">u</span><span class="p">]</span><span class="o">.</span><span class="n">items</span><span class="p">()))]</span>
<span class="n">in_stack</span> <span class="o">=</span> <span class="p">{</span><span class="n">u</span><span class="p">}</span>
<span class="n">neg_count</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">while</span> <span class="n">stack</span><span class="p">:</span>
<span class="n">u</span><span class="p">,</span> <span class="n">it</span> <span class="o">=</span> <span class="n">stack</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">v</span><span class="p">,</span> <span class="n">e</span> <span class="o">=</span> <span class="nb">next</span><span class="p">(</span><span class="n">it</span><span class="p">)</span>
<span class="k">except</span> <span class="ne">StopIteration</span><span class="p">:</span>
<span class="n">to_scan</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
<span class="n">stack</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="n">in_stack</span><span class="o">.</span><span class="n">remove</span><span class="p">(</span><span class="n">u</span><span class="p">)</span>
<span class="k">continue</span>
<span class="n">t</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="o">+</span> <span class="n">weight</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
<span class="n">d_v</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="n">v</span><span class="p">]</span>
<span class="k">if</span> <span class="n">t</span> <span class="o"><=</span> <span class="n">d_v</span><span class="p">:</span>
<span class="n">is_neg</span> <span class="o">=</span> <span class="n">t</span> <span class="o"><</span> <span class="n">d_v</span>
<span class="n">d</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">t</span>
<span class="n">pred</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">u</span>
<span class="k">if</span> <span class="n">v</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">neg_count</span><span class="p">:</span>
<span class="n">neg_count</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">neg_count</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="o">+</span> <span class="nb">int</span><span class="p">(</span><span class="n">is_neg</span><span class="p">)</span>
<span class="n">stack</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">v</span><span class="p">,</span> <span class="nb">iter</span><span class="p">(</span><span class="n">G_succ</span><span class="p">[</span><span class="n">v</span><span class="p">]</span><span class="o">.</span><span class="n">items</span><span class="p">())))</span>
<span class="n">in_stack</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="k">elif</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">in_stack</span> <span class="ow">and</span> <span class="n">neg_count</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="o">+</span> <span class="nb">int</span><span class="p">(</span><span class="n">is_neg</span><span class="p">)</span> <span class="o">></span> <span class="n">neg_count</span><span class="p">[</span><span class="n">v</span><span class="p">]:</span>
<span class="c1"># (u, v) is a back edge, and the cycle formed by the</span>
<span class="c1"># path v to u and (u, v) contains at least one edge of</span>
<span class="c1"># negative reduced cost. The cycle must be of negative</span>
<span class="c1"># cost.</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXUnbounded</span><span class="p">(</span><span class="s2">"Negative cycle detected."</span><span class="p">)</span>
<span class="n">to_scan</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
<span class="k">return</span> <span class="n">to_scan</span>
<span class="k">def</span> <span class="nf">relax</span><span class="p">(</span><span class="n">to_scan</span><span class="p">):</span>
<span class="sd">"""Relax out-edges of relabeled nodes."""</span>
<span class="n">relabeled</span> <span class="o">=</span> <span class="nb">set</span><span class="p">()</span>
<span class="c1"># Scan nodes in to_scan in topological order and relax incident</span>
<span class="c1"># out-edges. Add the relabled nodes to labeled.</span>
<span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">to_scan</span><span class="p">:</span>
<span class="n">d_u</span> <span class="o">=</span> <span class="n">d</span><span class="p">[</span><span class="n">u</span><span class="p">]</span>
<span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">e</span> <span class="ow">in</span> <span class="n">G_succ</span><span class="p">[</span><span class="n">u</span><span class="p">]</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
<span class="n">w_e</span> <span class="o">=</span> <span class="n">weight</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span>
<span class="k">if</span> <span class="n">d_u</span> <span class="o">+</span> <span class="n">w_e</span> <span class="o"><</span> <span class="n">d</span><span class="p">[</span><span class="n">v</span><span class="p">]:</span>
<span class="n">d</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">d_u</span> <span class="o">+</span> <span class="n">w_e</span>
<span class="n">pred</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">u</span>
<span class="n">relabeled</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">v</span><span class="p">)</span>
<span class="k">return</span> <span class="n">relabeled</span>
<span class="c1"># Set of nodes relabled in the last round of scan operations. Denoted by B</span>
<span class="c1"># in Goldberg and Radzik's paper.</span>
<span class="n">relabeled</span> <span class="o">=</span> <span class="p">{</span><span class="n">source</span><span class="p">}</span>
<span class="k">while</span> <span class="n">relabeled</span><span class="p">:</span>
<span class="n">to_scan</span> <span class="o">=</span> <span class="n">topo_sort</span><span class="p">(</span><span class="n">relabeled</span><span class="p">)</span>
<span class="n">relabeled</span> <span class="o">=</span> <span class="n">relax</span><span class="p">(</span><span class="n">to_scan</span><span class="p">)</span>
<span class="n">d</span> <span class="o">=</span> <span class="p">{</span><span class="n">u</span><span class="p">:</span> <span class="n">d</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="k">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">pred</span><span class="p">}</span>
<span class="k">return</span> <span class="n">pred</span><span class="p">,</span> <span class="n">d</span></div>
<div class="viewcode-block" id="negative_edge_cycle"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.negative_edge_cycle.html#networkx.algorithms.shortest_paths.weighted.negative_edge_cycle">[docs]</a><span class="k">def</span> <span class="nf">negative_edge_cycle</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">,</span> <span class="n">heuristic</span><span class="o">=</span><span class="kc">True</span><span class="p">):</span>
<span class="sd">"""Returns True if there exists a negative edge cycle anywhere in G.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number.</span>
<span class="sd"> heuristic : bool</span>
<span class="sd"> Determines whether to use a heuristic to early detect negative</span>
<span class="sd"> cycles at a negligible cost. In case of graphs with a negative cycle,</span>
<span class="sd"> the performance of detection increases by at least an order of magnitude.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> negative_cycle : bool</span>
<span class="sd"> True if a negative edge cycle exists, otherwise False.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.cycle_graph(5, create_using=nx.DiGraph())</span>
<span class="sd"> >>> print(nx.negative_edge_cycle(G))</span>
<span class="sd"> False</span>
<span class="sd"> >>> G[1][2]["weight"] = -7</span>
<span class="sd"> >>> print(nx.negative_edge_cycle(G))</span>
<span class="sd"> True</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> This algorithm uses bellman_ford_predecessor_and_distance() but finds</span>
<span class="sd"> negative cycles on any component by first adding a new node connected to</span>
<span class="sd"> every node, and starting bellman_ford_predecessor_and_distance on that</span>
<span class="sd"> node. It then removes that extra node.</span>
<span class="sd"> """</span>
<span class="c1"># find unused node to use temporarily</span>
<span class="n">newnode</span> <span class="o">=</span> <span class="o">-</span><span class="mi">1</span>
<span class="k">while</span> <span class="n">newnode</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="n">newnode</span> <span class="o">-=</span> <span class="mi">1</span>
<span class="c1"># connect it to all nodes</span>
<span class="n">G</span><span class="o">.</span><span class="n">add_edges_from</span><span class="p">([(</span><span class="n">newnode</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">G</span><span class="p">])</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">bellman_ford_predecessor_and_distance</span><span class="p">(</span>
<span class="n">G</span><span class="p">,</span> <span class="n">newnode</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">,</span> <span class="n">heuristic</span><span class="o">=</span><span class="n">heuristic</span>
<span class="p">)</span>
<span class="k">except</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXUnbounded</span><span class="p">:</span>
<span class="k">return</span> <span class="kc">True</span>
<span class="k">finally</span><span class="p">:</span>
<span class="n">G</span><span class="o">.</span><span class="n">remove_node</span><span class="p">(</span><span class="n">newnode</span><span class="p">)</span>
<span class="k">return</span> <span class="kc">False</span></div>
<div class="viewcode-block" id="find_negative_cycle"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.find_negative_cycle.html#networkx.algorithms.shortest_paths.weighted.find_negative_cycle">[docs]</a><span class="k">def</span> <span class="nf">find_negative_cycle</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sd">"""Returns a cycle with negative total weight if it exists.</span>
<span class="sd"> Bellman-Ford is used to find shortest_paths. That algorithm</span>
<span class="sd"> stops if there exists a negative cycle. This algorithm</span>
<span class="sd"> picks up from there and returns the found negative cycle.</span>
<span class="sd"> The cycle consists of a list of nodes in the cycle order. The last</span>
<span class="sd"> node equals the first to make it a cycle.</span>
<span class="sd"> You can look up the edge weights in the original graph. In the case</span>
<span class="sd"> of multigraphs the relevant edge is the minimal weight edge between</span>
<span class="sd"> the nodes in the 2-tuple.</span>
<span class="sd"> If the graph has no negative cycle, a NetworkXError is raised.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> source: node label</span>
<span class="sd"> The search for the negative cycle will start from this node.</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.DiGraph()</span>
<span class="sd"> >>> G.add_weighted_edges_from([(0, 1, 2), (1, 2, 2), (2, 0, 1), (1, 4, 2), (4, 0, -5)])</span>
<span class="sd"> >>> nx.find_negative_cycle(G, 0)</span>
<span class="sd"> [4, 0, 1, 4]</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> cycle : list</span>
<span class="sd"> A list of nodes in the order of the cycle found. The last node</span>
<span class="sd"> equals the first to indicate a cycle.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> If no negative cycle is found.</span>
<span class="sd"> """</span>
<span class="n">weight</span> <span class="o">=</span> <span class="n">_weight_function</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="n">pred</span> <span class="o">=</span> <span class="p">{</span><span class="n">source</span><span class="p">:</span> <span class="p">[]}</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">_inner_bellman_ford</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="p">[</span><span class="n">source</span><span class="p">],</span> <span class="n">weight</span><span class="p">,</span> <span class="n">pred</span><span class="o">=</span><span class="n">pred</span><span class="p">)</span>
<span class="k">if</span> <span class="n">v</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"No negative cycles detected."</span><span class="p">)</span>
<span class="c1"># negative cycle detected... find it</span>
<span class="n">neg_cycle</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">stack</span> <span class="o">=</span> <span class="p">[(</span><span class="n">v</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">pred</span><span class="p">[</span><span class="n">v</span><span class="p">]))]</span>
<span class="n">seen</span> <span class="o">=</span> <span class="p">{</span><span class="n">v</span><span class="p">}</span>
<span class="k">while</span> <span class="n">stack</span><span class="p">:</span>
<span class="n">node</span><span class="p">,</span> <span class="n">preds</span> <span class="o">=</span> <span class="n">stack</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="k">if</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">preds</span><span class="p">:</span>
<span class="c1"># found the cycle</span>
<span class="n">neg_cycle</span><span class="o">.</span><span class="n">extend</span><span class="p">([</span><span class="n">node</span><span class="p">,</span> <span class="n">v</span><span class="p">])</span>
<span class="n">neg_cycle</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="nb">reversed</span><span class="p">(</span><span class="n">neg_cycle</span><span class="p">))</span>
<span class="k">return</span> <span class="n">neg_cycle</span>
<span class="k">if</span> <span class="n">preds</span><span class="p">:</span>
<span class="n">nbr</span> <span class="o">=</span> <span class="n">preds</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="k">if</span> <span class="n">nbr</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">seen</span><span class="p">:</span>
<span class="n">stack</span><span class="o">.</span><span class="n">append</span><span class="p">((</span><span class="n">nbr</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">pred</span><span class="p">[</span><span class="n">nbr</span><span class="p">])))</span>
<span class="n">neg_cycle</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">node</span><span class="p">)</span>
<span class="n">seen</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">nbr</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">stack</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="k">if</span> <span class="n">neg_cycle</span><span class="p">:</span>
<span class="n">neg_cycle</span><span class="o">.</span><span class="n">pop</span><span class="p">()</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">if</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">v</span><span class="p">]</span> <span class="ow">and</span> <span class="n">weight</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="o"><</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="p">[</span><span class="n">v</span><span class="p">,</span> <span class="n">v</span><span class="p">]</span>
<span class="c1"># should not reach here</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"Negative cycle is detected but not found"</span><span class="p">)</span>
<span class="c1"># should not get here...</span>
<span class="n">msg</span> <span class="o">=</span> <span class="s2">"negative cycle detected but not identified"</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXUnbounded</span><span class="p">(</span><span class="n">msg</span><span class="p">)</span></div>
<div class="viewcode-block" id="bidirectional_dijkstra"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.bidirectional_dijkstra.html#networkx.algorithms.shortest_paths.weighted.bidirectional_dijkstra">[docs]</a><span class="k">def</span> <span class="nf">bidirectional_dijkstra</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">source</span><span class="p">,</span> <span class="n">target</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sa">r</span><span class="sd">"""Dijkstra's algorithm for shortest paths using bidirectional search.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> source : node</span>
<span class="sd"> Starting node.</span>
<span class="sd"> target : node</span>
<span class="sd"> Ending node.</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number or None to indicate a hidden edge.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> length, path : number and list</span>
<span class="sd"> length is the distance from source to target.</span>
<span class="sd"> path is a list of nodes on a path from source to target.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NodeNotFound</span>
<span class="sd"> If either `source` or `target` is not in `G`.</span>
<span class="sd"> NetworkXNoPath</span>
<span class="sd"> If no path exists between source and target.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> G = nx.path_graph(5)</span>
<span class="sd"> >>> length, path = nx.bidirectional_dijkstra(G, 0, 4)</span>
<span class="sd"> >>> print(length)</span>
<span class="sd"> 4</span>
<span class="sd"> >>> print(path)</span>
<span class="sd"> [0, 1, 2, 3, 4]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Edge weight attributes must be numerical.</span>
<span class="sd"> Distances are calculated as sums of weighted edges traversed.</span>
<span class="sd"> The weight function can be used to hide edges by returning None.</span>
<span class="sd"> So ``weight = lambda u, v, d: 1 if d['color']=="red" else None``</span>
<span class="sd"> will find the shortest red path.</span>
<span class="sd"> In practice bidirectional Dijkstra is much more than twice as fast as</span>
<span class="sd"> ordinary Dijkstra.</span>
<span class="sd"> Ordinary Dijkstra expands nodes in a sphere-like manner from the</span>
<span class="sd"> source. The radius of this sphere will eventually be the length</span>
<span class="sd"> of the shortest path. Bidirectional Dijkstra will expand nodes</span>
<span class="sd"> from both the source and the target, making two spheres of half</span>
<span class="sd"> this radius. Volume of the first sphere is `\pi*r*r` while the</span>
<span class="sd"> others are `2*\pi*r/2*r/2`, making up half the volume.</span>
<span class="sd"> This algorithm is not guaranteed to work if edge weights</span>
<span class="sd"> are negative or are floating point numbers</span>
<span class="sd"> (overflows and roundoff errors can cause problems).</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> shortest_path</span>
<span class="sd"> shortest_path_length</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="n">source</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span> <span class="ow">or</span> <span class="n">target</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="n">msg</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"Either source </span><span class="si">{</span><span class="n">source</span><span class="si">}</span><span class="s2"> or target </span><span class="si">{</span><span class="n">target</span><span class="si">}</span><span class="s2"> is not in G"</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NodeNotFound</span><span class="p">(</span><span class="n">msg</span><span class="p">)</span>
<span class="k">if</span> <span class="n">source</span> <span class="o">==</span> <span class="n">target</span><span class="p">:</span>
<span class="k">return</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="p">[</span><span class="n">source</span><span class="p">])</span>
<span class="n">weight</span> <span class="o">=</span> <span class="n">_weight_function</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="n">push</span> <span class="o">=</span> <span class="n">heappush</span>
<span class="n">pop</span> <span class="o">=</span> <span class="n">heappop</span>
<span class="c1"># Init: [Forward, Backward]</span>
<span class="n">dists</span> <span class="o">=</span> <span class="p">[{},</span> <span class="p">{}]</span> <span class="c1"># dictionary of final distances</span>
<span class="n">paths</span> <span class="o">=</span> <span class="p">[{</span><span class="n">source</span><span class="p">:</span> <span class="p">[</span><span class="n">source</span><span class="p">]},</span> <span class="p">{</span><span class="n">target</span><span class="p">:</span> <span class="p">[</span><span class="n">target</span><span class="p">]}]</span> <span class="c1"># dictionary of paths</span>
<span class="n">fringe</span> <span class="o">=</span> <span class="p">[[],</span> <span class="p">[]]</span> <span class="c1"># heap of (distance, node) for choosing node to expand</span>
<span class="n">seen</span> <span class="o">=</span> <span class="p">[{</span><span class="n">source</span><span class="p">:</span> <span class="mi">0</span><span class="p">},</span> <span class="p">{</span><span class="n">target</span><span class="p">:</span> <span class="mi">0</span><span class="p">}]</span> <span class="c1"># dict of distances to seen nodes</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">count</span><span class="p">()</span>
<span class="c1"># initialize fringe heap</span>
<span class="n">push</span><span class="p">(</span><span class="n">fringe</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">next</span><span class="p">(</span><span class="n">c</span><span class="p">),</span> <span class="n">source</span><span class="p">))</span>
<span class="n">push</span><span class="p">(</span><span class="n">fringe</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="nb">next</span><span class="p">(</span><span class="n">c</span><span class="p">),</span> <span class="n">target</span><span class="p">))</span>
<span class="c1"># neighs for extracting correct neighbor information</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">is_directed</span><span class="p">():</span>
<span class="n">neighs</span> <span class="o">=</span> <span class="p">[</span><span class="n">G</span><span class="o">.</span><span class="n">_succ</span><span class="p">,</span> <span class="n">G</span><span class="o">.</span><span class="n">_pred</span><span class="p">]</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">neighs</span> <span class="o">=</span> <span class="p">[</span><span class="n">G</span><span class="o">.</span><span class="n">_adj</span><span class="p">,</span> <span class="n">G</span><span class="o">.</span><span class="n">_adj</span><span class="p">]</span>
<span class="c1"># variables to hold shortest discovered path</span>
<span class="c1"># finaldist = 1e30000</span>
<span class="n">finalpath</span> <span class="o">=</span> <span class="p">[]</span>
<span class="nb">dir</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">while</span> <span class="n">fringe</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">and</span> <span class="n">fringe</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
<span class="c1"># choose direction</span>
<span class="c1"># dir == 0 is forward direction and dir == 1 is back</span>
<span class="nb">dir</span> <span class="o">=</span> <span class="mi">1</span> <span class="o">-</span> <span class="nb">dir</span>
<span class="c1"># extract closest to expand</span>
<span class="p">(</span><span class="n">dist</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span> <span class="o">=</span> <span class="n">pop</span><span class="p">(</span><span class="n">fringe</span><span class="p">[</span><span class="nb">dir</span><span class="p">])</span>
<span class="k">if</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">dists</span><span class="p">[</span><span class="nb">dir</span><span class="p">]:</span>
<span class="c1"># Shortest path to v has already been found</span>
<span class="k">continue</span>
<span class="c1"># update distance</span>
<span class="n">dists</span><span class="p">[</span><span class="nb">dir</span><span class="p">][</span><span class="n">v</span><span class="p">]</span> <span class="o">=</span> <span class="n">dist</span> <span class="c1"># equal to seen[dir][v]</span>
<span class="k">if</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">dists</span><span class="p">[</span><span class="mi">1</span> <span class="o">-</span> <span class="nb">dir</span><span class="p">]:</span>
<span class="c1"># if we have scanned v in both directions we are done</span>
<span class="c1"># we have now discovered the shortest path</span>
<span class="k">return</span> <span class="p">(</span><span class="n">finaldist</span><span class="p">,</span> <span class="n">finalpath</span><span class="p">)</span>
<span class="k">for</span> <span class="n">w</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">neighs</span><span class="p">[</span><span class="nb">dir</span><span class="p">][</span><span class="n">v</span><span class="p">]</span><span class="o">.</span><span class="n">items</span><span class="p">():</span>
<span class="c1"># weight(v, w, d) for forward and weight(w, v, d) for back direction</span>
<span class="n">cost</span> <span class="o">=</span> <span class="n">weight</span><span class="p">(</span><span class="n">v</span><span class="p">,</span> <span class="n">w</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span> <span class="k">if</span> <span class="nb">dir</span> <span class="o">==</span> <span class="mi">0</span> <span class="k">else</span> <span class="n">weight</span><span class="p">(</span><span class="n">w</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span>
<span class="k">if</span> <span class="n">cost</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">continue</span>
<span class="n">vwLength</span> <span class="o">=</span> <span class="n">dists</span><span class="p">[</span><span class="nb">dir</span><span class="p">][</span><span class="n">v</span><span class="p">]</span> <span class="o">+</span> <span class="n">cost</span>
<span class="k">if</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">dists</span><span class="p">[</span><span class="nb">dir</span><span class="p">]:</span>
<span class="k">if</span> <span class="n">vwLength</span> <span class="o"><</span> <span class="n">dists</span><span class="p">[</span><span class="nb">dir</span><span class="p">][</span><span class="n">w</span><span class="p">]:</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">"Contradictory paths found: negative weights?"</span><span class="p">)</span>
<span class="k">elif</span> <span class="n">w</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">seen</span><span class="p">[</span><span class="nb">dir</span><span class="p">]</span> <span class="ow">or</span> <span class="n">vwLength</span> <span class="o"><</span> <span class="n">seen</span><span class="p">[</span><span class="nb">dir</span><span class="p">][</span><span class="n">w</span><span class="p">]:</span>
<span class="c1"># relaxing</span>
<span class="n">seen</span><span class="p">[</span><span class="nb">dir</span><span class="p">][</span><span class="n">w</span><span class="p">]</span> <span class="o">=</span> <span class="n">vwLength</span>
<span class="n">push</span><span class="p">(</span><span class="n">fringe</span><span class="p">[</span><span class="nb">dir</span><span class="p">],</span> <span class="p">(</span><span class="n">vwLength</span><span class="p">,</span> <span class="nb">next</span><span class="p">(</span><span class="n">c</span><span class="p">),</span> <span class="n">w</span><span class="p">))</span>
<span class="n">paths</span><span class="p">[</span><span class="nb">dir</span><span class="p">][</span><span class="n">w</span><span class="p">]</span> <span class="o">=</span> <span class="n">paths</span><span class="p">[</span><span class="nb">dir</span><span class="p">][</span><span class="n">v</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="n">w</span><span class="p">]</span>
<span class="k">if</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">seen</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="ow">and</span> <span class="n">w</span> <span class="ow">in</span> <span class="n">seen</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
<span class="c1"># see if this path is better than the already</span>
<span class="c1"># discovered shortest path</span>
<span class="n">totaldist</span> <span class="o">=</span> <span class="n">seen</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">w</span><span class="p">]</span> <span class="o">+</span> <span class="n">seen</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="n">w</span><span class="p">]</span>
<span class="k">if</span> <span class="n">finalpath</span> <span class="o">==</span> <span class="p">[]</span> <span class="ow">or</span> <span class="n">finaldist</span> <span class="o">></span> <span class="n">totaldist</span><span class="p">:</span>
<span class="n">finaldist</span> <span class="o">=</span> <span class="n">totaldist</span>
<span class="n">revpath</span> <span class="o">=</span> <span class="n">paths</span><span class="p">[</span><span class="mi">1</span><span class="p">][</span><span class="n">w</span><span class="p">][:]</span>
<span class="n">revpath</span><span class="o">.</span><span class="n">reverse</span><span class="p">()</span>
<span class="n">finalpath</span> <span class="o">=</span> <span class="n">paths</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="n">w</span><span class="p">]</span> <span class="o">+</span> <span class="n">revpath</span><span class="p">[</span><span class="mi">1</span><span class="p">:]</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXNoPath</span><span class="p">(</span><span class="sa">f</span><span class="s2">"No path between </span><span class="si">{</span><span class="n">source</span><span class="si">}</span><span class="s2"> and </span><span class="si">{</span><span class="n">target</span><span class="si">}</span><span class="s2">."</span><span class="p">)</span></div>
<div class="viewcode-block" id="johnson"><a class="viewcode-back" href="../../../../reference/algorithms/generated/networkx.algorithms.shortest_paths.weighted.johnson.html#networkx.algorithms.shortest_paths.weighted.johnson">[docs]</a><span class="k">def</span> <span class="nf">johnson</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="s2">"weight"</span><span class="p">):</span>
<span class="sa">r</span><span class="sd">"""Uses Johnson's Algorithm to compute shortest paths.</span>
<span class="sd"> Johnson's Algorithm finds a shortest path between each pair of</span>
<span class="sd"> nodes in a weighted graph even if negative weights are present.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> weight : string or function</span>
<span class="sd"> If this is a string, then edge weights will be accessed via the</span>
<span class="sd"> edge attribute with this key (that is, the weight of the edge</span>
<span class="sd"> joining `u` to `v` will be ``G.edges[u, v][weight]``). If no</span>
<span class="sd"> such edge attribute exists, the weight of the edge is assumed to</span>
<span class="sd"> be one.</span>
<span class="sd"> If this is a function, the weight of an edge is the value</span>
<span class="sd"> returned by the function. The function must accept exactly three</span>
<span class="sd"> positional arguments: the two endpoints of an edge and the</span>
<span class="sd"> dictionary of edge attributes for that edge. The function must</span>
<span class="sd"> return a number.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> distance : dictionary</span>
<span class="sd"> Dictionary, keyed by source and target, of shortest paths.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> NetworkXError</span>
<span class="sd"> If given graph is not weighted.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> graph = nx.DiGraph()</span>
<span class="sd"> >>> graph.add_weighted_edges_from(</span>
<span class="sd"> ... [("0", "3", 3), ("0", "1", -5), ("0", "2", 2), ("1", "2", 4), ("2", "3", 1)]</span>
<span class="sd"> ... )</span>
<span class="sd"> >>> paths = nx.johnson(graph, weight="weight")</span>
<span class="sd"> >>> paths["0"]["2"]</span>
<span class="sd"> ['0', '1', '2']</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Johnson's algorithm is suitable even for graphs with negative weights. It</span>
<span class="sd"> works by using the Bellman–Ford algorithm to compute a transformation of</span>
<span class="sd"> the input graph that removes all negative weights, allowing Dijkstra's</span>
<span class="sd"> algorithm to be used on the transformed graph.</span>
<span class="sd"> The time complexity of this algorithm is $O(n^2 \log n + n m)$,</span>
<span class="sd"> where $n$ is the number of nodes and $m$ the number of edges in the</span>
<span class="sd"> graph. For dense graphs, this may be faster than the Floyd–Warshall</span>
<span class="sd"> algorithm.</span>
<span class="sd"> See Also</span>
<span class="sd"> --------</span>
<span class="sd"> floyd_warshall_predecessor_and_distance</span>
<span class="sd"> floyd_warshall_numpy</span>
<span class="sd"> all_pairs_shortest_path</span>
<span class="sd"> all_pairs_shortest_path_length</span>
<span class="sd"> all_pairs_dijkstra_path</span>
<span class="sd"> bellman_ford_predecessor_and_distance</span>
<span class="sd"> all_pairs_bellman_ford_path</span>
<span class="sd"> all_pairs_bellman_ford_path_length</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">nx</span><span class="o">.</span><span class="n">is_weighted</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="o">=</span><span class="n">weight</span><span class="p">):</span>
<span class="k">raise</span> <span class="n">nx</span><span class="o">.</span><span class="n">NetworkXError</span><span class="p">(</span><span class="s2">"Graph is not weighted."</span><span class="p">)</span>
<span class="n">dist</span> <span class="o">=</span> <span class="p">{</span><span class="n">v</span><span class="p">:</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">G</span><span class="p">}</span>
<span class="n">pred</span> <span class="o">=</span> <span class="p">{</span><span class="n">v</span><span class="p">:</span> <span class="p">[]</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">G</span><span class="p">}</span>
<span class="n">weight</span> <span class="o">=</span> <span class="n">_weight_function</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">weight</span><span class="p">)</span>
<span class="c1"># Calculate distance of shortest paths</span>
<span class="n">dist_bellman</span> <span class="o">=</span> <span class="n">_bellman_ford</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="p">),</span> <span class="n">weight</span><span class="p">,</span> <span class="n">pred</span><span class="o">=</span><span class="n">pred</span><span class="p">,</span> <span class="n">dist</span><span class="o">=</span><span class="n">dist</span><span class="p">)</span>
<span class="c1"># Update the weight function to take into account the Bellman--Ford</span>
<span class="c1"># relaxation distances.</span>
<span class="k">def</span> <span class="nf">new_weight</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span><span class="p">):</span>
<span class="k">return</span> <span class="n">weight</span><span class="p">(</span><span class="n">u</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span> <span class="o">+</span> <span class="n">dist_bellman</span><span class="p">[</span><span class="n">u</span><span class="p">]</span> <span class="o">-</span> <span class="n">dist_bellman</span><span class="p">[</span><span class="n">v</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">dist_path</span><span class="p">(</span><span class="n">v</span><span class="p">):</span>
<span class="n">paths</span> <span class="o">=</span> <span class="p">{</span><span class="n">v</span><span class="p">:</span> <span class="p">[</span><span class="n">v</span><span class="p">]}</span>
<span class="n">_dijkstra</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">v</span><span class="p">,</span> <span class="n">new_weight</span><span class="p">,</span> <span class="n">paths</span><span class="o">=</span><span class="n">paths</span><span class="p">)</span>
<span class="k">return</span> <span class="n">paths</span>
<span class="k">return</span> <span class="p">{</span><span class="n">v</span><span class="p">:</span> <span class="n">dist_path</span><span class="p">(</span><span class="n">v</span><span class="p">)</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">G</span><span class="p">}</span></div>
</pre></div>
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