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<h1>Source code for networkx.algorithms.tournament</h1><div class="highlight"><pre>
<span></span><span class="sd">"""Functions concerning tournament graphs.</span>
<span class="sd">A `tournament graph`_ is a complete oriented graph. In other words, it</span>
<span class="sd">is a directed graph in which there is exactly one directed edge joining</span>
<span class="sd">each pair of distinct nodes. For each function in this module that</span>
<span class="sd">accepts a graph as input, you must provide a tournament graph. The</span>
<span class="sd">responsibility is on the caller to ensure that the graph is a tournament</span>
<span class="sd">graph.</span>
<span class="sd">To access the functions in this module, you must access them through the</span>
<span class="sd">:mod:`networkx.algorithms.tournament` module::</span>
<span class="sd"> >>> from networkx.algorithms import tournament</span>
<span class="sd"> >>> G = nx.DiGraph([(0, 1), (1, 2), (2, 0)])</span>
<span class="sd"> >>> tournament.is_tournament(G)</span>
<span class="sd"> True</span>
<span class="sd">.. _tournament graph: https://en.wikipedia.org/wiki/Tournament_%28graph_theory%29</span>
<span class="sd">"""</span>
<span class="kn">from</span> <span class="nn">itertools</span> <span class="kn">import</span> <span class="n">combinations</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.simple_paths</span> <span class="kn">import</span> <span class="n">is_simple_path</span> <span class="k">as</span> <span class="n">is_path</span>
<span class="kn">from</span> <span class="nn">networkx.utils</span> <span class="kn">import</span> <span class="n">arbitrary_element</span><span class="p">,</span> <span class="n">not_implemented_for</span><span class="p">,</span> <span class="n">py_random_state</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span>
<span class="s2">"hamiltonian_path"</span><span class="p">,</span>
<span class="s2">"is_reachable"</span><span class="p">,</span>
<span class="s2">"is_strongly_connected"</span><span class="p">,</span>
<span class="s2">"is_tournament"</span><span class="p">,</span>
<span class="s2">"random_tournament"</span><span class="p">,</span>
<span class="s2">"score_sequence"</span><span class="p">,</span>
<span class="p">]</span>
<span class="k">def</span> <span class="nf">index_satisfying</span><span class="p">(</span><span class="n">iterable</span><span class="p">,</span> <span class="n">condition</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns the index of the first element in `iterable` that</span>
<span class="sd"> satisfies the given condition.</span>
<span class="sd"> If no such element is found (that is, when the iterable is</span>
<span class="sd"> exhausted), this returns the length of the iterable (that is, one</span>
<span class="sd"> greater than the last index of the iterable).</span>
<span class="sd"> `iterable` must not be empty. If `iterable` is empty, this</span>
<span class="sd"> function raises :exc:`ValueError`.</span>
<span class="sd"> """</span>
<span class="c1"># Pre-condition: iterable must not be empty.</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">x</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">iterable</span><span class="p">):</span>
<span class="k">if</span> <span class="n">condition</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="n">i</span>
<span class="c1"># If we reach the end of the iterable without finding an element</span>
<span class="c1"># that satisfies the condition, return the length of the iterable,</span>
<span class="c1"># which is one greater than the index of its last element. If the</span>
<span class="c1"># iterable was empty, `i` will not be defined, so we raise an</span>
<span class="c1"># exception.</span>
<span class="k">try</span><span class="p">:</span>
<span class="k">return</span> <span class="n">i</span> <span class="o">+</span> <span class="mi">1</span>
<span class="k">except</span> <span class="ne">NameError</span> <span class="k">as</span> <span class="n">err</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">"iterable must be non-empty"</span><span class="p">)</span> <span class="kn">from</span> <span class="nn">err</span>
<div class="viewcode-block" id="is_tournament"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.tournament.is_tournament.html#networkx.algorithms.tournament.is_tournament">[docs]</a><span class="nd">@nx</span><span class="o">.</span><span class="n">_dispatch</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"undirected"</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"multigraph"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">is_tournament</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns True if and only if `G` is a tournament.</span>
<span class="sd"> A tournament is a directed graph, with neither self-loops nor</span>
<span class="sd"> multi-edges, in which there is exactly one directed edge joining</span>
<span class="sd"> each pair of distinct nodes.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> A directed graph representing a tournament.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> bool</span>
<span class="sd"> Whether the given graph is a tournament graph.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> from networkx.algorithms import tournament</span>
<span class="sd"> >>> G = nx.DiGraph([(0, 1), (1, 2), (2, 0)])</span>
<span class="sd"> >>> tournament.is_tournament(G)</span>
<span class="sd"> True</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Some definitions require a self-loop on each node, but that is not</span>
<span class="sd"> the convention used here.</span>
<span class="sd"> """</span>
<span class="c1"># In a tournament, there is exactly one directed edge joining each pair.</span>
<span class="k">return</span> <span class="p">(</span>
<span class="nb">all</span><span class="p">((</span><span class="n">v</span> <span class="ow">in</span> <span class="n">G</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">u</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="k">for</span> <span class="n">u</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">combinations</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="ow">and</span> <span class="n">nx</span><span class="o">.</span><span class="n">number_of_selfloops</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span>
<span class="p">)</span></div>
<div class="viewcode-block" id="hamiltonian_path"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.tournament.hamiltonian_path.html#networkx.algorithms.tournament.hamiltonian_path">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"undirected"</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"multigraph"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">hamiltonian_path</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns a Hamiltonian path in the given tournament graph.</span>
<span class="sd"> Each tournament has a Hamiltonian path. If furthermore, the</span>
<span class="sd"> tournament is strongly connected, then the returned Hamiltonian path</span>
<span class="sd"> is a Hamiltonian cycle (by joining the endpoints of the path).</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> A directed graph representing a tournament.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> path : list</span>
<span class="sd"> A list of nodes which form a Hamiltonian path in `G`.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> from networkx.algorithms import tournament</span>
<span class="sd"> >>> G = nx.DiGraph([(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)])</span>
<span class="sd"> >>> tournament.hamiltonian_path(G)</span>
<span class="sd"> [0, 1, 2, 3]</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This is a recursive implementation with an asymptotic running time</span>
<span class="sd"> of $O(n^2)$, ignoring multiplicative polylogarithmic factors, where</span>
<span class="sd"> $n$ is the number of nodes in the graph.</span>
<span class="sd"> """</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">0</span><span class="p">:</span>
<span class="k">return</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">arbitrary_element</span><span class="p">(</span><span class="n">G</span><span class="p">)]</span>
<span class="n">v</span> <span class="o">=</span> <span class="n">arbitrary_element</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="n">hampath</span> <span class="o">=</span> <span class="n">hamiltonian_path</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">subgraph</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">-</span> <span class="p">{</span><span class="n">v</span><span class="p">}))</span>
<span class="c1"># Get the index of the first node in the path that does *not* have</span>
<span class="c1"># an edge to `v`, then insert `v` before that node.</span>
<span class="n">index</span> <span class="o">=</span> <span class="n">index_satisfying</span><span class="p">(</span><span class="n">hampath</span><span class="p">,</span> <span class="k">lambda</span> <span class="n">u</span><span class="p">:</span> <span class="n">v</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">[</span><span class="n">u</span><span class="p">])</span>
<span class="n">hampath</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="n">index</span><span class="p">,</span> <span class="n">v</span><span class="p">)</span>
<span class="k">return</span> <span class="n">hampath</span></div>
<div class="viewcode-block" id="random_tournament"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.tournament.random_tournament.html#networkx.algorithms.tournament.random_tournament">[docs]</a><span class="nd">@py_random_state</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">random_tournament</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="w"> </span><span class="sa">r</span><span class="sd">"""Returns a random tournament graph on `n` nodes.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> n : int</span>
<span class="sd"> The number of nodes in the returned graph.</span>
<span class="sd"> seed : integer, random_state, or None (default)</span>
<span class="sd"> Indicator of random number generation state.</span>
<span class="sd"> See :ref:`Randomness<randomness>`.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> G : DiGraph</span>
<span class="sd"> A tournament on `n` nodes, with exactly one directed edge joining</span>
<span class="sd"> each pair of distinct nodes.</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> This algorithm adds, for each pair of distinct nodes, an edge with</span>
<span class="sd"> uniformly random orientation. In other words, `\binom{n}{2}` flips</span>
<span class="sd"> of an unbiased coin decide the orientations of the edges in the</span>
<span class="sd"> graph.</span>
<span class="sd"> """</span>
<span class="c1"># Flip an unbiased coin for each pair of distinct nodes.</span>
<span class="n">coins</span> <span class="o">=</span> <span class="p">(</span><span class="n">seed</span><span class="o">.</span><span class="n">random</span><span class="p">()</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">((</span><span class="n">n</span> <span class="o">*</span> <span class="p">(</span><span class="n">n</span> <span class="o">-</span> <span class="mi">1</span><span class="p">))</span> <span class="o">//</span> <span class="mi">2</span><span class="p">))</span>
<span class="n">pairs</span> <span class="o">=</span> <span class="n">combinations</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="mi">2</span><span class="p">)</span>
<span class="n">edges</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">r</span> <span class="o"><</span> <span class="mf">0.5</span> <span class="k">else</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="k">for</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">r</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">pairs</span><span class="p">,</span> <span class="n">coins</span><span class="p">))</span>
<span class="k">return</span> <span class="n">nx</span><span class="o">.</span><span class="n">DiGraph</span><span class="p">(</span><span class="n">edges</span><span class="p">)</span></div>
<div class="viewcode-block" id="score_sequence"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.tournament.score_sequence.html#networkx.algorithms.tournament.score_sequence">[docs]</a><span class="nd">@nx</span><span class="o">.</span><span class="n">_dispatch</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"undirected"</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"multigraph"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">score_sequence</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Returns the score sequence for the given tournament graph.</span>
<span class="sd"> The score sequence is the sorted list of the out-degrees of the</span>
<span class="sd"> nodes of the graph.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> A directed graph representing a tournament.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> list</span>
<span class="sd"> A sorted list of the out-degrees of the nodes of `G`.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> from networkx.algorithms import tournament</span>
<span class="sd"> >>> G = nx.DiGraph([(1, 0), (1, 3), (0, 2), (0, 3), (2, 1), (3, 2)])</span>
<span class="sd"> >>> tournament.score_sequence(G)</span>
<span class="sd"> [1, 1, 2, 2]</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">d</span> <span class="k">for</span> <span class="n">v</span><span class="p">,</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">G</span><span class="o">.</span><span class="n">out_degree</span><span class="p">())</span></div>
<span class="nd">@nx</span><span class="o">.</span><span class="n">_dispatch</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"undirected"</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"multigraph"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">tournament_matrix</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sa">r</span><span class="sd">"""Returns the tournament matrix for the given tournament graph.</span>
<span class="sd"> This function requires SciPy.</span>
<span class="sd"> The *tournament matrix* of a tournament graph with edge set *E* is</span>
<span class="sd"> the matrix *T* defined by</span>
<span class="sd"> .. math::</span>
<span class="sd"> T_{i j} =</span>
<span class="sd"> \begin{cases}</span>
<span class="sd"> +1 & \text{if } (i, j) \in E \\</span>
<span class="sd"> -1 & \text{if } (j, i) \in E \\</span>
<span class="sd"> 0 & \text{if } i == j.</span>
<span class="sd"> \end{cases}</span>
<span class="sd"> An equivalent definition is `T = A - A^T`, where *A* is the</span>
<span class="sd"> adjacency matrix of the graph `G`.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> A directed graph representing a tournament.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> SciPy sparse array</span>
<span class="sd"> The tournament matrix of the tournament graph `G`.</span>
<span class="sd"> Raises</span>
<span class="sd"> ------</span>
<span class="sd"> ImportError</span>
<span class="sd"> If SciPy is not available.</span>
<span class="sd"> """</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">adjacency_matrix</span><span class="p">(</span><span class="n">G</span><span class="p">)</span>
<span class="k">return</span> <span class="n">A</span> <span class="o">-</span> <span class="n">A</span><span class="o">.</span><span class="n">T</span>
<div class="viewcode-block" id="is_reachable"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.tournament.is_reachable.html#networkx.algorithms.tournament.is_reachable">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"undirected"</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"multigraph"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">is_reachable</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Decides whether there is a path from `s` to `t` in the</span>
<span class="sd"> tournament.</span>
<span class="sd"> This function is more theoretically efficient than the reachability</span>
<span class="sd"> checks than the shortest path algorithms in</span>
<span class="sd"> :mod:`networkx.algorithms.shortest_paths`.</span>
<span class="sd"> The given graph **must** be a tournament, otherwise this function's</span>
<span class="sd"> behavior is undefined.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> A directed graph representing a tournament.</span>
<span class="sd"> s : node</span>
<span class="sd"> A node in the graph.</span>
<span class="sd"> t : node</span>
<span class="sd"> A node in the graph.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> bool</span>
<span class="sd"> Whether there is a path from `s` to `t` in `G`.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> from networkx.algorithms import tournament</span>
<span class="sd"> >>> G = nx.DiGraph([(1, 0), (1, 3), (1, 2), (2, 3), (2, 0), (3, 0)])</span>
<span class="sd"> >>> tournament.is_reachable(G, 1, 3)</span>
<span class="sd"> True</span>
<span class="sd"> >>> tournament.is_reachable(G, 3, 2)</span>
<span class="sd"> False</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Although this function is more theoretically efficient than the</span>
<span class="sd"> generic shortest path functions, a speedup requires the use of</span>
<span class="sd"> parallelism. Though it may in the future, the current implementation</span>
<span class="sd"> does not use parallelism, thus you may not see much of a speedup.</span>
<span class="sd"> This algorithm comes from [1].</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Tantau, Till.</span>
<span class="sd"> "A note on the complexity of the reachability problem for</span>
<span class="sd"> tournaments."</span>
<span class="sd"> *Electronic Colloquium on Computational Complexity*. 2001.</span>
<span class="sd"> <http://eccc.hpi-web.de/report/2001/092/></span>
<span class="sd"> """</span>
<span class="k">def</span> <span class="nf">two_neighborhood</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="w"> </span><span class="sd">"""Returns the set of nodes at distance at most two from `v`.</span>
<span class="sd"> `G` must be a graph and `v` a node in that graph.</span>
<span class="sd"> The returned set includes the nodes at distance zero (that is,</span>
<span class="sd"> the node `v` itself), the nodes at distance one (that is, the</span>
<span class="sd"> out-neighbors of `v`), and the nodes at distance two.</span>
<span class="sd"> """</span>
<span class="c1"># TODO This is trivially parallelizable.</span>
<span class="k">return</span> <span class="p">{</span>
<span class="n">x</span> <span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="n">G</span> <span class="k">if</span> <span class="n">x</span> <span class="o">==</span> <span class="n">v</span> <span class="ow">or</span> <span class="n">x</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">or</span> <span class="nb">any</span><span class="p">(</span><span class="n">is_path</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="p">[</span><span class="n">v</span><span class="p">,</span> <span class="n">z</span><span class="p">,</span> <span class="n">x</span><span class="p">])</span> <span class="k">for</span> <span class="n">z</span> <span class="ow">in</span> <span class="n">G</span><span class="p">)</span>
<span class="p">}</span>
<span class="k">def</span> <span class="nf">is_closed</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">nodes</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Decides whether the given set of nodes is closed.</span>
<span class="sd"> A set *S* of nodes is *closed* if for each node *u* in the graph</span>
<span class="sd"> not in *S* and for each node *v* in *S*, there is an edge from</span>
<span class="sd"> *u* to *v*.</span>
<span class="sd"> """</span>
<span class="c1"># TODO This is trivially parallelizable.</span>
<span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="n">v</span> <span class="ow">in</span> <span class="n">G</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="nb">set</span><span class="p">(</span><span class="n">G</span><span class="p">)</span> <span class="o">-</span> <span class="n">nodes</span> <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">nodes</span><span class="p">)</span>
<span class="c1"># TODO This is trivially parallelizable.</span>
<span class="n">neighborhoods</span> <span class="o">=</span> <span class="p">[</span><span class="n">two_neighborhood</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="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">G</span><span class="p">]</span>
<span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="ow">not</span> <span class="p">(</span><span class="n">is_closed</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="n">S</span><span class="p">)</span> <span class="ow">and</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">S</span> <span class="ow">and</span> <span class="n">t</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">S</span><span class="p">)</span> <span class="k">for</span> <span class="n">S</span> <span class="ow">in</span> <span class="n">neighborhoods</span><span class="p">)</span></div>
<div class="viewcode-block" id="is_strongly_connected"><a class="viewcode-back" href="../../../reference/algorithms/generated/networkx.algorithms.tournament.is_strongly_connected.html#networkx.algorithms.tournament.is_strongly_connected">[docs]</a><span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"undirected"</span><span class="p">)</span>
<span class="nd">@not_implemented_for</span><span class="p">(</span><span class="s2">"multigraph"</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">is_strongly_connected</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Decides whether the given tournament is strongly connected.</span>
<span class="sd"> This function is more theoretically efficient than the</span>
<span class="sd"> :func:`~networkx.algorithms.components.is_strongly_connected`</span>
<span class="sd"> function.</span>
<span class="sd"> The given graph **must** be a tournament, otherwise this function's</span>
<span class="sd"> behavior is undefined.</span>
<span class="sd"> Parameters</span>
<span class="sd"> ----------</span>
<span class="sd"> G : NetworkX graph</span>
<span class="sd"> A directed graph representing a tournament.</span>
<span class="sd"> Returns</span>
<span class="sd"> -------</span>
<span class="sd"> bool</span>
<span class="sd"> Whether the tournament is strongly connected.</span>
<span class="sd"> Examples</span>
<span class="sd"> --------</span>
<span class="sd"> >>> from networkx.algorithms import tournament</span>
<span class="sd"> >>> G = nx.DiGraph([(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3), (3, 0)])</span>
<span class="sd"> >>> tournament.is_strongly_connected(G)</span>
<span class="sd"> True</span>
<span class="sd"> >>> G.remove_edge(1, 3)</span>
<span class="sd"> >>> tournament.is_strongly_connected(G)</span>
<span class="sd"> False</span>
<span class="sd"> Notes</span>
<span class="sd"> -----</span>
<span class="sd"> Although this function is more theoretically efficient than the</span>
<span class="sd"> generic strong connectivity function, a speedup requires the use of</span>
<span class="sd"> parallelism. Though it may in the future, the current implementation</span>
<span class="sd"> does not use parallelism, thus you may not see much of a speedup.</span>
<span class="sd"> This algorithm comes from [1].</span>
<span class="sd"> References</span>
<span class="sd"> ----------</span>
<span class="sd"> .. [1] Tantau, Till.</span>
<span class="sd"> "A note on the complexity of the reachability problem for</span>
<span class="sd"> tournaments."</span>
<span class="sd"> *Electronic Colloquium on Computational Complexity*. 2001.</span>
<span class="sd"> <http://eccc.hpi-web.de/report/2001/092/></span>
<span class="sd"> """</span>
<span class="c1"># TODO This is trivially parallelizable.</span>
<span class="k">return</span> <span class="nb">all</span><span class="p">(</span><span class="n">is_reachable</span><span class="p">(</span><span class="n">G</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">for</span> <span class="n">u</span> <span class="ow">in</span> <span class="n">G</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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