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