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