354 lines
10 KiB
Python
354 lines
10 KiB
Python
|
"""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
|
||
|
|
|
||
|
|
|
||
|
|
@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
|
||
|
|
)
|
||
|
|
|
||
|
|
|
||
|
|
@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
|
||
|
|
|
||
|
|
|
||
|
|
@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)
|
||
|
|
|
||
|
|
|
||
|
|
@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
|
||
|
|
|
||
|
|
|
||
|
|
@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)
|
||
|
|
|
||
|
|
|
||
|
|
@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)
|