xhs-note-crawling/python/py/Lib/site-packages/networkx/algorithms/distance_regular.py

"""
=======================
Distance-regular graphs
=======================
"""

from collections import defaultdict
from itertools import combinations_with_replacement
from math import log

import networkx as nx
from networkx.utils import not_implemented_for

from .distance_measures import diameter

__all__ = [
    "is_distance_regular",
    "is_strongly_regular",
    "intersection_array",
    "global_parameters",
]


@nx._dispatchable
def is_distance_regular(G):
    """Returns True if the graph is distance regular, False otherwise.

    A connected graph G is distance-regular if for any nodes x,y
    and any integers i,j=0,1,...,d (where d is the graph
    diameter), the number of vertices at distance i from x and
    distance j from y depends only on i,j and the graph distance
    between x and y, independently of the choice of x and y.

    Parameters
    ----------
    G: Networkx graph (undirected)

    Returns
    -------
    bool
      True if the graph is Distance Regular, False otherwise

    Examples
    --------
    >>> G = nx.hypercube_graph(6)
    >>> nx.is_distance_regular(G)
    True

    See Also
    --------
    intersection_array, global_parameters

    Notes
    -----
    For undirected and simple graphs only

    References
    ----------
    .. [1] Brouwer, A. E.; Cohen, A. M.; and Neumaier, A.
        Distance-Regular Graphs. New York: Springer-Verlag, 1989.
    .. [2] Weisstein, Eric W. "Distance-Regular Graph."
        http://mathworld.wolfram.com/Distance-RegularGraph.html

    """
    try:
        intersection_array(G)
        return True
    except nx.NetworkXError:
        return False


def global_parameters(b, c):
    """Returns global parameters for a given intersection array.

    Given a distance-regular graph G with diameter d and integers b_i,
    c_i,i = 0,....,d such that for any 2 vertices x,y in G at a distance
    i=d(x,y), there are exactly c_i neighbors of y at a distance of i-1 from x
    and b_i neighbors of y at a distance of i+1 from x.

    Thus, a distance regular graph has the global parameters,
    [[c_0,a_0,b_0],[c_1,a_1,b_1],......,[c_d,a_d,b_d]] for the
    intersection array  [b_0,b_1,.....b_{d-1};c_1,c_2,.....c_d]
    where a_i+b_i+c_i=k , k= degree of every vertex.

    Parameters
    ----------
    b : list

    c : list

    Returns
    -------
    iterable
       An iterable over three tuples.

    Examples
    --------
    >>> G = nx.dodecahedral_graph()
    >>> b, c = nx.intersection_array(G)
    >>> list(nx.global_parameters(b, c))
    [(0, 0, 3), (1, 0, 2), (1, 1, 1), (1, 1, 1), (2, 0, 1), (3, 0, 0)]

    References
    ----------
    .. [1] Weisstein, Eric W. "Global Parameters."
       From MathWorld--A Wolfram Web Resource.
       http://mathworld.wolfram.com/GlobalParameters.html

    See Also
    --------
    intersection_array
    """
    return ((y, b[0] - x - y, x) for x, y in zip(b + [0], [0] + c))


@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable
def intersection_array(G):
    """Returns the intersection array of a distance-regular graph.

    Given a distance-regular graph G with integers b_i, c_i,i = 0,....,d
    such that for any 2 vertices x,y in G at a distance i=d(x,y), there
    are exactly c_i neighbors of y at a distance of i-1 from x and b_i
    neighbors of y at a distance of i+1 from x.

    A distance regular graph's intersection array is given by,
    [b_0,b_1,.....b_{d-1};c_1,c_2,.....c_d]

    Parameters
    ----------
    G: Networkx graph (undirected)

    Returns
    -------
    b,c: tuple of lists

    Examples
    --------
    >>> G = nx.icosahedral_graph()
    >>> nx.intersection_array(G)
    ([5, 2, 1], [1, 2, 5])

    References
    ----------
    .. [1] Weisstein, Eric W. "Intersection Array."
       From MathWorld--A Wolfram Web Resource.
       http://mathworld.wolfram.com/IntersectionArray.html

    See Also
    --------
    global_parameters
    """
    # the input graph is very unlikely to be distance-regular: here are the
    # number a(n) of connected simple graphs, and the number b(n) of
    # distance-regular graphs among them:
    #
    #    n  | 1 2 3 4  5   6   7     8      9       10
    #  -----+------------------------------------------------------------------
    #  a(n) | 1 1 2 6 21 112 853 11117 261080 11716571 https://oeis.org/A001349
    #  b(n) | 1 1 1 2  2   4   2     5      4        7 https://oeis.org/A241814
    #
    # in light of this, let's compute shortest path lengths as we go instead of
    # precomputing them all
    # test for regular graph (all degrees must be equal)
    if not nx.is_regular(G) or not nx.is_connected(G):
        raise nx.NetworkXError("Graph is not distance regular.")

    path_length = defaultdict(dict)
    bint = {}  # 'b' intersection array
    cint = {}  # 'c' intersection array

    # see https://doi.org/10.1016/j.ejc.2004.07.004, Theorem 1.5, page 81:
    # the diameter of a distance-regular graph is at most (8 log_2 n) / 3,
    # so let's compute it as we go in the hope that we can stop early
    diam = 0
    max_diameter_for_dr_graphs = (8 * log(len(G), 2)) / 3
    for u, v in combinations_with_replacement(G, 2):
        # compute needed shortest path lengths
        pl_u = path_length[u]
        if v not in pl_u:
            pl_u.update(nx.single_source_shortest_path_length(G, u))
            for x, distance in pl_u.items():
                path_length[x][u] = distance

        i = path_length[u][v]
        diam = max(diam, i)

        # diameter too large: graph can't be distance-regular
        if diam > max_diameter_for_dr_graphs:
            raise nx.NetworkXError("Graph is not distance regular.")

        vnbrs = G[v]
        # compute needed path lengths
        for n in vnbrs:
            pl_n = path_length[n]
            if u not in pl_n:
                pl_n.update(nx.single_source_shortest_path_length(G, n))
                for x, distance in pl_n.items():
                    path_length[x][n] = distance

        # number of neighbors of v at a distance of i-1 from u
        c = sum(1 for n in vnbrs if pl_u[n] == i - 1)
        # number of neighbors of v at a distance of i+1 from u
        b = sum(1 for n in vnbrs if pl_u[n] == i + 1)
        # b, c are independent of u and v
        if cint.get(i, c) != c or bint.get(i, b) != b:
            raise nx.NetworkXError("Graph is not distance regular")
        bint[i] = b
        cint[i] = c

    return (
        [bint.get(j, 0) for j in range(diam)],
        [cint.get(j + 1, 0) for j in range(diam)],
    )


# TODO There is a definition for directed strongly regular graphs.
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable
def is_strongly_regular(G):
    """Returns True if and only if the given graph is strongly
    regular.

    An undirected graph is *strongly regular* if

    * it is regular,
    * each pair of adjacent vertices has the same number of neighbors in
      common,
    * each pair of nonadjacent vertices has the same number of neighbors
      in common.

    Each strongly regular graph is a distance-regular graph.
    Conversely, if a distance-regular graph has diameter two, then it is
    a strongly regular graph. For more information on distance-regular
    graphs, see :func:`is_distance_regular`.

    Parameters
    ----------
    G : NetworkX graph
        An undirected graph.

    Returns
    -------
    bool
        Whether `G` is strongly regular.

    Examples
    --------

    The cycle graph on five vertices is strongly regular. It is
    two-regular, each pair of adjacent vertices has no shared neighbors,
    and each pair of nonadjacent vertices has one shared neighbor::

        >>> G = nx.cycle_graph(5)
        >>> nx.is_strongly_regular(G)
        True

    """
    # Here is an alternate implementation based directly on the
    # definition of strongly regular graphs:
    #
    #     return (all_equal(G.degree().values())
    #             and all_equal(len(common_neighbors(G, u, v))
    #                           for u, v in G.edges())
    #             and all_equal(len(common_neighbors(G, u, v))
    #                           for u, v in non_edges(G)))
    #
    # We instead use the fact that a distance-regular graph of diameter
    # two is strongly regular.
    return is_distance_regular(G) and diameter(G) == 2