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author shellac
date Mon, 22 Mar 2021 18:12:50 +0000
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Greedy graph coloring using various strategies.
from collections import defaultdict, deque
import itertools

import networkx as nx
from networkx.utils import arbitrary_element
from networkx.utils import py_random_state
from . import greedy_coloring_with_interchange as _interchange

__all__ = [

def strategy_largest_first(G, colors):
    """Returns a list of the nodes of ``G`` in decreasing order by

    ``G`` is a NetworkX graph. ``colors`` is ignored.

    return sorted(G,, reverse=True)

def strategy_random_sequential(G, colors, seed=None):
    """Returns a random permutation of the nodes of ``G`` as a list.

    ``G`` is a NetworkX graph. ``colors`` is ignored.

    seed : integer, random_state, or None (default)
        Indicator of random number generation state.
        See :ref:`Randomness<randomness>`.
    nodes = list(G)
    return nodes

def strategy_smallest_last(G, colors):
    """Returns a deque of the nodes of ``G``, "smallest" last.

    Specifically, the degrees of each node are tracked in a bucket queue.
    From this, the node of minimum degree is repeatedly popped from the
    graph, updating its neighbors' degrees.

    ``G`` is a NetworkX graph. ``colors`` is ignored.

    This implementation of the strategy runs in $O(n + m)$ time
    (ignoring polylogarithmic factors), where $n$ is the number of nodes
    and $m$ is the number of edges.

    This strategy is related to :func:`strategy_independent_set`: if we
    interpret each node removed as an independent set of size one, then
    this strategy chooses an independent set of size one instead of a
    maximal independent set.

    H = G.copy()
    result = deque()

    # Build initial degree list (i.e. the bucket queue data structure)
    degrees = defaultdict(set)  # set(), for fast random-access removals
    lbound = float("inf")
    for node, d in
        lbound = min(lbound, d)  # Lower bound on min-degree.

    def find_min_degree():
        # Save time by starting the iterator at `lbound`, not 0.
        # The value that we find will be our new `lbound`, which we set later.
        return next(d for d in itertools.count(lbound) if d in degrees)

    for _ in G:
        # Pop a min-degree node and add it to the list.
        min_degree = find_min_degree()
        u = degrees[min_degree].pop()
        if not degrees[min_degree]:  # Clean up the degree list.
            del degrees[min_degree]

        # Update degrees of removed node's neighbors.
        for v in H[u]:
            degree =
            if not degrees[degree]:  # Clean up the degree list.
                del degrees[degree]
            degrees[degree - 1].add(v)

        # Finally, remove the node.
        lbound = min_degree - 1  # Subtract 1 in case of tied neighbors.

    return result

def _maximal_independent_set(G):
    """Returns a maximal independent set of nodes in ``G`` by repeatedly
    choosing an independent node of minimum degree (with respect to the
    subgraph of unchosen nodes).

    result = set()
    remaining = set(G)
    while remaining:
        G = G.subgraph(remaining)
        v = min(remaining,
        remaining -= set(G[v]) | {v}
    return result

def strategy_independent_set(G, colors):
    """Uses a greedy independent set removal strategy to determine the

    This function updates ``colors`` **in-place** and return ``None``,
    unlike the other strategy functions in this module.

    This algorithm repeatedly finds and removes a maximal independent
    set, assigning each node in the set an unused color.

    ``G`` is a NetworkX graph.

    This strategy is related to :func:`strategy_smallest_last`: in that
    strategy, an independent set of size one is chosen at each step
    instead of a maximal independent set.

    remaining_nodes = set(G)
    while len(remaining_nodes) > 0:
        nodes = _maximal_independent_set(G.subgraph(remaining_nodes))
        remaining_nodes -= nodes
        yield from nodes

def strategy_connected_sequential_bfs(G, colors):
    """Returns an iterable over nodes in ``G`` in the order given by a
    breadth-first traversal.

    The generated sequence has the property that for each node except
    the first, at least one neighbor appeared earlier in the sequence.

    ``G`` is a NetworkX graph. ``colors`` is ignored.

    return strategy_connected_sequential(G, colors, "bfs")

def strategy_connected_sequential_dfs(G, colors):
    """Returns an iterable over nodes in ``G`` in the order given by a
    depth-first traversal.

    The generated sequence has the property that for each node except
    the first, at least one neighbor appeared earlier in the sequence.

    ``G`` is a NetworkX graph. ``colors`` is ignored.

    return strategy_connected_sequential(G, colors, "dfs")

def strategy_connected_sequential(G, colors, traversal="bfs"):
    """Returns an iterable over nodes in ``G`` in the order given by a
    breadth-first or depth-first traversal.

    ``traversal`` must be one of the strings ``'dfs'`` or ``'bfs'``,
    representing depth-first traversal or breadth-first traversal,

    The generated sequence has the property that for each node except
    the first, at least one neighbor appeared earlier in the sequence.

    ``G`` is a NetworkX graph. ``colors`` is ignored.

    if traversal == "bfs":
        traverse = nx.bfs_edges
    elif traversal == "dfs":
        traverse = nx.dfs_edges
        raise nx.NetworkXError(
            "Please specify one of the strings 'bfs' or"
            " 'dfs' for connected sequential ordering"
    for component in nx.connected_components(G):
        source = arbitrary_element(component)
        # Yield the source node, then all the nodes in the specified
        # traversal order.
        yield source
        for (_, end) in traverse(G.subgraph(component), source):
            yield end

def strategy_saturation_largest_first(G, colors):
    """Iterates over all the nodes of ``G`` in "saturation order" (also
    known as "DSATUR").

    ``G`` is a NetworkX graph. ``colors`` is a dictionary mapping nodes of
    ``G`` to colors, for those nodes that have already been colored.

    distinct_colors = {v: set() for v in G}
    for i in range(len(G)):
        # On the first time through, simply choose the node of highest degree.
        if i == 0:
            node = max(G,
            yield node
            # Add the color 0 to the distinct colors set for each
            # neighbors of that node.
            for v in G[node]:
            # Compute the maximum saturation and the set of nodes that
            # achieve that saturation.
            saturation = {
                v: len(c) for v, c in distinct_colors.items() if v not in colors
            # Yield the node with the highest saturation, and break ties by
            # degree.
            node = max(saturation, key=lambda v: (saturation[v],
            yield node
            # Update the distinct color sets for the neighbors.
            color = colors[node]
            for v in G[node]:

#: Dictionary mapping name of a strategy as a string to the strategy function.
    "largest_first": strategy_largest_first,
    "random_sequential": strategy_random_sequential,
    "smallest_last": strategy_smallest_last,
    "independent_set": strategy_independent_set,
    "connected_sequential_bfs": strategy_connected_sequential_bfs,
    "connected_sequential_dfs": strategy_connected_sequential_dfs,
    "connected_sequential": strategy_connected_sequential,
    "saturation_largest_first": strategy_saturation_largest_first,
    "DSATUR": strategy_saturation_largest_first,

def greedy_color(G, strategy="largest_first", interchange=False):
    """Color a graph using various strategies of greedy graph coloring.

    Attempts to color a graph using as few colors as possible, where no
    neighbours of a node can have same color as the node itself. The
    given strategy determines the order in which nodes are colored.

    The strategies are described in [1]_, and smallest-last is based on

    G : NetworkX graph

    strategy : string or function(G, colors)
       A function (or a string representing a function) that provides
       the coloring strategy, by returning nodes in the ordering they
       should be colored. ``G`` is the graph, and ``colors`` is a
       dictionary of the currently assigned colors, keyed by nodes. The
       function must return an iterable over all the nodes in ``G``.

       If the strategy function is an iterator generator (that is, a
       function with ``yield`` statements), keep in mind that the
       ``colors`` dictionary will be updated after each ``yield``, since
       this function chooses colors greedily.

       If ``strategy`` is a string, it must be one of the following,
       each of which represents one of the built-in strategy functions.

       * ``'largest_first'``
       * ``'random_sequential'``
       * ``'smallest_last'``
       * ``'independent_set'``
       * ``'connected_sequential_bfs'``
       * ``'connected_sequential_dfs'``
       * ``'connected_sequential'`` (alias for the previous strategy)
       * ``'saturation_largest_first'``
       * ``'DSATUR'`` (alias for the previous strategy)

    interchange: bool
       Will use the color interchange algorithm described by [3]_ if set
       to ``True``.

       Note that ``saturation_largest_first`` and ``independent_set``
       do not work with interchange. Furthermore, if you use
       interchange with your own strategy function, you cannot rely
       on the values in the ``colors`` argument.

    A dictionary with keys representing nodes and values representing
    corresponding coloring.

    >>> G = nx.cycle_graph(4)
    >>> d = nx.coloring.greedy_color(G, strategy="largest_first")
    >>> d in [{0: 0, 1: 1, 2: 0, 3: 1}, {0: 1, 1: 0, 2: 1, 3: 0}]

        If ``strategy`` is ``saturation_largest_first`` or
        ``independent_set`` and ``interchange`` is ``True``.

    .. [1] Adrian Kosowski, and Krzysztof Manuszewski,
       Classical Coloring of Graphs, Graph Colorings, 2-19, 2004.
       ISBN 0-8218-3458-4.
    .. [2] David W. Matula, and Leland L. Beck, "Smallest-last
       ordering and clustering and graph coloring algorithms." *J. ACM* 30,
       3 (July 1983), 417–427. <>
    .. [3] Maciej M. Sysło, Marsingh Deo, Janusz S. Kowalik,
       Discrete Optimization Algorithms with Pascal Programs, 415-424, 1983.
       ISBN 0-486-45353-7.

    if len(G) == 0:
        return {}
    # Determine the strategy provided by the caller.
    strategy = STRATEGIES.get(strategy, strategy)
    if not callable(strategy):
        raise nx.NetworkXError(
            "strategy must be callable or a valid string. " f"{strategy} not valid."
    # Perform some validation on the arguments before executing any
    # strategy functions.
    if interchange:
        if strategy is strategy_independent_set:
            msg = "interchange cannot be used with independent_set"
            raise nx.NetworkXPointlessConcept(msg)
        if strategy is strategy_saturation_largest_first:
            msg = "interchange cannot be used with" " saturation_largest_first"
            raise nx.NetworkXPointlessConcept(msg)
    colors = {}
    nodes = strategy(G, colors)
    if interchange:
        return _interchange.greedy_coloring_with_interchange(G, nodes)
    for u in nodes:
        # Set to keep track of colors of neighbours
        neighbour_colors = {colors[v] for v in G[u] if v in colors}
        # Find the first unused color.
        for color in itertools.count():
            if color not in neighbour_colors:
        # Assign the new color to the current node.
        colors[u] = color
    return colors