613 lines
20 KiB
Python
613 lines
20 KiB
Python
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"""
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Algorithms for calculating min/max spanning trees/forests.
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"""
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from heapq import heappop, heappush
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from operator import itemgetter
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from itertools import count
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from math import isnan
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import networkx as nx
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from networkx.utils import UnionFind, not_implemented_for
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__all__ = [
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"minimum_spanning_edges",
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"maximum_spanning_edges",
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"minimum_spanning_tree",
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"maximum_spanning_tree",
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]
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@not_implemented_for("multigraph")
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def boruvka_mst_edges(
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G, minimum=True, weight="weight", keys=False, data=True, ignore_nan=False
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):
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"""Iterate over edges of a Borůvka's algorithm min/max spanning tree.
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Parameters
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----------
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G : NetworkX Graph
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The edges of `G` must have distinct weights,
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otherwise the edges may not form a tree.
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minimum : bool (default: True)
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Find the minimum (True) or maximum (False) spanning tree.
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weight : string (default: 'weight')
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The name of the edge attribute holding the edge weights.
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keys : bool (default: True)
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This argument is ignored since this function is not
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implemented for multigraphs; it exists only for consistency
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with the other minimum spanning tree functions.
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data : bool (default: True)
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Flag for whether to yield edge attribute dicts.
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If True, yield edges `(u, v, d)`, where `d` is the attribute dict.
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If False, yield edges `(u, v)`.
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ignore_nan : bool (default: False)
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If a NaN is found as an edge weight normally an exception is raised.
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If `ignore_nan is True` then that edge is ignored instead.
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"""
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# Initialize a forest, assuming initially that it is the discrete
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# partition of the nodes of the graph.
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forest = UnionFind(G)
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def best_edge(component):
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"""Returns the optimum (minimum or maximum) edge on the edge
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boundary of the given set of nodes.
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A return value of ``None`` indicates an empty boundary.
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"""
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sign = 1 if minimum else -1
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minwt = float("inf")
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boundary = None
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for e in nx.edge_boundary(G, component, data=True):
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wt = e[-1].get(weight, 1) * sign
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if isnan(wt):
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if ignore_nan:
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continue
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msg = f"NaN found as an edge weight. Edge {e}"
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raise ValueError(msg)
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if wt < minwt:
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minwt = wt
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boundary = e
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return boundary
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# Determine the optimum edge in the edge boundary of each component
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# in the forest.
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best_edges = (best_edge(component) for component in forest.to_sets())
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best_edges = [edge for edge in best_edges if edge is not None]
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# If each entry was ``None``, that means the graph was disconnected,
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# so we are done generating the forest.
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while best_edges:
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# Determine the optimum edge in the edge boundary of each
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# component in the forest.
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#
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# This must be a sequence, not an iterator. In this list, the
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# same edge may appear twice, in different orientations (but
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# that's okay, since a union operation will be called on the
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# endpoints the first time it is seen, but not the second time).
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#
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# Any ``None`` indicates that the edge boundary for that
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# component was empty, so that part of the forest has been
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# completed.
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#
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# TODO This can be parallelized, both in the outer loop over
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# each component in the forest and in the computation of the
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# minimum. (Same goes for the identical lines outside the loop.)
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best_edges = (best_edge(component) for component in forest.to_sets())
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best_edges = [edge for edge in best_edges if edge is not None]
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# Join trees in the forest using the best edges, and yield that
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# edge, since it is part of the spanning tree.
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#
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# TODO This loop can be parallelized, to an extent (the union
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# operation must be atomic).
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for u, v, d in best_edges:
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if forest[u] != forest[v]:
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if data:
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yield u, v, d
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else:
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yield u, v
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forest.union(u, v)
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def kruskal_mst_edges(
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G, minimum, weight="weight", keys=True, data=True, ignore_nan=False
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):
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"""Iterate over edges of a Kruskal's algorithm min/max spanning tree.
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Parameters
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----------
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G : NetworkX Graph
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The graph holding the tree of interest.
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minimum : bool (default: True)
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Find the minimum (True) or maximum (False) spanning tree.
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weight : string (default: 'weight')
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The name of the edge attribute holding the edge weights.
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keys : bool (default: True)
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If `G` is a multigraph, `keys` controls whether edge keys ar yielded.
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Otherwise `keys` is ignored.
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data : bool (default: True)
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Flag for whether to yield edge attribute dicts.
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If True, yield edges `(u, v, d)`, where `d` is the attribute dict.
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If False, yield edges `(u, v)`.
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ignore_nan : bool (default: False)
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If a NaN is found as an edge weight normally an exception is raised.
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If `ignore_nan is True` then that edge is ignored instead.
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"""
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subtrees = UnionFind()
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if G.is_multigraph():
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edges = G.edges(keys=True, data=True)
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def filter_nan_edges(edges=edges, weight=weight):
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sign = 1 if minimum else -1
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for u, v, k, d in edges:
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wt = d.get(weight, 1) * sign
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if isnan(wt):
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if ignore_nan:
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continue
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msg = f"NaN found as an edge weight. Edge {(u, v, k, d)}"
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raise ValueError(msg)
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yield wt, u, v, k, d
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else:
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edges = G.edges(data=True)
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def filter_nan_edges(edges=edges, weight=weight):
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sign = 1 if minimum else -1
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for u, v, d in edges:
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wt = d.get(weight, 1) * sign
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if isnan(wt):
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if ignore_nan:
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continue
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msg = f"NaN found as an edge weight. Edge {(u, v, d)}"
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raise ValueError(msg)
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yield wt, u, v, d
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edges = sorted(filter_nan_edges(), key=itemgetter(0))
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# Multigraphs need to handle edge keys in addition to edge data.
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if G.is_multigraph():
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for wt, u, v, k, d in edges:
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if subtrees[u] != subtrees[v]:
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if keys:
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if data:
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yield u, v, k, d
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else:
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yield u, v, k
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else:
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if data:
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yield u, v, d
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else:
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yield u, v
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subtrees.union(u, v)
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else:
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for wt, u, v, d in edges:
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if subtrees[u] != subtrees[v]:
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if data:
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yield (u, v, d)
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else:
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yield (u, v)
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subtrees.union(u, v)
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def prim_mst_edges(G, minimum, weight="weight", keys=True, data=True, ignore_nan=False):
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"""Iterate over edges of Prim's algorithm min/max spanning tree.
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Parameters
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----------
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G : NetworkX Graph
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The graph holding the tree of interest.
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minimum : bool (default: True)
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Find the minimum (True) or maximum (False) spanning tree.
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weight : string (default: 'weight')
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The name of the edge attribute holding the edge weights.
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keys : bool (default: True)
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If `G` is a multigraph, `keys` controls whether edge keys ar yielded.
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Otherwise `keys` is ignored.
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data : bool (default: True)
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Flag for whether to yield edge attribute dicts.
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If True, yield edges `(u, v, d)`, where `d` is the attribute dict.
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If False, yield edges `(u, v)`.
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ignore_nan : bool (default: False)
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If a NaN is found as an edge weight normally an exception is raised.
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If `ignore_nan is True` then that edge is ignored instead.
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"""
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is_multigraph = G.is_multigraph()
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push = heappush
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pop = heappop
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nodes = set(G)
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c = count()
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sign = 1 if minimum else -1
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while nodes:
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u = nodes.pop()
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frontier = []
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visited = {u}
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if is_multigraph:
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for v, keydict in G.adj[u].items():
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for k, d in keydict.items():
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wt = d.get(weight, 1) * sign
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if isnan(wt):
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if ignore_nan:
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continue
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msg = f"NaN found as an edge weight. Edge {(u, v, k, d)}"
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raise ValueError(msg)
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push(frontier, (wt, next(c), u, v, k, d))
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else:
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for v, d in G.adj[u].items():
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wt = d.get(weight, 1) * sign
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if isnan(wt):
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if ignore_nan:
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continue
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msg = f"NaN found as an edge weight. Edge {(u, v, d)}"
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raise ValueError(msg)
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push(frontier, (wt, next(c), u, v, d))
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while frontier:
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if is_multigraph:
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W, _, u, v, k, d = pop(frontier)
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else:
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W, _, u, v, d = pop(frontier)
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if v in visited or v not in nodes:
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continue
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# Multigraphs need to handle edge keys in addition to edge data.
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if is_multigraph and keys:
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if data:
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yield u, v, k, d
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else:
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yield u, v, k
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else:
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if data:
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yield u, v, d
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else:
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yield u, v
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# update frontier
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visited.add(v)
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nodes.discard(v)
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if is_multigraph:
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for w, keydict in G.adj[v].items():
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if w in visited:
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continue
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for k2, d2 in keydict.items():
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new_weight = d2.get(weight, 1) * sign
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push(frontier, (new_weight, next(c), v, w, k2, d2))
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else:
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for w, d2 in G.adj[v].items():
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if w in visited:
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continue
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new_weight = d2.get(weight, 1) * sign
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push(frontier, (new_weight, next(c), v, w, d2))
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ALGORITHMS = {
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"boruvka": boruvka_mst_edges,
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"borůvka": boruvka_mst_edges,
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"kruskal": kruskal_mst_edges,
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"prim": prim_mst_edges,
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}
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@not_implemented_for("directed")
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def minimum_spanning_edges(
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G, algorithm="kruskal", weight="weight", keys=True, data=True, ignore_nan=False
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):
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"""Generate edges in a minimum spanning forest of an undirected
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weighted graph.
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A minimum spanning tree is a subgraph of the graph (a tree)
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with the minimum sum of edge weights. A spanning forest is a
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union of the spanning trees for each connected component of the graph.
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Parameters
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----------
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G : undirected Graph
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An undirected graph. If `G` is connected, then the algorithm finds a
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spanning tree. Otherwise, a spanning forest is found.
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algorithm : string
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The algorithm to use when finding a minimum spanning tree. Valid
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choices are 'kruskal', 'prim', or 'boruvka'. The default is 'kruskal'.
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weight : string
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Edge data key to use for weight (default 'weight').
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keys : bool
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Whether to yield edge key in multigraphs in addition to the edge.
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If `G` is not a multigraph, this is ignored.
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data : bool, optional
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If True yield the edge data along with the edge.
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ignore_nan : bool (default: False)
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If a NaN is found as an edge weight normally an exception is raised.
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If `ignore_nan is True` then that edge is ignored instead.
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Returns
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-------
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edges : iterator
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An iterator over edges in a maximum spanning tree of `G`.
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Edges connecting nodes `u` and `v` are represented as tuples:
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`(u, v, k, d)` or `(u, v, k)` or `(u, v, d)` or `(u, v)`
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If `G` is a multigraph, `keys` indicates whether the edge key `k` will
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be reported in the third position in the edge tuple. `data` indicates
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whether the edge datadict `d` will appear at the end of the edge tuple.
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If `G` is not a multigraph, the tuples are `(u, v, d)` if `data` is True
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or `(u, v)` if `data` is False.
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Examples
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--------
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>>> from networkx.algorithms import tree
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Find minimum spanning edges by Kruskal's algorithm
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>>> G = nx.cycle_graph(4)
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>>> G.add_edge(0, 3, weight=2)
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>>> mst = tree.minimum_spanning_edges(G, algorithm="kruskal", data=False)
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>>> edgelist = list(mst)
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>>> sorted(sorted(e) for e in edgelist)
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[[0, 1], [1, 2], [2, 3]]
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Find minimum spanning edges by Prim's algorithm
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>>> G = nx.cycle_graph(4)
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>>> G.add_edge(0, 3, weight=2)
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>>> mst = tree.minimum_spanning_edges(G, algorithm="prim", data=False)
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>>> edgelist = list(mst)
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>>> sorted(sorted(e) for e in edgelist)
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[[0, 1], [1, 2], [2, 3]]
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Notes
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-----
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||
|
|
For Borůvka's algorithm, each edge must have a weight attribute, and
|
||
|
|
each edge weight must be distinct.
|
||
|
|
|
||
|
|
For the other algorithms, if the graph edges do not have a weight
|
||
|
|
attribute a default weight of 1 will be used.
|
||
|
|
|
||
|
|
Modified code from David Eppstein, April 2006
|
||
|
|
http://www.ics.uci.edu/~eppstein/PADS/
|
||
|
|
|
||
|
|
"""
|
||
|
|
try:
|
||
|
|
algo = ALGORITHMS[algorithm]
|
||
|
|
except KeyError as e:
|
||
|
|
msg = f"{algorithm} is not a valid choice for an algorithm."
|
||
|
|
raise ValueError(msg) from e
|
||
|
|
|
||
|
|
return algo(
|
||
|
|
G, minimum=True, weight=weight, keys=keys, data=data, ignore_nan=ignore_nan
|
||
|
|
)
|
||
|
|
|
||
|
|
|
||
|
|
@not_implemented_for("directed")
|
||
|
|
def maximum_spanning_edges(
|
||
|
|
G, algorithm="kruskal", weight="weight", keys=True, data=True, ignore_nan=False
|
||
|
|
):
|
||
|
|
"""Generate edges in a maximum spanning forest of an undirected
|
||
|
|
weighted graph.
|
||
|
|
|
||
|
|
A maximum spanning tree is a subgraph of the graph (a tree)
|
||
|
|
with the maximum possible sum of edge weights. A spanning forest is a
|
||
|
|
union of the spanning trees for each connected component of the graph.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : undirected Graph
|
||
|
|
An undirected graph. If `G` is connected, then the algorithm finds a
|
||
|
|
spanning tree. Otherwise, a spanning forest is found.
|
||
|
|
|
||
|
|
algorithm : string
|
||
|
|
The algorithm to use when finding a maximum spanning tree. Valid
|
||
|
|
choices are 'kruskal', 'prim', or 'boruvka'. The default is 'kruskal'.
|
||
|
|
|
||
|
|
weight : string
|
||
|
|
Edge data key to use for weight (default 'weight').
|
||
|
|
|
||
|
|
keys : bool
|
||
|
|
Whether to yield edge key in multigraphs in addition to the edge.
|
||
|
|
If `G` is not a multigraph, this is ignored.
|
||
|
|
|
||
|
|
data : bool, optional
|
||
|
|
If True yield the edge data along with the edge.
|
||
|
|
|
||
|
|
ignore_nan : bool (default: False)
|
||
|
|
If a NaN is found as an edge weight normally an exception is raised.
|
||
|
|
If `ignore_nan is True` then that edge is ignored instead.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
edges : iterator
|
||
|
|
An iterator over edges in a maximum spanning tree of `G`.
|
||
|
|
Edges connecting nodes `u` and `v` are represented as tuples:
|
||
|
|
`(u, v, k, d)` or `(u, v, k)` or `(u, v, d)` or `(u, v)`
|
||
|
|
|
||
|
|
If `G` is a multigraph, `keys` indicates whether the edge key `k` will
|
||
|
|
be reported in the third position in the edge tuple. `data` indicates
|
||
|
|
whether the edge datadict `d` will appear at the end of the edge tuple.
|
||
|
|
|
||
|
|
If `G` is not a multigraph, the tuples are `(u, v, d)` if `data` is True
|
||
|
|
or `(u, v)` if `data` is False.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from networkx.algorithms import tree
|
||
|
|
|
||
|
|
Find maximum spanning edges by Kruskal's algorithm
|
||
|
|
|
||
|
|
>>> G = nx.cycle_graph(4)
|
||
|
|
>>> G.add_edge(0, 3, weight=2)
|
||
|
|
>>> mst = tree.maximum_spanning_edges(G, algorithm="kruskal", data=False)
|
||
|
|
>>> edgelist = list(mst)
|
||
|
|
>>> sorted(sorted(e) for e in edgelist)
|
||
|
|
[[0, 1], [0, 3], [1, 2]]
|
||
|
|
|
||
|
|
Find maximum spanning edges by Prim's algorithm
|
||
|
|
|
||
|
|
>>> G = nx.cycle_graph(4)
|
||
|
|
>>> G.add_edge(0, 3, weight=2) # assign weight 2 to edge 0-3
|
||
|
|
>>> mst = tree.maximum_spanning_edges(G, algorithm="prim", data=False)
|
||
|
|
>>> edgelist = list(mst)
|
||
|
|
>>> sorted(sorted(e) for e in edgelist)
|
||
|
|
[[0, 1], [0, 3], [2, 3]]
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
For Borůvka's algorithm, each edge must have a weight attribute, and
|
||
|
|
each edge weight must be distinct.
|
||
|
|
|
||
|
|
For the other algorithms, if the graph edges do not have a weight
|
||
|
|
attribute a default weight of 1 will be used.
|
||
|
|
|
||
|
|
Modified code from David Eppstein, April 2006
|
||
|
|
http://www.ics.uci.edu/~eppstein/PADS/
|
||
|
|
"""
|
||
|
|
try:
|
||
|
|
algo = ALGORITHMS[algorithm]
|
||
|
|
except KeyError as e:
|
||
|
|
msg = f"{algorithm} is not a valid choice for an algorithm."
|
||
|
|
raise ValueError(msg) from e
|
||
|
|
|
||
|
|
return algo(
|
||
|
|
G, minimum=False, weight=weight, keys=keys, data=data, ignore_nan=ignore_nan
|
||
|
|
)
|
||
|
|
|
||
|
|
|
||
|
|
def minimum_spanning_tree(G, weight="weight", algorithm="kruskal", ignore_nan=False):
|
||
|
|
"""Returns a minimum spanning tree or forest on an undirected graph `G`.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : undirected graph
|
||
|
|
An undirected graph. If `G` is connected, then the algorithm finds a
|
||
|
|
spanning tree. Otherwise, a spanning forest is found.
|
||
|
|
|
||
|
|
weight : str
|
||
|
|
Data key to use for edge weights.
|
||
|
|
|
||
|
|
algorithm : string
|
||
|
|
The algorithm to use when finding a minimum spanning tree. Valid
|
||
|
|
choices are 'kruskal', 'prim', or 'boruvka'. The default is
|
||
|
|
'kruskal'.
|
||
|
|
|
||
|
|
ignore_nan : bool (default: False)
|
||
|
|
If a NaN is found as an edge weight normally an exception is raised.
|
||
|
|
If `ignore_nan is True` then that edge is ignored instead.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
G : NetworkX Graph
|
||
|
|
A minimum spanning tree or forest.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> G = nx.cycle_graph(4)
|
||
|
|
>>> G.add_edge(0, 3, weight=2)
|
||
|
|
>>> T = nx.minimum_spanning_tree(G)
|
||
|
|
>>> sorted(T.edges(data=True))
|
||
|
|
[(0, 1, {}), (1, 2, {}), (2, 3, {})]
|
||
|
|
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
For Borůvka's algorithm, each edge must have a weight attribute, and
|
||
|
|
each edge weight must be distinct.
|
||
|
|
|
||
|
|
For the other algorithms, if the graph edges do not have a weight
|
||
|
|
attribute a default weight of 1 will be used.
|
||
|
|
|
||
|
|
There may be more than one tree with the same minimum or maximum weight.
|
||
|
|
See :mod:`networkx.tree.recognition` for more detailed definitions.
|
||
|
|
|
||
|
|
Isolated nodes with self-loops are in the tree as edgeless isolated nodes.
|
||
|
|
|
||
|
|
"""
|
||
|
|
edges = minimum_spanning_edges(
|
||
|
|
G, algorithm, weight, keys=True, data=True, ignore_nan=ignore_nan
|
||
|
|
)
|
||
|
|
T = G.__class__() # Same graph class as G
|
||
|
|
T.graph.update(G.graph)
|
||
|
|
T.add_nodes_from(G.nodes.items())
|
||
|
|
T.add_edges_from(edges)
|
||
|
|
return T
|
||
|
|
|
||
|
|
|
||
|
|
def maximum_spanning_tree(G, weight="weight", algorithm="kruskal", ignore_nan=False):
|
||
|
|
"""Returns a maximum spanning tree or forest on an undirected graph `G`.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : undirected graph
|
||
|
|
An undirected graph. If `G` is connected, then the algorithm finds a
|
||
|
|
spanning tree. Otherwise, a spanning forest is found.
|
||
|
|
|
||
|
|
weight : str
|
||
|
|
Data key to use for edge weights.
|
||
|
|
|
||
|
|
algorithm : string
|
||
|
|
The algorithm to use when finding a maximum spanning tree. Valid
|
||
|
|
choices are 'kruskal', 'prim', or 'boruvka'. The default is
|
||
|
|
'kruskal'.
|
||
|
|
|
||
|
|
ignore_nan : bool (default: False)
|
||
|
|
If a NaN is found as an edge weight normally an exception is raised.
|
||
|
|
If `ignore_nan is True` then that edge is ignored instead.
|
||
|
|
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
G : NetworkX Graph
|
||
|
|
A maximum spanning tree or forest.
|
||
|
|
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> G = nx.cycle_graph(4)
|
||
|
|
>>> G.add_edge(0, 3, weight=2)
|
||
|
|
>>> T = nx.maximum_spanning_tree(G)
|
||
|
|
>>> sorted(T.edges(data=True))
|
||
|
|
[(0, 1, {}), (0, 3, {'weight': 2}), (1, 2, {})]
|
||
|
|
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
For Borůvka's algorithm, each edge must have a weight attribute, and
|
||
|
|
each edge weight must be distinct.
|
||
|
|
|
||
|
|
For the other algorithms, if the graph edges do not have a weight
|
||
|
|
attribute a default weight of 1 will be used.
|
||
|
|
|
||
|
|
There may be more than one tree with the same minimum or maximum weight.
|
||
|
|
See :mod:`networkx.tree.recognition` for more detailed definitions.
|
||
|
|
|
||
|
|
Isolated nodes with self-loops are in the tree as edgeless isolated nodes.
|
||
|
|
|
||
|
|
"""
|
||
|
|
edges = maximum_spanning_edges(
|
||
|
|
G, algorithm, weight, keys=True, data=True, ignore_nan=ignore_nan
|
||
|
|
)
|
||
|
|
edges = list(edges)
|
||
|
|
T = G.__class__() # Same graph class as G
|
||
|
|
T.graph.update(G.graph)
|
||
|
|
T.add_nodes_from(G.nodes.items())
|
||
|
|
T.add_edges_from(edges)
|
||
|
|
return T
|