def kruskal_mst(G):
"""Finds MST using Kruskal's algoritm
Params
--------
G: NetworkX Graph
An input weighted graph to find MST
Returns
--------
T: NetworkX Graph
A minimum spanning tree
"""
T = nx.Graph()
T.add_nodes_from(G.nodes)
forest = UnionFind(G)
edges = G.edges(data=True)
def get_weights():
"""Unpacks the weights from a dictionary"""
for u, v, dct in edges:
w = dct.get('weight', 1)
yield w, u, v, dct
edges = sorted(get_weights(), key=itemgetter(0))
for w, u, v, _ in edges:
if forest[u] != forest[v]:
T.add_edge(u, v, weight=w)
forest.union(u, v)
return T
def kruskal(G: "networkx Graph object") -> (list, list, list):
"""
Function that recieves a graph an returns the history of the algorithm,
the Mininum Spanning Tree and the edges that causes cycles
"""
# List for the minimum spanning tree
mst = []
# List for the history of the algorithm
history = []
# List for the edges tat causes cycles
cycles = []
# Disjoint structure for calculating the edges that causes cycles
subtrees = UnionFind()
# Sort the edges in ascendant order
edges = sorted(G.edges(data=True), key=lambda t: t[2].get('weight', 1))
# For node A, Node B, Weight in the sorted edges
for u, v, d in edges:
# If the edge doesn't make a cycle append it to the mst
if subtrees[u] != subtrees[v]:
mst.append((u, v))
subtrees.union(u, v)
# Else append it to a cycle
else:
cycles.append((u, v))
history.append((u, v))
return history, mst, cycles
def max_spacing(self, features=['curvature'], num_clusters=2):
"""
Adds cluster group attribute to example based on features and numbe of clusters
using max spcaing algorithm
"""
example_pairs = list(combinations(range(len(self.examples)), 2))
pair_dist = lambda x: self.dist(x, features)
pair_to_edge = lambda x: [x[0], x[1], pair_dist(x)]
edges = list(map(pair_to_edge, example_pairs))
edges.sort(key=lambda x: x[2])
union_find = UnionFind()
#list(map(lambda x: union_find[x], range(len(self.examples))))
current_num_clusters = len(self.examples)
while current_num_clusters != num_clusters:
edge = edges.pop(0)
if union_find[edge[0]] != union_find[edge[1]]:
union_find.union(edge[0], edge[1])
current_num_clusters -= 1
for (index, example) in enumerate(self.examples):
example.cluster_group = union_find[index]
print(list(map(lambda x: x.cluster_group, self.examples)))
def _min_span_edges(self, data=True):
'''
Computes minumum spanning edges between nodes in Graph
based on Kruskal algorithm. Only works for undirected graphs.
Directed graphs will throw an error.
Parameters
==========
data: Bool
Returns
=======
generator
'''
from networkx.utils import UnionFind
##############################################################
# networkx implementation for finding minimum spanning edges #
##############################################################
if self.graph.is_directed():
raise nx.NetworkXError(
"Mimimum spanning tree not defined for directed graphs.")
subtrees = UnionFind()
edges = sorted(self.graph.edges(data=True))
for u, v, d in edges:
if subtrees[u] != subtrees[v]:
if data:
yield (u, v, d)
else:
yield (u, v)
subtrees.union(u, v)
def kruskal_mst_edges(G, minimum, weight='weight', keys=True, data=True):
subtrees = UnionFind()
if G.is_multigraph():
edges = G.edges(keys=True, data=True)
else:
edges = G.edges(data=True)
getweight = lambda t: t[-1].get(weight, 1)
edges = sorted(edges, key=getweight, reverse=not minimum)
is_multigraph = G.is_multigraph()
# Multigraphs need to handle edge keys in addition to edge data.
if is_multigraph:
for u, v, k, d in edges:
if subtrees[u] != subtrees[v]:
if keys:
if data:
yield (u, v, k, d)
else:
yield (u, v, k)
else:
if data:
yield (u, v, d)
else:
yield (u, v)
subtrees.union(u, v)
else:
for u, v, d in edges:
if subtrees[u] != subtrees[v]:
if data:
yield (u, v, d)
else:
yield (u, v)
subtrees.union(u, v)
def get_steiner_tree(self):
if self.steiner_tree is None:
# Path expansion from scratch
self.steiner_tree = nx.Graph()
self.steiner_cost = 0
edges = []
for parent, timestamp in self.terminals.items():
for child in self.metric_steiner_tree[parent]:
if child in self.terminals and self.terminals[
child] > timestamp:
new_path = []
self._fill_path(child, parent, new_path)
for i in range(len(new_path)):
prev_node = new_path[i]
next_node = new_path[i + 1] if (
i + 1) < len(new_path) else parent
heapq.heappush(edges, (self._get_edge_weight(
prev_node, next_node), prev_node, next_node))
# Kruskal's algorithm to break cycles
subtrees = UnionFind()
while edges:
d, u, v = heapq.heappop(edges)
if subtrees[u] != subtrees[v]:
self.steiner_tree.add_edge(u, v, weight=d)
self.steiner_cost += d
subtrees.union(u, v)
return self.steiner_tree, self.steiner_cost
def connectAllNodesInRadius(self, graph, kdTree, radius):
# keep a union-find data structure to improve search performance by not
# allowing cycles in the graph
graphPartitions = UnionFind()
for currNodeLabel, currNodeData in list(graph.nodes(data=True)):
currPos = currNodeData['pos']
# search for all nodes in radius of the current node in question
pointToCheck = currPos.flatten()
neighborLabelsInRadius = kdTree.query_ball_point(
pointToCheck, radius)
# adding all NEW edges that don't collide to the graph
for neighborLabel in neighborLabelsInRadius:
goalPos = graph.getNodeData(neighborLabel, 'pos')
collides = self.checkCollision(currPos, goalPos)
notInSameComponent = self.checkConnectivity(
graphPartitions, currNodeLabel, neighborLabel)
if (not collides) and notInSameComponent:
weight = self.calculateDist(currPos, goalPos)
graph.add_edge(currNodeLabel, neighborLabel, weight=weight)
# need to update union-find data with the new edge
graphPartitions.union(currNodeLabel, neighborLabel)
return graph
def kruskal_mst_edges(G, minimum, weight='weight', keys=True, data=True):
subtrees = UnionFind()
if G.is_multigraph():
edges = G.edges(keys=True, data=True)
else:
edges = G.edges(data=True)
getweight = lambda t: t[-1].get(weight, 1)
edges = sorted(edges, key=getweight, reverse=not minimum)
is_multigraph = G.is_multigraph()
# Multigraphs need to handle edge keys in addition to edge data.
if is_multigraph:
for u, v, k, d in edges:
if subtrees[u] != subtrees[v]:
if keys:
if data:
yield (u, v, k, d)
else:
yield (u, v, k)
else:
if data:
yield (u, v, d)
else:
yield (u, v)
subtrees.union(u, v)
else:
for u, v, d in edges:
if subtrees[u] != subtrees[v]:
if data:
yield (u, v, d)
else:
yield (u, v)
subtrees.union(u, v)
def kruskal_mst_edges(G, weight='weight', data=True):
"""Generate edges in a minimum spanning forest of an undirected
weighted graph.
Parameters
----------
G : NetworkX Graph
weight : string
Edge data key to use for weight (default 'weight').
data : bool, optional
If True yield the edge data along with the edge.
Returns
-------
edges : iterator
A generator that produces edges in the minimum spanning tree.
The edges are three-tuples (u,v,w) where w is the weight.
"""
if G.is_directed():
raise nx.NetworkXError(
"Mimimum spanning tree not defined for directed graphs.")
subtrees = UnionFind()
#average sum of incident edges
track_sum = {}
for edge in G.edges(data=True):
edge = list(edge)
incident_degree = G.degree(edge[0]) + G.degree(edge[1]) - 2
sum_of_incident = sum([
v3['weight'] for v1, v2, v3 in G.edges.data()
if v1 == edge[0] or v2 == edge[0]
])
sum_of_incident += sum([
v3['weight'] for v1, v2, v3 in G.edges.data()
if v1 == edge[1] or v2 == edge[1]
])
average_sum_of_incident = sum_of_incident / incident_degree
edge[2]['weight'] += average_sum_of_incident
track_sum[str(edge[0]) + str(edge[1]) +
str(edge[2]['weight'])] = average_sum_of_incident
"""
track_sum = {}
for edge in G.edges(data=True):
edge = list(edge)
incident_degree = G.degree(edge[0]) + G.degree(edge[1]) - 2
sum_of_incident = sum([v3['weight'] for v1,v2,v3 in G.edges.data() if v1==edge[0] or v2==edge[0]])
sum_of_incident += sum([v3['weight'] for v1,v2,v3 in G.edges.data() if v1==edge[1] or v2==edge[1]])
edge[2]['weight'] += incident_degree
track_sum[str(edge[0]) + str(edge[1]) + str(edge[2]['weight'])] = incident_degree
"""
edges = sorted(G.edges(data=True), key=lambda t: t[2].get(weight, 1))
for u, v, d in edges:
if subtrees[u] != subtrees[v]:
if data:
remove = track_sum.get(str(u) + str(v) + str(d['weight']))
d['weight'] -= remove
yield (u, v, d)
else:
yield (u, v)
subtrees.union(u, v)
def minimum_spanning_edges(G):
"""Generate edges in a minimum spanning forest of an undirected
weighted graph.
A minimum spanning tree is a subgraph of the graph (a tree)
with the minimum sum of edge weights. A spanning forest is a
union of the spanning trees for each connected component of the graph.
Parameters
----------
G : NetworkX Graph
Returns
-------
edges : iterator
A generator that produces edges in the minimum spanning tree.
The edges are three-tuples (u,v,w) where w is the weight.
Examples
--------
>>> G=nx.cycle_graph(4)
>>> G.add_edge(0,3,weight=2) # assign weight 2 to edge 0-3
>>> mst=nx.minimum_spanning_edges(G) # a generator of MST edges
>>> edgelist=list(mst) # make a list of the edges
>>> print(sorted(edgelist))
[(0, 1, {'weight': 1}), (1, 2, {'weight': 1}), (2, 3, {'weight': 1})]
>>> T=nx.Graph(edgelist) # build a graph of the MST.
>>> print(sorted(T.edges(data=True)))
[(0, 1, {'weight': 1}), (1, 2, {'weight': 1}), (2, 3, {'weight': 1})]
Notes
-----
Uses Kruskal's algorithm.
If the graph edges do not have a weight attribute a default weight of 1
will be assigned.
Modified code from David Eppstein, April 2006
http://www.ics.uci.edu/~eppstein/PADS/
"""
# Modified code from David Eppstein, April 2006
# http://www.ics.uci.edu/~eppstein/PADS/
# Kruskal's algorithm: sort edges by weight, and add them one at a time.
# We use Kruskal's algorithm, first because it is very simple to
# implement once UnionFind exists, and second, because the only slow
# part (the sort) is sped up by being built in to Python.
from networkx.utils import UnionFind
if G.is_directed():
raise NetworkXError(\
"Mimimum spanning tree not defined for directed graphs.")
subtrees = UnionFind()
edges = sorted((G[u][v].get('weight', 1), u, v) for u in G for v in G[u])
for W, u, v in edges:
if subtrees[u] != subtrees[v]:
yield (u, v, {'weight': W})
subtrees.union(u, v)
def minimum_spanning_edges(G):
"""Generate edges in a minimum spanning forest of an undirected
weighted graph.
A minimum spanning tree is a subgraph of the graph (a tree)
with the minimum sum of edge weights. A spanning forest is a
union of the spanning trees for each connected component of the graph.
Parameters
----------
G : NetworkX Graph
Returns
-------
edges : iterator
A generator that produces edges in the minimum spanning tree.
The edges are three-tuples (u,v,w) where w is the weight.
Examples
--------
>>> G=nx.cycle_graph(4)
>>> G.add_edge(0,3,weight=2) # assign weight 2 to edge 0-3
>>> mst=nx.minimum_spanning_edges(G) # a generator of MST edges
>>> edgelist=list(mst) # make a list of the edges
>>> print(sorted(edgelist))
[(0, 1, {'weight': 1}), (1, 2, {'weight': 1}), (2, 3, {'weight': 1})]
>>> T=nx.Graph(edgelist) # build a graph of the MST.
>>> print(sorted(T.edges(data=True)))
[(0, 1, {'weight': 1}), (1, 2, {'weight': 1}), (2, 3, {'weight': 1})]
Notes
-----
Uses Kruskal's algorithm.
If the graph edges do not have a weight attribute a default weight of 1
will be assigned.
Modified code from David Eppstein, April 2006
http://www.ics.uci.edu/~eppstein/PADS/
"""
# Modified code from David Eppstein, April 2006
# http://www.ics.uci.edu/~eppstein/PADS/
# Kruskal's algorithm: sort edges by weight, and add them one at a time.
# We use Kruskal's algorithm, first because it is very simple to
# implement once UnionFind exists, and second, because the only slow
# part (the sort) is sped up by being built in to Python.
from networkx.utils import UnionFind
if G.is_directed():
raise NetworkXError(\
"Mimimum spanning tree not defined for directed graphs.")
subtrees = UnionFind()
edges = sorted((G[u][v].get('weight',1),u,v) for u in G for v in G[u])
for W,u,v in edges:
if subtrees[u] != subtrees[v]:
yield (u,v,{'weight':W})
subtrees.union(u,v)
def minimum_spanning_edges(G, weight='weight', data=True):
"""
Generate edges in a minimum spanning forest of an undirected
weighted graph.
A minimum spanning tree is a subgraph of the graph (a tree)
with the minimum sum of edge weights. A spanning forest is a
union of the spanning trees for each connected component of the graph.
:param G: NetworkX Graph
:param weight: Edge data key to use for weight (default 'weight')
:param data: If True yield the edge data along with the edge (optional)
:return edges: iterator. A generator that produces edges in the minimum spanning tree.
The edges are three-tuples (u,v,w) where w is the weight.
Examples
--------
>>> G=nx.cycle_graph(4)
>>> G.add_edge(0,3,weight=2) # assign weight 2 to edge 0-3
>>> mst=nx.minimum_spanning_edges(G,data=False) # a generator of MST edges
>>> edgelist=list(mst) # make a list of the edges
>>> print(sorted(edgelist))
[(0, 1), (1, 2), (2, 3)]
Notes
-----
Uses Kruskal's algorithm.
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/
"""
# Modified code from David Eppstein, April 2006
# http://www.ics.uci.edu/~eppstein/PADS/
# Kruskal's algorithm: sort edges by weight, and add them one at a time.
# We use Kruskal's algorithm, first because it is very simple to
# implement once UnionFind exists, and second, because the only slow
# part (the sort) is sped up by being built in to Python.
from networkx.utils import UnionFind
if G.is_directed():
raise nx.NetworkXError(
"Mimimum spanning tree not defined for directed graphs.")
subtrees = UnionFind()
edges = sorted(G.edges(data=True), key=lambda t: t[2].get(weight, 1.0))
for u, v, d in edges:
if subtrees[u] != subtrees[v]:
if data:
yield (u, v, d)
else:
yield (u, v)
subtrees.union(u, v)
def kruskal(maze):
walls = []
for node in maze.graph.nodes():
walls.extend(w for w in maze.walls(node) if w not in walls)
cells = UnionFind()
for c in maze.graph.nodes(): cells[c]
random.shuffle(walls)
for c1, c2 in walls:
if cells[c1] != cells[c2]:
yield 'connect', (c1, c2)
cells.union(c1, c2)
def minimum_spanning_edges(G, weight='weight', data=False):
if G.is_directed():
raise nx.NetworkXError(
"Mimimum spanning tree not defined for directed graphs.")
subtrees = UnionFind()
edges = list(G.edges())
random.shuffle(edges)
for u, v in edges:
if subtrees[u] != subtrees[v]:
yield (u, v)
subtrees.union(u, v)
def kruskal_mst_edges(G, minimum, weight='weight', data=True):
subtrees = UnionFind()
edges = sorted(G.edges(data=True), key=lambda t: t[2].get(weight, 1),
reverse=not minimum)
for u, v, d in edges:
if subtrees[u] != subtrees[v]:
if data:
yield (u, v, d)
else:
yield (u, v)
subtrees.union(u, v)
def kruskal(maze):
walls = []
for node in maze.graph.nodes():
walls.extend(w for w in maze.walls(node) if w not in walls)
cells = UnionFind()
for c in maze.graph.nodes():
cells[c]
random.shuffle(walls)
for c1, c2 in walls:
if cells[c1] != cells[c2]:
yield 'connect', (c1, c2)
cells.union(c1, c2)
def kruskal_mst_edges(G, minimum, weight='weight', data=True):
subtrees = UnionFind()
edges = sorted(G.edges(data=True),
key=lambda t: t[2].get(weight, 1),
reverse=not minimum)
for u, v, d in edges:
if subtrees[u] != subtrees[v]:
if data:
yield (u, v, d)
else:
yield (u, v)
subtrees.union(u, v)
def mst(g, weight='weight', data=True):
# Initialize a UnionFind() object to store the spanning edges of the MST
mst = UnionFind()
# Sort the edges in ascending order of their weights
edges = sorted(g.edges(data = True), key = lambda t: t[2].get(weight, 1))
# For each edge (u, v) by ordered weight, check if adding the edge
# will create a cycle; if not, add to the mst, else discard the edge
for (u, v, w) in edges:
if mst[u] != mst[v]:
yield (u, v, w)
mst.union(u, v)
def minimum_spanning_edges(G,weight='weight',data=True):
from networkx.utils import UnionFind
if G.is_directed():
raise nx.NetworkXError(
"Mimimum spanning tree not defined for directed graphs.")
subtrees = UnionFind()
edges = sorted(G.edges(data=True),key=lambda t: t[2].get(weight,1))
for u,v,d in edges:
if subtrees[u] != subtrees[v]:
if data:
yield (u,v,d)
else:
yield (u,v)
subtrees.union(u,v)
def boruvka_mst(G):
"""Finds MST using Boruvka's algoritm
Params
--------
G: NetworkX Graph
An input weighted graph to find MST
Returns
--------
T: NetworkX Graph
A minimum spanning tree
"""
T = nx.Graph()
T.add_nodes_from(G.nodes)
forest = UnionFind(G)
def find_edge(comp):
"""Finds the minimum edge for the given connected component"""
minw = np.inf
border = None
for e in nx.edge_boundary(G, comp, data=True):
w = e[-1].get('weight', 1)
if w < minw:
minw = w
border = e
return border
min_edges = (find_edge(comp) for comp in forest.to_sets())
min_edges = [edge for edge in min_edges if edge is not None]
while min_edges:
min_edges = (find_edge(comp) for comp in forest.to_sets())
min_edges = [edge for edge in min_edges if edge is not None]
for u, v, w in min_edges:
if forest[u] != forest[v]:
T.add_edge(u, v, weight=w['weight'])
forest.union(u, v)
return T
def kruskal_mst(G):
"""Generate a minimum spanning tree of an undirected graph.
Uses Kruskal's algorithm.
Parameters
----------
G : NetworkX Graph
Returns
-------
A generator that produces edges in the minimum spanning tree.
The edges are three-tuples (u,v,w) where w is the weight.
Examples
--------
>>> G=nx.cycle_graph(4)
>>> G.add_edge(0,3,weight=2) # assign weight 2 to edge 0-3
>>> mst=nx.kruskal_mst(G) # a generator of MST edges
>>> edgelist=list(mst) # make a list of the edges
>>> print sorted(edgelist)
[(0, 1, {'weight': 1}), (1, 2, {'weight': 1}), (2, 3, {'weight': 1})]
>>> T=nx.Graph(edgelist) # build a graph of the MST.
>>> print sorted(T.edges(data=True))
[(0, 1, {'weight': 1}), (1, 2, {'weight': 1}), (2, 3, {'weight': 1})]
Notes
-----
Modified code from David Eppstein, April 2006
http://www.ics.uci.edu/~eppstein/PADS/
"""
# Modified code from David Eppstein, April 2006
# http://www.ics.uci.edu/~eppstein/PADS/
# Kruskal's algorithm: sort edges by weight, and add them one at a time.
# We use Kruskal's algorithm, first because it is very simple to
# implement once UnionFind exists, and second, because the only slow
# part (the sort) is sped up by being built in to Python.
from networkx.utils import UnionFind
subtrees = UnionFind()
edges = sorted((G[u][v].get('weight', 1), u, v) for u in G for v in G[u])
for W, u, v in edges:
if subtrees[u] != subtrees[v]:
yield (u, v, {'weight': W})
subtrees.union(u, v)
def minimum_spanning_edges(G, weight='weight', data=True):
"""Generate edges in a minimum spanning forest of an undirected
weighted graph.
A minimum spanning tree is a subgraph of the graph (a tree)
with the minimum sum of edge weights. A spanning forest is a
union of the spanning trees for each connected component of the graph.
Parameters
----------
G : NetworkX Graph
weight : string
Edge data key to use for weight (default 'weight').
data : bool, optional
If True yield the edge data along with the edge.
Returns
-------
edges : iterator
A generator that produces edges in the minimum spanning tree.
The edges are three-tuples (u,v,w) where w is the weight.
Notes
-----
Uses Kruskal's algorithm.
"""
from networkx.utils import UnionFind
if G.is_directed():
raise nx.NetworkXError(
"Mimimum spanning tree not defined for directed graphs.")
subtrees = UnionFind()
edges = sorted(G.edges(data=True), key=lambda t: t[2].get(weight, 1))
for u, v, d in edges:
if subtrees[u] != subtrees[v]:
if data:
yield (u, v, d)
else:
yield (u, v)
subtrees.union(u, v)
def kruskal_mst(G):
"""Generate a minimum spanning tree of an undirected graph.
Uses Kruskal's algorithm.
Parameters
----------
G : NetworkX Graph
Returns
-------
A generator that produces edges in the minimum spanning tree.
The edges are three-tuples (u,v,w) where w is the weight.
Examples
--------
>>> G=nx.cycle_graph(4)
>>> G.add_edge(0,3,weight=2) # assign weight 2 to edge 0-3
>>> mst=nx.kruskal_mst(G) # a generator of MST edges
>>> edgelist=list(mst) # make a list of the edges
>>> print sorted(edgelist)
[(0, 1, {'weight': 1}), (1, 2, {'weight': 1}), (2, 3, {'weight': 1})]
>>> T=nx.Graph(edgelist) # build a graph of the MST.
>>> print sorted(T.edges(data=True))
[(0, 1, {'weight': 1}), (1, 2, {'weight': 1}), (2, 3, {'weight': 1})]
Notes
-----
Modified code from David Eppstein, April 2006
http://www.ics.uci.edu/~eppstein/PADS/
"""
# Modified code from David Eppstein, April 2006
# http://www.ics.uci.edu/~eppstein/PADS/
# Kruskal's algorithm: sort edges by weight, and add them one at a time.
# We use Kruskal's algorithm, first because it is very simple to
# implement once UnionFind exists, and second, because the only slow
# part (the sort) is sped up by being built in to Python.
from networkx.utils import UnionFind
subtrees = UnionFind()
edges = sorted((G[u][v].get('weight',1),u,v) for u in G for v in G[u])
for W,u,v in edges:
if subtrees[u] != subtrees[v]:
yield (u,v,{'weight':W})
subtrees.union(u,v)
def krustal(G):
T = nx.Graph()
subtrees = UnionFind()
# make list of all edges in G
edge_pool = list(E(G))
#loop until no more edges left to consider adding
while len(edge_pool) > 0:
#get the shortest edge in the edge list, I don't know how to sort yet
edge = get_min_edge(G, edge_pool)
#remove the edge from the pool for the next iteration
edge_pool.remove(edge)
#check if the edge will form a cycle, if not then add it to the tree
if subtrees[edge[0]] != subtrees[edge[1]]:
subtrees.union(edge[0], edge[1])
T.add_edge(*edge)
return T
def kruskal(G, pos):
subG = nx.empty_graph()
subsets = UnionFind()
edgelist = []
for edge in G.edges:
edgelist.append([edge[0], edge[1], G[edge[0]][edge[1]]['weight']])
edgelist.sort(key=lambda x: x[2])
i = 0
for edge in edgelist:
edge = edgelist[i]
if subsets[edge[0]] != subsets[edge[1]]:
subsets.union(edge[0], edge[1])
subG.add_edge(edge[0], edge[1], weight=edge[2], color='g')
drawGraph(G, pos, subG)
plt.show()
#time.sleep(.5)
i += 1
drawGraph(subG, pos, subG)
plt.show()
return subG
def minimum_spanning_edges(G, weight='weight', data=True):
"""
Examples
--------
>>> G=nx.cycle_graph(4)
>>> G.add_edge(0,3,weight=2) # assign weight 2 to edge 0-3
>>> mst=nx.minimum_spanning_edges(G,data=False) # a generator of MST edges
>>> edgelist=list(mst) # make a list of the edges
>>> print(sorted(edgelist))
[(0, 1), (1, 2), (2, 3)]
Notes
-----
Uses Kruskal's algorithm.
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/
"""
from networkx.utils import UnionFind
if G.is_directed():
raise nx.NetworkXError(
"Mimimum spanning tree not defined for directed graphs.")
subtrees = UnionFind()
edges = sorted(G.edges(data=True), key=lambda t: t[2].get(weight, 1))
probability = 99
for u, v, d in edges:
r = rand.randint(1, 101)
if subtrees[u] != subtrees[v]:
if data and (r in range(probability)):
yield (u, v, d)
else:
yield (u, v)
subtrees.union(u, v)
probability -= 1
def calculate_hamming_clusters(numbers):
numbers_map = defaultdict(list)
for index, node in enumerate(numbers):
numbers_map[node].append(index)
# no duplicates in the union find
union_find = UnionFind(numbers_map)
hamming_distance_one = [1 << i for i in range(24)]
hamming_distance_two = [
1 << i ^ 1 << j for i, j in itertools.combinations(range(24), 2)
]
hamming_distances = [*hamming_distance_one, *hamming_distance_two]
keys = list(numbers_map)
for distance_mask in hamming_distances:
for key in keys:
key2 = key ^ distance_mask
if numbers_map[key2]:
union_find.union(key, key2)
return len(list(union_find.to_sets()))
def lca_networkx(G, root, pairs):
"""
[`networkx.algorithms.lowest_common_ancestor`][nx]
Implemented according to CLRS page 584 (3rd edition).
Compare to [epp]
[nx]: https://github.com/networkx/networkx/master/
[epp]: https://www.ics.uci.edu/~eppstein/PADS/LCA.py
"""
from collections import defaultdict
from networkx.utils import UnionFind, arbitrary_element
from networkx import dfs_postorder_nodes
pair_dict = defaultdict(set)
for u, v in pairs:
pair_dict[u].add(v)
pair_dict[v].add(u)
# Iterative implementation of Tarjan's offline lca algorithm
# as described in CLRS on page 521.
uf = UnionFind()
ancestors = {}
for node in G:
ancestors[node] = uf[node]
colors = defaultdict(bool)
for node in dfs_postorder_nodes(G, root):
colors[node] = True
for v in pair_dict[node]:
if colors[v]:
if (v, node) in pairs:
yield (v, node), ancestors[uf[v]]
if node != root:
parent = arbitrary_element(G.pred[node])
uf.union(parent, node)
ancestors[uf[parent]] = parent
def main():
H, W = pin(2)
Q = pin(1)
uf = UnionFind()
C = Counter()
for _ in [0] * Q:
t = pin(1)
if t == 1:
x, y = pin(2)
x -= 1
y -= 1
C[(x, y)] = 1
for i in [(0, 1), (0, -1), (1, 0), (-1, 0)]:
if C[(x + i[0], y + i[1])] == 1:
uf.union((x, y), ((x + i[0], y + i[1])))
else:
xa, ya, xb, yb = pin(4)
if C[(xa - 1, ya - 1)] == 0:
print("No")
elif uf[(xa - 1, ya - 1)] == uf[(xb - 1, yb - 1)]:
print("Yes")
else:
print("No")
return
def boruvka_mst_edges(G,
minimum=True,
weight='weight',
keys=False,
data=True,
ignore_nan=False):
"""Iterate over edges of a Borůvka's algorithm min/max spanning tree.
Parameters
----------
G : NetworkX Graph
The edges of `G` must have distinct weights,
otherwise the edges may not form a tree.
minimum : bool (default: True)
Find the minimum (True) or maximum (False) spanning tree.
weight : string (default: 'weight')
The name of the edge attribute holding the edge weights.
keys : bool (default: True)
This argument is ignored since this function is not
implemented for multigraphs; it exists only for consistency
with the other minimum spanning tree functions.
data : bool (default: True)
Flag for whether to yield edge attribute dicts.
If True, yield edges `(u, v, d)`, where `d` is the attribute dict.
If False, yield edges `(u, v)`.
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.
"""
# Initialize a forest, assuming initially that it is the discrete
# partition of the nodes of the graph.
forest = UnionFind(G)
def best_edge(component):
"""Returns the optimum (minimum or maximum) edge on the edge
boundary of the given set of nodes.
A return value of ``None`` indicates an empty boundary.
"""
sign = 1 if minimum else -1
minwt = float('inf')
boundary = None
for e in nx.edge_boundary(G, component, data=True):
wt = e[-1].get(weight, 1) * sign
if isnan(wt):
if ignore_nan:
continue
msg = "NaN found as an edge weight. Edge %s"
raise ValueError(msg % (e, ))
if wt < minwt:
minwt = wt
boundary = e
return boundary
# Determine the optimum edge in the edge boundary of each component
# in the forest.
best_edges = (best_edge(component) for component in forest.to_sets())
best_edges = [edge for edge in best_edges if edge is not None]
# If each entry was ``None``, that means the graph was disconnected,
# so we are done generating the forest.
while best_edges:
# Determine the optimum edge in the edge boundary of each
# component in the forest.
#
# This must be a sequence, not an iterator. In this list, the
# same edge may appear twice, in different orientations (but
# that's okay, since a union operation will be called on the
# endpoints the first time it is seen, but not the second time).
#
# Any ``None`` indicates that the edge boundary for that
# component was empty, so that part of the forest has been
# completed.
#
# TODO This can be parallelized, both in the outer loop over
# each component in the forest and in the computation of the
# minimum. (Same goes for the identical lines outside the loop.)
best_edges = (best_edge(component) for component in forest.to_sets())
best_edges = [edge for edge in best_edges if edge is not None]
# Join trees in the forest using the best edges, and yield that
# edge, since it is part of the spanning tree.
#
# TODO This loop can be parallelized, to an extent (the union
# operation must be atomic).
for u, v, d in best_edges:
if forest[u] != forest[v]:
if data:
yield u, v, d
else:
yield u, v
forest.union(u, v)
def tree_all_pairs_lowest_common_ancestor(G, root=None, pairs=None):
r"""Yield the lowest common ancestor for sets of pairs in a tree.
Parameters
----------
G : NetworkX directed graph (must be a tree)
root : node, optional (default: None)
The root of the subtree to operate on.
If None, assume the entire graph has exactly one source and use that.
pairs : iterable or iterator of pairs of nodes, optional (default: None)
The pairs of interest. If None, Defaults to all pairs of nodes
under `root` that have a lowest common ancestor.
Returns
-------
lcas : generator of tuples `((u, v), lca)` where `u` and `v` are nodes
in `pairs` and `lca` is their lowest common ancestor.
Notes
-----
Only defined on non-null trees represented with directed edges from
parents to children. Uses Tarjan's off-line lowest-common-ancestors
algorithm. Runs in time $O(4 \times (V + E + P))$ time, where 4 is the largest
value of the inverse Ackermann function likely to ever come up in actual
use, and $P$ is the number of pairs requested (or $V^2$ if all are needed).
Tarjan, R. E. (1979), "Applications of path compression on balanced trees",
Journal of the ACM 26 (4): 690-715, doi:10.1145/322154.322161.
See Also
--------
all_pairs_lowest_common_ancestor (similar routine for general DAGs)
lowest_common_ancestor (just a single pair for general DAGs)
"""
if len(G) == 0:
raise nx.NetworkXPointlessConcept("LCA meaningless on null graphs.")
elif None in G:
raise nx.NetworkXError("None is not a valid node.")
# Index pairs of interest for efficient lookup from either side.
if pairs is not None:
pair_dict = defaultdict(set)
# See note on all_pairs_lowest_common_ancestor.
if not isinstance(pairs, (Mapping, Set)):
pairs = set(pairs)
for u, v in pairs:
for n in (u, v):
if n not in G:
msg = f"The node {str(n)} is not in the digraph."
raise nx.NodeNotFound(msg)
pair_dict[u].add(v)
pair_dict[v].add(u)
# If root is not specified, find the exactly one node with in degree 0 and
# use it. Raise an error if none are found, or more than one is. Also check
# for any nodes with in degree larger than 1, which would imply G is not a
# tree.
if root is None:
for n, deg in G.in_degree:
if deg == 0:
if root is not None:
msg = "No root specified and tree has multiple sources."
raise nx.NetworkXError(msg)
root = n
elif deg > 1:
msg = "Tree LCA only defined on trees; use DAG routine."
raise nx.NetworkXError(msg)
if root is None:
raise nx.NetworkXError("Graph contains a cycle.")
# Iterative implementation of Tarjan's offline lca algorithm
# as described in CLRS on page 521 (2nd edition)/page 584 (3rd edition)
uf = UnionFind()
ancestors = {}
for node in G:
ancestors[node] = uf[node]
colors = defaultdict(bool)
for node in nx.dfs_postorder_nodes(G, root):
colors[node] = True
for v in (pair_dict[node] if pairs is not None else G):
if colors[v]:
# If the user requested both directions of a pair, give it.
# Otherwise, just give one.
if pairs is not None and (node, v) in pairs:
yield (node, v), ancestors[uf[v]]
if pairs is None or (v, node) in pairs:
yield (v, node), ancestors[uf[v]]
if node != root:
parent = arbitrary_element(G.pred[node])
uf.union(parent, node)
ancestors[uf[parent]] = parent
def kruskal_mst_edges(G, minimum, weight='weight',
keys=True, data=True, ignore_nan=False):
"""Iterate over edges of a Kruskal's algorithm min/max spanning tree.
Parameters
----------
G : NetworkX Graph
The graph holding the tree of interest.
minimum : bool (default: True)
Find the minimum (True) or maximum (False) spanning tree.
weight : string (default: 'weight')
The name of the edge attribute holding the edge weights.
keys : bool (default: True)
If `G` is a multigraph, `keys` controls whether edge keys ar yielded.
Otherwise `keys` is ignored.
data : bool (default: True)
Flag for whether to yield edge attribute dicts.
If True, yield edges `(u, v, d)`, where `d` is the attribute dict.
If False, yield edges `(u, v)`.
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.
"""
subtrees = UnionFind()
if G.is_multigraph():
edges = G.edges(keys=True, data=True)
def filter_nan_edges(edges=edges, weight=weight):
sign = 1 if minimum else -1
for u, v, k, d in edges:
wt = d.get(weight, 1) * sign
if isnan(wt):
if ignore_nan:
continue
msg = "NaN found as an edge weight. Edge %s"
raise ValueError(msg % ((u, v, k, d),))
yield wt, u, v, k, d
else:
edges = G.edges(data=True)
def filter_nan_edges(edges=edges, weight=weight):
sign = 1 if minimum else -1
for u, v, d in edges:
wt = d.get(weight, 1) * sign
if isnan(wt):
if ignore_nan:
continue
msg = "NaN found as an edge weight. Edge %s"
raise ValueError(msg % ((u, v, d),))
yield wt, u, v, d
edges = sorted(filter_nan_edges(), key=itemgetter(0))
# Multigraphs need to handle edge keys in addition to edge data.
if G.is_multigraph():
for wt, u, v, k, d in edges:
if subtrees[u] != subtrees[v]:
if keys:
if data:
yield u, v, k, d
else:
yield u, v, k
else:
if data:
yield u, v, d
else:
yield u, v
subtrees.union(u, v)
else:
for wt, u, v, d in edges:
if subtrees[u] != subtrees[v]:
if data:
yield (u, v, d)
else:
yield (u, v)
subtrees.union(u, v)
def kruskal_mst_edges(G,
minimum,
weight='weight',
keys=True,
data=True,
ignore_nan=False):
"""Iterate over edges of a Kruskal's algorithm min/max spanning tree.
Parameters
----------
G : NetworkX Graph
The graph holding the tree of interest.
minimum : bool (default: True)
Find the minimum (True) or maximum (False) spanning tree.
weight : string (default: 'weight')
The name of the edge attribute holding the edge weights.
keys : bool (default: True)
If `G` is a multigraph, `keys` controls whether edge keys ar yielded.
Otherwise `keys` is ignored.
data : bool (default: True)
Flag for whether to yield edge attribute dicts.
If True, yield edges `(u, v, d)`, where `d` is the attribute dict.
If False, yield edges `(u, v)`.
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.
"""
subtrees = UnionFind()
if G.is_multigraph():
edges = G.edges(keys=True, data=True)
def filter_nan_edges(edges=edges, weight=weight):
sign = 1 if minimum else -1
for u, v, k, d in edges:
wt = d.get(weight, 1) * sign
if isnan(wt):
if ignore_nan:
continue
msg = "NaN found as an edge weight. Edge %s"
raise ValueError(msg % ((u, v, f, k, d), ))
yield wt, u, v, k, d
else:
edges = G.edges(data=True)
def filter_nan_edges(edges=edges, weight=weight):
sign = 1 if minimum else -1
for u, v, d in edges:
wt = d.get(weight, 1) * sign
if isnan(wt):
if ignore_nan:
continue
msg = "NaN found as an edge weight. Edge %s"
raise ValueError(msg % ((u, v, d), ))
yield wt, u, v, d
edges = sorted(filter_nan_edges(), key=itemgetter(0))
# Multigraphs need to handle edge keys in addition to edge data.
if G.is_multigraph():
for wt, u, v, k, d in edges:
if subtrees[u] != subtrees[v]:
if keys:
if data:
yield u, v, k, d
else:
yield u, v, k
else:
if data:
yield u, v, d
else:
yield u, v
subtrees.union(u, v)
else:
for wt, u, v, d in edges:
if subtrees[u] != subtrees[v]:
if data:
yield (u, v, d)
else:
yield (u, v)
subtrees.union(u, v)
if __name__ == "__main__":
filename = "clustering_big.txt"
with open(filename, "r") as f:
lines = f.readlines()
n_nodes, n_bits = map(int, lines[0].split())
print(f'{n_nodes} nodes')
print(f'{n_bits} bits per node')
numbers = [int(''.join(line.split()), 2) for line in lines[1:]]
nodes = {}
for node, num in enumerate(numbers):
if num not in nodes:
nodes[num] = set()
nodes[num].add(node)
uf = UnionFind(range(n_nodes))
distances = [1 << i for i in range(n_bits)]
distances += [(1 << ix_1) ^ (1 << ix_2) for (ix_1, ix_2) in itertools.combinations(range(n_bits), 2)]
distances.append(0)
for distance in distances:
for number in nodes.keys():
if (number ^ distance) in nodes:
for node_from in nodes[number]:
for node_to in nodes[number ^ distance]:
uf.union(node_from, node_to)
print(len(list(uf.to_sets()))) # 6118
def boruvka_mst_edges(G, minimum=True, weight='weight', keys=False, data=True):
"""Iterates over the edges of a minimum spanning tree as computed by
Borůvka's algorithm.
`G` is a NetworkX graph. Also, the edges must have distinct weights,
otherwise the edges may not form a tree.
`weight` is the edge attribute that stores the edge weights. Each
edge in the graph must have such an attribute, otherwise a
:exc:`KeyError` will be raised.
If `data` is True, this iterator yields edges of the form
``(u, v, d)``, where ``u`` and ``v`` are nodes and ``d`` is the edge
attribute dictionary. Otherwise, it yields edges of the form
``(u, v)``.
The `keys` argument is ignored, since this function is not
implemented for multigraphs; it exists only for consistency with the
other minimum spanning tree functions.
"""
opt = min if minimum else max
# Initialize a forest, assuming initially that it is the discrete
# partition of the nodes of the graph.
forest = UnionFind(G)
def best_edge(component):
"""Returns the optimum (minimum or maximum) edge on the edge
boundary of the given set of nodes.
A return value of ``None`` indicates an empty boundary.
"""
# TODO In Python 3.4 and later, we can just do
#
# boundary = nx.edge_boundary(G, component, data=weight)
# return opt(boundary, key=lambda e: e[-1][weight], default=None)
#
# which is better because it doesn't require creating a list.
boundary = list(nx.edge_boundary(G, component, data=True))
if not boundary:
return None
return opt(boundary, key=lambda e: e[-1][weight])
# Determine the optimum edge in the edge boundary of each component
# in the forest.
best_edges = (best_edge(component) for component in forest.to_sets())
best_edges = [edge for edge in best_edges if edge is not None]
# If each entry was ``None``, that means the graph was disconnected,
# so we are done generating the forest.
while best_edges:
# Determine the optimum edge in the edge boundary of each
# component in the forest.
#
# This must be a sequence, not an iterator. In this list, the
# same edge may appear twice, in different orientations (but
# that's okay, since a union operation will be called on the
# endpoints the first time it is seen, but not the second time).
#
# Any ``None`` indicates that the edge boundary for that
# component was empty, so that part of the forest has been
# completed.
#
# TODO This can be parallelized, both in the outer loop over
# each component in the forest and in the computation of the
# minimum. (Same goes for the identical lines outside the loop.)
best_edges = (best_edge(component) for component in forest.to_sets())
best_edges = [edge for edge in best_edges if edge is not None]
# Join trees in the forest using the best edges, and yield that
# edge, since it is part of the spanning tree.
#
# TODO This loop can be parallelized, to an extent (the union
# operation must be atomic).
for u, v, d in best_edges:
if forest[u] != forest[v]:
if data:
yield u, v, d
else:
yield u, v
forest.union(u, v)
def kruskal_mst_edges(G, weight='weight', data=True):
"""Generate edges in a minimum spanning forest of an undirected
weighted graph.
Parameters
----------
G : NetworkX Graph
weight : string
Edge data key to use for weight (default 'weight').
data : bool, optional
If True yield the edge data along with the edge.
Returns
-------
edges : iterator
A generator that produces edges in the minimum spanning tree.
The edges are three-tuples (u,v,w) where w is the weight.
"""
subtrees = UnionFind()
edges = sorted(
G.edges(data=True),
key=lambda t: t[2].get("weight")) #sorted by edge weights first
edges_no_weights = [e[0:2] for e in edges] #same order as edges
edges_copy = edges.copy()
available_edges = len(G.nodes) - 1
dominated = set() #set of dominated vertices
num_edges_mst = 0
while num_edges_mst < len(G.nodes):
i = 0
u, v, d = edges_copy[0]
while (u in dominated and v in dominated and i + 1 < len(edges_copy)
and subtrees[u] == subtrees[v]):
i += 1
u, v, d = edges_copy[i]
if subtrees[u] != subtrees[v]:
dominated.add(u)
dominated.add(v)
new_v_reached = 0
u_subtree = []
v_subtree = []
#for loop to find the number of new vertices reached:
for x, y, w in edges_copy:
if x in set(G.__getitem__(u)) or y in set(
G.__getitem__(v)): #neighbors of u and v
if x not in dominated:
new_v_reached += 1
if y not in dominated:
new_v_reached += 1
#all parts of current mst
subtrees.union(u, v)
curr_mst_vertices = [list(s) for s in subtrees.to_sets() if u in s]
curr_mst_vertices = curr_mst_vertices[0]
current_tree = nx.Graph()
for v1, v2, v3 in G.edges.data():
if v1 in curr_mst_vertices and v2 in curr_mst_vertices:
edges_ = edges[edges_no_weights.index((v1, v2))]
current_tree.add_edge(v1, v2, weight=v3['weight'])
before = average_pairwise_distance(current_tree)
#add one edge to find increase in cost
for v1, v2, v3 in edges_copy:
#print(v1,v2,v3)
new_tree = current_tree.copy()
#edge (u, X) (v, X) (X, u) (X, v)
if (v1 == u and v2 != v) or (v1 == v and v2 != v) or (
v2 == u and v1 != v) or (v2 == v and v1 != u):
new_tree.add_edge(v1, v2, weight=v3['weight'])
after = average_pairwise_distance(new_tree)
edge_update = edges_copy[edges_no_weights.index((v1, v2))]
edge_update_list = list(edge_update[0:2])
if after - before > 0 and new_v_reached != 0:
edge_update_list.append(
{'weight': ((after - before) / new_v_reached)})
edges_copy[edges_no_weights.index(
(v1, v2))] = edge_update_list
#update edges_copy
edges_copy = sorted(edges_copy, key=lambda x: x[2]['weight'])
#available_edges -= 1
num_edges_mst += 1
yield (u, v, edges[edges_no_weights.index((u, v))][2])
if num_edges_mst == len(G.nodes) - 1:
break
def minimum_spanning_edges(G, weight="weight", data=True):
"""Generate edges in a minimum spanning forest of an undirected
weighted graph.
A minimum spanning tree is a subgraph of the graph (a tree)
with the minimum sum of edge weights. A spanning forest is a
union of the spanning trees for each connected component of the graph.
Parameters
----------
G : NetworkX Graph
weight : string
Edge data key to use for weight (default 'weight').
data : bool, optional
If True yield the edge data along with the edge.
Returns
-------
edges : iterator
A generator that produces edges in the minimum spanning tree.
The edges are three-tuples (u,v,w) where w is the weight.
Examples
--------
>>> G=nx.cycle_graph(4)
>>> G.add_edge(0,3,weight=2) # assign weight 2 to edge 0-3
>>> mst=nx.minimum_spanning_edges(G,data=False) # a generator of MST edges
>>> edgelist=list(mst) # make a list of the edges
>>> print(sorted(edgelist))
[(0, 1), (1, 2), (2, 3)]
Notes
-----
Uses Kruskal's algorithm.
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/
"""
# Modified code from David Eppstein, April 2006
# http://www.ics.uci.edu/~eppstein/PADS/
# Kruskal's algorithm: sort edges by weight, and add them one at a time.
# We use Kruskal's algorithm, first because it is very simple to
# implement once UnionFind exists, and second, because the only slow
# part (the sort) is sped up by being built in to Python.
from networkx.utils import UnionFind
if G.is_directed():
raise nx.NetworkXError("Mimimum spanning tree not defined for directed graphs.")
subtrees = UnionFind()
edges = sorted(G.edges(data=True), key=lambda t: t[2].get(weight, 1))
for u, v, d in edges:
if subtrees[u] != subtrees[v]:
if data:
yield (u, v, d)
else:
yield (u, v)
subtrees.union(u, v)
def boruvka_mst_edges(G, minimum=True, weight='weight',
keys=False, data=True, ignore_nan=False):
"""Iterate over edges of a Borůvka's algorithm min/max spanning tree.
Parameters
----------
G : NetworkX Graph
The edges of `G` must have distinct weights,
otherwise the edges may not form a tree.
minimum : bool (default: True)
Find the minimum (True) or maximum (False) spanning tree.
weight : string (default: 'weight')
The name of the edge attribute holding the edge weights.
keys : bool (default: True)
This argument is ignored since this function is not
implemented for multigraphs; it exists only for consistency
with the other minimum spanning tree functions.
data : bool (default: True)
Flag for whether to yield edge attribute dicts.
If True, yield edges `(u, v, d)`, where `d` is the attribute dict.
If False, yield edges `(u, v)`.
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.
"""
# Initialize a forest, assuming initially that it is the discrete
# partition of the nodes of the graph.
forest = UnionFind(G)
def best_edge(component):
"""Returns the optimum (minimum or maximum) edge on the edge
boundary of the given set of nodes.
A return value of ``None`` indicates an empty boundary.
"""
sign = 1 if minimum else -1
minwt = float('inf')
boundary = None
for e in nx.edge_boundary(G, component, data=True):
wt = e[-1].get(weight, 1) * sign
if isnan(wt):
if ignore_nan:
continue
msg = "NaN found as an edge weight. Edge %s"
raise ValueError(msg % (e,))
if wt < minwt:
minwt = wt
boundary = e
return boundary
# Determine the optimum edge in the edge boundary of each component
# in the forest.
best_edges = (best_edge(component) for component in forest.to_sets())
best_edges = [edge for edge in best_edges if edge is not None]
# If each entry was ``None``, that means the graph was disconnected,
# so we are done generating the forest.
while best_edges:
# Determine the optimum edge in the edge boundary of each
# component in the forest.
#
# This must be a sequence, not an iterator. In this list, the
# same edge may appear twice, in different orientations (but
# that's okay, since a union operation will be called on the
# endpoints the first time it is seen, but not the second time).
#
# Any ``None`` indicates that the edge boundary for that
# component was empty, so that part of the forest has been
# completed.
#
# TODO This can be parallelized, both in the outer loop over
# each component in the forest and in the computation of the
# minimum. (Same goes for the identical lines outside the loop.)
best_edges = (best_edge(component) for component in forest.to_sets())
best_edges = [edge for edge in best_edges if edge is not None]
# Join trees in the forest using the best edges, and yield that
# edge, since it is part of the spanning tree.
#
# TODO This loop can be parallelized, to an extent (the union
# operation must be atomic).
for u, v, d in best_edges:
if forest[u] != forest[v]:
if data:
yield u, v, d
else:
yield u, v
forest.union(u, v)
def kruskal_mst_edges(G,
minimum,
weight="weight",
keys=True,
data=True,
ignore_nan=False,
partition=None):
"""
Iterate over edge of a Kruskal's algorithm min/max spanning tree.
Parameters
----------
G : NetworkX Graph
The graph holding the tree of interest.
minimum : bool (default: True)
Find the minimum (True) or maximum (False) spanning tree.
weight : string (default: 'weight')
The name of the edge attribute holding the edge weights.
keys : bool (default: True)
If `G` is a multigraph, `keys` controls whether edge keys ar yielded.
Otherwise `keys` is ignored.
data : bool (default: True)
Flag for whether to yield edge attribute dicts.
If True, yield edges `(u, v, d)`, where `d` is the attribute dict.
If False, yield edges `(u, v)`.
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.
partition : string (default: None)
The name of the edge attribute holding the partition data, if it exists.
Partition data is written to the edges using the `EdgePartition` enum.
If a partition exists, all included edges and none of the excluded edges
will appear in the final tree. Open edges may or may not be used.
Yields
------
edge tuple
The edges as discovered by Kruskal's method. Each edge can
take the following forms: `(u, v)`, `(u, v, d)` or `(u, v, k, d)`
depending on the `key` and `data` parameters
"""
subtrees = UnionFind()
if G.is_multigraph():
edges = G.edges(keys=True, data=True)
else:
edges = G.edges(data=True)
"""
Sort the edges of the graph with respect to the partition data.
Edges are returned in the following order:
* Included edges
* Open edges from smallest to largest weight
* Excluded edges
"""
included_edges = []
open_edges = []
for e in edges:
d = e[-1]
wt = d.get(weight, 1)
if isnan(wt):
if ignore_nan:
continue
raise ValueError(f"NaN found as an edge weight. Edge {e}")
edge = (wt, ) + e
if d.get(partition) == EdgePartition.INCLUDED:
included_edges.append(edge)
elif d.get(partition) == EdgePartition.EXCLUDED:
continue
else:
open_edges.append(edge)
if minimum:
sorted_open_edges = sorted(open_edges, key=itemgetter(0))
else:
sorted_open_edges = sorted(open_edges, key=itemgetter(0), reverse=True)
# Condense the lists into one
included_edges.extend(sorted_open_edges)
sorted_edges = included_edges
del open_edges, sorted_open_edges, included_edges
# Multigraphs need to handle edge keys in addition to edge data.
if G.is_multigraph():
for wt, u, v, k, d in sorted_edges:
if subtrees[u] != subtrees[v]:
if keys:
if data:
yield u, v, k, d
else:
yield u, v, k
else:
if data:
yield u, v, d
else:
yield u, v
subtrees.union(u, v)
else:
for wt, u, v, d in sorted_edges:
if subtrees[u] != subtrees[v]:
if data:
yield u, v, d
else:
yield u, v
subtrees.union(u, v)
def boruvka_mst_edges(G, minimum=True, weight='weight', keys=False, data=True):
"""Iterates over the edges of a minimum spanning tree as computed by
Borůvka's algorithm.
`G` is a NetworkX graph. Also, the edges must have distinct weights,
otherwise the edges may not form a tree.
`weight` is the edge attribute that stores the edge weights. Each
edge in the graph must have such an attribute, otherwise a
:exc:`KeyError` will be raised.
If `data` is True, this iterator yields edges of the form
``(u, v, d)``, where ``u`` and ``v`` are nodes and ``d`` is the edge
attribute dictionary. Otherwise, it yields edges of the form
``(u, v)``.
The `keys` argument is ignored, since this function is not
implemented for multigraphs; it exists only for consistency with the
other minimum spanning tree functions.
"""
opt = min if minimum else max
# Initialize a forest, assuming initially that it is the discrete
# partition of the nodes of the graph.
forest = UnionFind(G)
def best_edge(component):
"""Returns the optimum (minimum or maximum) edge on the edge
boundary of the given set of nodes.
A return value of ``None`` indicates an empty boundary.
"""
# TODO In Python 3.4 and later, we can just do
#
# boundary = nx.edge_boundary(G, component, data=weight)
# return opt(boundary, key=lambda e: e[-1][weight], default=None)
#
# which is better because it doesn't require creating a list.
boundary = list(nx.edge_boundary(G, component, data=True))
if not boundary:
return None
return opt(boundary, key=lambda e: e[-1][weight])
# Determine the optimum edge in the edge boundary of each component
# in the forest.
best_edges = (best_edge(component) for component in forest.to_sets())
best_edges = [edge for edge in best_edges if edge is not None]
# If each entry was ``None``, that means the graph was disconnected,
# so we are done generating the forest.
while best_edges:
# Determine the optimum edge in the edge boundary of each
# component in the forest.
#
# This must be a sequence, not an iterator. In this list, the
# same edge may appear twice, in different orientations (but
# that's okay, since a union operation will be called on the
# endpoints the first time it is seen, but not the second time).
#
# Any ``None`` indicates that the edge boundary for that
# component was empty, so that part of the forest has been
# completed.
#
# TODO This can be parallelized, both in the outer loop over
# each component in the forest and in the computation of the
# minimum. (Same goes for the identical lines outside the loop.)
best_edges = (best_edge(component) for component in forest.to_sets())
best_edges = [edge for edge in best_edges if edge is not None]
# Join trees in the forest using the best edges, and yield that
# edge, since it is part of the spanning tree.
#
# TODO This loop can be parallelized, to an extent (the union
# operation must be atomic).
for u, v, d in best_edges:
if forest[u] != forest[v]:
if data:
yield u, v, d
else:
yield u, v
forest.union(u, v)
from networkx.utils import UnionFind
H, W = map(int, input().split())
Q = int(input())
P = set()
uf = UnionFind()
for _ in range(Q):
t, *q = map(int, input().split())
if t == 1:
x, y = (a-1 for a in q)
P.add((x, y))
for u, v in [(x, y+1), (x, y-1), (x+1, y), (x-1, y)]:
if (u, v) in P:
uf.union((x, y), (u, v))
else:
xa, ya, xb, yb = (a-1 for a in q)
if {(xa, ya), (xb, yb)}<=P and uf[(xa, ya)]==uf[(xb, yb)]:
print("Yes")
else:
print("No")
def tree_all_pairs_lowest_common_ancestor(G, root=None, pairs=None):
r"""Yield the lowest common ancestor for sets of pairs in a tree.
Parameters
----------
G : NetworkX directed graph (must be a tree)
root : node, optional (default: None)
The root of the subtree to operate on.
If None, assume the entire graph has exactly one source and use that.
pairs : iterable or iterator of pairs of nodes, optional (default: None)
The pairs of interest. If None, Defaults to all pairs of nodes
under `root` that have a lowest common ancestor.
Returns
-------
lcas : generator of tuples `((u, v), lca)` where `u` and `v` are nodes
in `pairs` and `lca` is their lowest common ancestor.
Notes
-----
Only defined on non-null trees represented with directed edges from
parents to children. Uses Tarjan's off-line lowest-common-ancestors
algorithm. Runs in time $O(4 \times (V + E + P))$ time, where 4 is the largest
value of the inverse Ackermann function likely to ever come up in actual
use, and $P$ is the number of pairs requested (or $V^2$ if all are needed).
Tarjan, R. E. (1979), "Applications of path compression on balanced trees",
Journal of the ACM 26 (4): 690-715, doi:10.1145/322154.322161.
See Also
--------
all_pairs_lowest_common_ancestor (similar routine for general DAGs)
lowest_common_ancestor (just a single pair for general DAGs)
"""
if len(G) == 0:
raise nx.NetworkXPointlessConcept("LCA meaningless on null graphs.")
elif None in G:
raise nx.NetworkXError("None is not a valid node.")
# Index pairs of interest for efficient lookup from either side.
if pairs is not None:
pair_dict = defaultdict(set)
# See note on all_pairs_lowest_common_ancestor.
if not isinstance(pairs, (Mapping, Set)):
pairs = set(pairs)
for u, v in pairs:
for n in (u, v):
if n not in G:
msg = "The node %s is not in the digraph." % str(n)
raise nx.NodeNotFound(msg)
pair_dict[u].add(v)
pair_dict[v].add(u)
# If root is not specified, find the exactly one node with in degree 0 and
# use it. Raise an error if none are found, or more than one is. Also check
# for any nodes with in degree larger than 1, which would imply G is not a
# tree.
if root is None:
for n, deg in G.in_degree:
if deg == 0:
if root is not None:
msg = "No root specified and tree has multiple sources."
raise nx.NetworkXError(msg)
root = n
elif deg > 1:
msg = "Tree LCA only defined on trees; use DAG routine."
raise nx.NetworkXError(msg)
if root is None:
raise nx.NetworkXError("Graph contains a cycle.")
# Iterative implementation of Tarjan's offline lca algorithm
# as described in CLRS on page 521 (2nd edition)/page 584 (3rd edition)
uf = UnionFind()
ancestors = {}
for node in G:
ancestors[node] = uf[node]
colors = defaultdict(bool)
for node in nx.dfs_postorder_nodes(G, root):
colors[node] = True
for v in (pair_dict[node] if pairs is not None else G):
if colors[v]:
# If the user requested both directions of a pair, give it.
# Otherwise, just give one.
if pairs is not None and (node, v) in pairs:
yield (node, v), ancestors[uf[v]]
if pairs is None or (v, node) in pairs:
yield (v, node), ancestors[uf[v]]
if node != root:
parent = arbitrary_element(G.pred[node])
uf.union(parent, node)
ancestors[uf[parent]] = parent