def max_independent_set(tree, root, weight_attribute='w'):
''' Finds the maximum independent set in rooted tree.
The weight of every node is taken from each node's
weight_attribute attribute. Returns (|MIS|, {MIS}),
where |MIS| is the total weight of nodes in the
set and {MIS} is the set of nodes. '''
if tree.is_directed():
raise NotUndirectedGraph("Tree must be undirected graph!")
f_plus, f_minus = {}, {}
max_set = set()
children, levels, _ = _get_children(tree, root)
weight = lambda u: tree.nodes[u][weight_attribute]
for level in levels[::-1]:
for u in level:
if u in children:
f_plus[u] = weight(u) + sum(f_minus[v]
for v in children[u])
f_minus[u] = sum(max(f_plus[v], f_minus[v])
for v in children[u])
else:
f_plus[u], f_minus[u] = weight(u), 0
def node_is_added(predecessor_is_added, node):
return (not predecessor_is_added and
f_plus[node] >= f_minus[node])
stack = deque([(root, node_is_added(False, root))])
while stack:
u, added = stack.pop()
if added:
max_set.add(u)
if u not in children:
continue
for v in children[u]:
stack.append((v, node_is_added(added, v)))
return max(f_plus[root], f_minus[root]), max_set
def _assign_level_labels(tree, root):
''' Assings labels to each node. Yeilds the different labels on
each level of the tree. For more information, search for AHU
algorithm for three isomorphism. '''
labels = {u: 0 for u in tree if tree.degree(u) == 1 and u != root}
children_labels = {}
children, levels, height = _get_children(tree, root)
sort = lambda x: tuple(sorted(x)) # helper
for level in levels[:-1][::-1]:
nodes_with_label = defaultdict(list)
different_labels = set()
for node in level:
if node in children: # if node has children
for child in children[node]: # for each children
if node in children_labels:
children_labels[node] += (labels[child],)
else:
children_labels[node] = (labels[child],)
children_labels[node] = sort(children_labels[node])
# map the children labels to the node
nodes_with_label[children_labels[node]].append(node)
# to compute different labels on the current level
different_labels.add(children_labels[node])
different_labels = list(different_labels)
_tuple_sort(different_labels)
yield different_labels
for k, node_children_label in enumerate(different_labels):
for node in nodes_with_label[node_children_label]:
labels[node] = k # k-th distinct tuple
