228 lines
7.9 KiB
Python
228 lines
7.9 KiB
Python
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"""Lukes Algorithm for exact optimal weighted tree partitioning."""
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from copy import deepcopy
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from functools import lru_cache
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from random import choice
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import networkx as nx
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from networkx.utils import not_implemented_for
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__all__ = ["lukes_partitioning"]
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D_EDGE_W = "weight"
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D_EDGE_VALUE = 1.0
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D_NODE_W = "weight"
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D_NODE_VALUE = 1
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PKEY = "partitions"
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CLUSTER_EVAL_CACHE_SIZE = 2048
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def _split_n_from(n, min_size_of_first_part):
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# splits j in two parts of which the first is at least
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# the second argument
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assert n >= min_size_of_first_part
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for p1 in range(min_size_of_first_part, n + 1):
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yield p1, n - p1
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def lukes_partitioning(G, max_size, node_weight=None, edge_weight=None):
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"""Optimal partitioning of a weighted tree using the Lukes algorithm.
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This algorithm partitions a connected, acyclic graph featuring integer
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node weights and float edge weights. The resulting clusters are such
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that the total weight of the nodes in each cluster does not exceed
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max_size and that the weight of the edges that are cut by the partition
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is minimum. The algorithm is based on LUKES[1].
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Parameters
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----------
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G : graph
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max_size : int
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Maximum weight a partition can have in terms of sum of
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node_weight for all nodes in the partition
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edge_weight : key
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Edge data key to use as weight. If None, the weights are all
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set to one.
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node_weight : key
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Node data key to use as weight. If None, the weights are all
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set to one. The data must be int.
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Returns
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-------
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partition : list
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A list of sets of nodes representing the clusters of the
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partition.
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Raises
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------
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NotATree
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If G is not a tree.
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TypeError
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If any of the values of node_weight is not int.
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References
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----------
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.. Lukes, J. A. (1974).
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"Efficient Algorithm for the Partitioning of Trees."
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IBM Journal of Research and Development, 18(3), 217–224.
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"""
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# First sanity check and tree preparation
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if not nx.is_tree(G):
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raise nx.NotATree("lukes_partitioning works only on trees")
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else:
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if nx.is_directed(G):
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root = [n for n, d in G.in_degree() if d == 0]
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assert len(root) == 1
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root = root[0]
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t_G = deepcopy(G)
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else:
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root = choice(list(G.nodes))
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# this has the desirable side effect of not inheriting attributes
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t_G = nx.dfs_tree(G, root)
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# Since we do not want to screw up the original graph,
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# if we have a blank attribute, we make a deepcopy
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if edge_weight is None or node_weight is None:
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safe_G = deepcopy(G)
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if edge_weight is None:
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nx.set_edge_attributes(safe_G, D_EDGE_VALUE, D_EDGE_W)
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edge_weight = D_EDGE_W
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if node_weight is None:
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nx.set_node_attributes(safe_G, D_NODE_VALUE, D_NODE_W)
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node_weight = D_NODE_W
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else:
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safe_G = G
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# Second sanity check
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# The values of node_weight MUST BE int.
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# I cannot see any room for duck typing without incurring serious
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# danger of subtle bugs.
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all_n_attr = nx.get_node_attributes(safe_G, node_weight).values()
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for x in all_n_attr:
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if not isinstance(x, int):
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raise TypeError(
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"lukes_partitioning needs integer "
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f"values for node_weight ({node_weight})"
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)
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# SUBROUTINES -----------------------
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# these functions are defined here for two reasons:
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# - brevity: we can leverage global "safe_G"
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# - caching: signatures are hashable
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@not_implemented_for("undirected")
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# this is intended to be called only on t_G
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def _leaves(gr):
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for x in gr.nodes:
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if not nx.descendants(gr, x):
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yield x
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@not_implemented_for("undirected")
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def _a_parent_of_leaves_only(gr):
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tleaves = set(_leaves(gr))
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for n in set(gr.nodes) - tleaves:
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if all([x in tleaves for x in nx.descendants(gr, n)]):
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return n
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@lru_cache(CLUSTER_EVAL_CACHE_SIZE)
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def _value_of_cluster(cluster):
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valid_edges = [e for e in safe_G.edges if e[0] in cluster and e[1] in cluster]
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return sum(safe_G.edges[e][edge_weight] for e in valid_edges)
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def _value_of_partition(partition):
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return sum(_value_of_cluster(frozenset(c)) for c in partition)
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@lru_cache(CLUSTER_EVAL_CACHE_SIZE)
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def _weight_of_cluster(cluster):
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return sum(safe_G.nodes[n][node_weight] for n in cluster)
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def _pivot(partition, node):
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ccx = [c for c in partition if node in c]
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assert len(ccx) == 1
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return ccx[0]
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def _concatenate_or_merge(partition_1, partition_2, x, i, ref_weigth):
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ccx = _pivot(partition_1, x)
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cci = _pivot(partition_2, i)
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merged_xi = ccx.union(cci)
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# We first check if we can do the merge.
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# If so, we do the actual calculations, otherwise we concatenate
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if _weight_of_cluster(frozenset(merged_xi)) <= ref_weigth:
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cp1 = list(filter(lambda x: x != ccx, partition_1))
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cp2 = list(filter(lambda x: x != cci, partition_2))
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option_2 = [merged_xi] + cp1 + cp2
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return option_2, _value_of_partition(option_2)
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else:
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option_1 = partition_1 + partition_2
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return option_1, _value_of_partition(option_1)
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# INITIALIZATION -----------------------
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leaves = set(_leaves(t_G))
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for lv in leaves:
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t_G.nodes[lv][PKEY] = dict()
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slot = safe_G.nodes[lv][node_weight]
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t_G.nodes[lv][PKEY][slot] = [{lv}]
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t_G.nodes[lv][PKEY][0] = [{lv}]
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for inner in [x for x in t_G.nodes if x not in leaves]:
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t_G.nodes[inner][PKEY] = dict()
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slot = safe_G.nodes[inner][node_weight]
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t_G.nodes[inner][PKEY][slot] = [{inner}]
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# CORE ALGORITHM -----------------------
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while True:
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x_node = _a_parent_of_leaves_only(t_G)
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weight_of_x = safe_G.nodes[x_node][node_weight]
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best_value = 0
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best_partition = None
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bp_buffer = dict()
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x_descendants = nx.descendants(t_G, x_node)
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for i_node in x_descendants:
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for j in range(weight_of_x, max_size + 1):
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for a, b in _split_n_from(j, weight_of_x):
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if (
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a not in t_G.nodes[x_node][PKEY].keys()
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or b not in t_G.nodes[i_node][PKEY].keys()
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):
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# it's not possible to form this particular weight sum
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continue
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part1 = t_G.nodes[x_node][PKEY][a]
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part2 = t_G.nodes[i_node][PKEY][b]
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part, value = _concatenate_or_merge(part1, part2, x_node, i_node, j)
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if j not in bp_buffer.keys() or bp_buffer[j][1] < value:
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# we annotate in the buffer the best partition for j
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bp_buffer[j] = part, value
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# we also keep track of the overall best partition
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if best_value <= value:
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best_value = value
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best_partition = part
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# as illustrated in Lukes, once we finished a child, we can
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# discharge the partitions we found into the graph
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# (the key phrase is make all x == x')
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# so that they are used by the subsequent children
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for w, (best_part_for_vl, vl) in bp_buffer.items():
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t_G.nodes[x_node][PKEY][w] = best_part_for_vl
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bp_buffer.clear()
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# the absolute best partition for this node
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# across all weights has to be stored at 0
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t_G.nodes[x_node][PKEY][0] = best_partition
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t_G.remove_nodes_from(x_descendants)
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if x_node == root:
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# the 0-labeled partition of root
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# is the optimal one for the whole tree
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return t_G.nodes[root][PKEY][0]
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