Source code for

"""Lukes Algorithm for exact optimal weighted tree partitioning."""

from copy import deepcopy
from functools import lru_cache
from random import choice

import networkx as nx
from networkx.utils import not_implemented_for

__all__ = ["lukes_partitioning"]

D_EDGE_W = "weight"
D_NODE_W = "weight"
PKEY = "partitions"

def _split_n_from(n, min_size_of_first_part):
    # splits j in two parts of which the first is at least
    # the second argument
    assert n >= min_size_of_first_part
    for p1 in range(min_size_of_first_part, n + 1):
        yield p1, n - p1

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