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solver.py
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solver.py
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import numpy as np
import networkx as nx
import torch
from queue import Queue
from tqdm import tqdm
from utils import add_vertex_cost_to_edge
from graph import replace_subgraph, Segment, parse_computation_graph
def optimal_grad_checkpointing(net, input):
'''
:param net: nn.Module, pytorch model
:param input: an input tensor
:return: run_segment: nn.Module, wrapper for optimal grad checkpointing with forward() callable for training
'''
net.train()
inputs = [input]
try:
G, source, target = parse_computation_graph(net, inputs)
except:
print('Parsing Computation Graph with torch.jit failed, revert to manual parse_graph function')
if not hasattr(net, 'parse_graph'):
raise Exception("net.parse_graph(input) function needs to be provided")
with torch.no_grad():
G, source, target = net.parse_graph(input)
solver = ArbitrarySolver()
run_graph, best_cost = solver.solve(G, source, target)
run_segment = Segment(run_graph, source, target, do_checkpoint=True)
torch.cuda.empty_cache()
return run_segment
class ArbitrarySolver():
def __init__(self):
self.linear_solver = LinearSolver()
def solve(self, G, source, target, use_tqdm=True):
print('Building Division Tree')
graph, cost, division_type = self.build_division_tree(G, source, target)
print('Getting Max Terms')
max_terms = self.get_max_terms(graph, division_type, max_terms=set())
best_run_graph = None
best_total_cost = np.inf
max_terms = list(set(max_terms))
if 0 in max_terms:
max_terms.remove(0)
max_terms = sorted(max_terms, reverse=True)
best_max_term = -1
best_gc_cost = -1
print('Solving Optimal for Each Max Term')
if use_tqdm:
iterator = tqdm(max_terms)
else:
iterator = max_terms
for max_term in iterator:
run_graph, gc_cost = self.solve_with_max(graph, source, target, max_term, division_type)
total_cost = gc_cost + max_term
if total_cost < best_total_cost:
best_total_cost = total_cost
best_run_graph = run_graph
best_max_term = max_term
best_gc_cost = gc_cost
best_run_graph = self.optimize_run_graph(G, best_run_graph)
return best_run_graph, best_total_cost
def optimize_run_graph(self, G, run_graph):
# flatten run graph to avoid too many recursions, and other optimization to speed up
new_run_graph = run_graph.copy()
for edge_key in run_graph.edges:
source, target, id = edge_key
edge_nodes, edge_edges = self.nodes_edges_in_run_graph_edge(run_graph, edge_key)
subgraph = nx.MultiDiGraph()
for sub_node_key in edge_nodes:
subgraph.add_node(sub_node_key, **G.nodes[sub_node_key])
for sub_edge_key in edge_edges:
subgraph.add_edges_from({sub_edge_key: G.edges[sub_edge_key]}, **G.edges[sub_edge_key])
new_run_graph.edges[(source, target, id)]['graph'] = subgraph
new_run_graph.edges[(source, target, id)]['module'] = Segment(subgraph, source, target)
if 'transition' in new_run_graph.nodes[target]:
trans_complete = True
for (trans_source, trans_id) in new_run_graph.nodes[target]['transition_input_order']:
if trans_source not in edge_nodes:
trans_complete = False
break
if trans_complete:
# transition handled in subgraph
new_run_graph.nodes[target]['transition'] = None
new_run_graph.nodes[target]['transition_input_order'] = None
return new_run_graph
def nodes_edges_in_run_graph_edge(self, graph, edge_key):
start, end, id = edge_key
nodes = set()
nodes.add(start)
nodes.add(end)
edges = dict()
edge = graph.edges[edge_key]
subgraph = edge['module']
if type(subgraph) == Segment:
subgraph = subgraph.G
for e in subgraph.edges:
subnodes, subedges = self.nodes_edges_in_run_graph_edge(subgraph, e)
nodes = nodes.union(subnodes)
for key in subedges:
edges[key] = subedges[key]
return nodes, edges
else:
edges[edge_key] = edge
return nodes, edges
def edge_in_graph(self, run_graph, edge):
if edge in run_graph.edges:
return True
flag = False
for e in run_graph.edges:
subgraph = run_graph.edges[e]['module']
if type(subgraph) == Segment:
flag = flag or self.edge_in_graph(subgraph.G, edge)
if flag:
return flag
return flag
def solve_with_max(self, division_tree, source, target, max_term, division_type):
run_graph = division_tree.copy()
total_cost = 0
if division_type != 'linear':
for (s, t, id) in division_tree.edges:
cost = division_tree.edges[(s, t, id)]['cost']
subtree = division_tree.edges[(s, t, id)]['graph']
if cost > max_term:
sub_run_graph, GC_cost = self.solve_with_max(subtree, s, t, max_term, division_tree.edges[(s, t, id)]['division_type'])
run_graph = replace_subgraph(run_graph, sub_run_graph, s, t, id)
total_cost += GC_cost
for node in division_tree.nodes:
if node != source and node != target:
total_cost += division_tree.nodes[node]['cost']
return run_graph, total_cost
else:
nodes = list(nx.topological_sort(division_tree))
linear_subgraph_vertices = [source]
GC_set = set()
for i in range(len(nodes) - 1):
node1, node2 = nodes[i], nodes[i+1]
edge = division_tree.edges[(node1, node2, 0)]
if edge['cost'] > max_term:
sub_run_graph, GC_cost = self.solve_with_max(edge['graph'], node1, node2, max_term, edge['division_type'])
run_graph = replace_subgraph(run_graph, sub_run_graph, node1, node2, 0)
total_cost += GC_cost
GC_set.add(node1)
GC_set.add(node2)
if len(linear_subgraph_vertices) > 1:
linear_subgraph = division_tree.subgraph(linear_subgraph_vertices).copy()
GCPs = self.solve_linear(linear_subgraph, max_term)
GC_set = GC_set.union(set(GCPs))
start_node = GCPs[0]
linear_run_graph = nx.MultiDiGraph()
linear_run_graph.add_nodes_from({start_node: linear_subgraph.nodes[start_node]}, **linear_subgraph.nodes[start_node])
for j in range(1, len(GCPs)):
end_node = GCPs[j]
linear_run_graph.add_nodes_from({end_node: linear_subgraph.nodes[end_node]},
**linear_subgraph.nodes[end_node])
start_index = nodes.index(start_node)
end_index = nodes.index(end_node)
edge_subgraph = linear_subgraph.subgraph(nodes[start_index:(end_index+1)]).copy()
linear_run_graph.add_edge(start_node, end_node, module=Segment(edge_subgraph, start_node, end_node))
start_node = end_node
run_graph = replace_subgraph(run_graph, linear_run_graph, linear_subgraph_vertices[0], linear_subgraph_vertices[-1], id=None)
linear_subgraph_vertices = [node2]
else:
linear_subgraph_vertices = [node2]
else:
linear_subgraph_vertices.append(node2)
if len(linear_subgraph_vertices) > 1:
linear_subgraph = division_tree.subgraph(linear_subgraph_vertices).copy()
GCPs = self.solve_linear(linear_subgraph, max_term)
GC_set = GC_set.union(set(GCPs))
start_node = GCPs[0]
linear_run_graph = nx.MultiDiGraph()
linear_run_graph.add_nodes_from({start_node: linear_subgraph.nodes[start_node]},
**linear_subgraph.nodes[start_node])
for j in range(1, len(GCPs)):
end_node = GCPs[j]
linear_run_graph.add_nodes_from({end_node: linear_subgraph.nodes[end_node]},
**linear_subgraph.nodes[end_node])
start_index = nodes.index(start_node)
end_index = nodes.index(end_node)
edge_subgraph = linear_subgraph.subgraph(nodes[start_index:(end_index+1)]).copy()
linear_run_graph.add_edge(start_node, end_node, module=Segment(edge_subgraph, start_node, end_node))
start_node = end_node
run_graph = replace_subgraph(run_graph, linear_run_graph, linear_subgraph_vertices[0], linear_subgraph_vertices[-1], id=None)
for node in GC_set:
if node != source and node != target:
total_cost += division_tree.nodes[node]['cost']
return run_graph, total_cost
def solve_linear(self, linear_subgraph, max_term):
linear_node_num = len(linear_subgraph.nodes)
linear_nodes = list(nx.topological_sort(linear_subgraph))
AG = linear_subgraph.copy()
edge_costs = np.zeros([linear_node_num, linear_node_num])
cost_sums = np.zeros([linear_node_num])
cost_sums[0] = linear_subgraph.nodes[linear_nodes[0]]['cost']
for j in range(1, linear_node_num):
cost_sums[j] = cost_sums[j - 1] + linear_subgraph.nodes[linear_nodes[j]]['cost'] + \
linear_subgraph.edges[(linear_nodes[j - 1], linear_nodes[j], 0)]['cost']
for j in range(linear_node_num - 1):
for k in range(j + 1, linear_node_num):
edge_cost = cost_sums[k] - cost_sums[j] - linear_subgraph.nodes[linear_nodes[k]]['cost']
edge_costs[j, k] = edge_cost
for j in range(linear_node_num):
for k in range(j + 1, linear_node_num):
if edge_costs[j, k] <= max_term:
# add accessibility edge
AG.add_edge(linear_nodes[j], linear_nodes[k], cost=0)
# move vertex cost to edge cost to use networkx shortest path algorithm
AG = add_vertex_cost_to_edge(AG)
GCPs = nx.shortest_path(AG, source=linear_nodes[0], target=linear_nodes[-1], weight='weight')
return GCPs
def get_max_terms(self, division_tree, division_type, max_terms=set()):
if division_type == 'linear':
nodes = [n for n in nx.topological_sort(division_tree)]
node_num = len(nodes)
cost_sums = np.zeros([node_num])
cost_sums[0] = division_tree.nodes[nodes[0]]['cost']
for j in range(1, node_num):
cost_sums[j] = cost_sums[j - 1] + division_tree.nodes[nodes[j]]['cost'] + \
division_tree.edges[(nodes[j - 1], nodes[j], 0)]['cost']
for j in range(node_num - 1):
for k in range(j + 1, node_num):
edge_cost = cost_sums[k] - cost_sums[j] - division_tree.nodes[nodes[k]]['cost']
max_terms.add(edge_cost)
for (s, t, id) in division_tree.edges:
max_terms.add(division_tree.edges[(s, t, id)]['cost'])
graph = division_tree.edges[(s, t, id)].get('graph', None)
divi_type = division_tree.edges[(s, t, id)].get('division_type', None)
if graph is not None:
max_terms = self.get_max_terms(graph, divi_type, max_terms)
return max_terms
elif division_type == 'leave':
return max_terms
else:
for (s, t, id) in division_tree.edges:
max_terms.add(division_tree.edges[(s, t, id)]['cost'])
graph = division_tree.edges[(s, t, id)].get('graph', None)
divi_type = division_tree.edges[(s, t, id)].get('division_type', None)
if graph is not None:
max_terms = self.get_max_terms(graph, divi_type, max_terms)
return max_terms
def build_division_tree(self, G, source, target, known_division_type=None):
if len(G.nodes) == 2:
return G, 0, 'leave'
divisions, source_targets, division_type = self.get_division(G, source, target, division_type=None, known_division_type=known_division_type)
if len(divisions) == 1:
# leaves of division tree
return G, 0, 'leave'
division_tree = nx.MultiDiGraph()
total_cost = 0
for subgraph, source_target in zip(divisions, source_targets):
s, t = source_target
if s not in division_tree.nodes:
division_tree.add_nodes_from({s: G.nodes[s]}, **G.nodes[s])
if t not in division_tree.nodes:
division_tree.add_nodes_from({t: G.nodes[t]}, **G.nodes[t])
id = division_tree.add_edge(s, t)
graph, cost, sub_division_type = self.build_division_tree(subgraph, s, t, known_division_type=division_type)
division_tree.edges[(s, t, id)]['graph'] = graph
division_tree.edges[(s, t, id)]['module'] = Segment(graph, s, t)
division_tree.edges[(s, t, id)]['cost'] = cost
division_tree.edges[(s, t, id)]['division_type'] = sub_division_type
total_cost += cost
for node in division_tree.nodes:
if node != source and node != target:
total_cost += division_tree.nodes[node]['cost']
return division_tree, total_cost, division_type
def get_division(self, G, source, target, division_type=None, known_division_type=None):
if division_type is None:
nodes = list(nx.topological_sort(G))
nodes.remove(source)
nodes.remove(target)
if known_division_type == 'linear':
# skipped finding splitting vertex
pass
else:
for node in nodes:
ancestors = nx.ancestors(G, node)
descendants = nx.descendants(G, node)
subgraph1 = G.subgraph(list(ancestors) + [node])
subgraph2 = G.subgraph(list(descendants) + [node])
edges1 = set(subgraph1.edges)
edges2 = set(subgraph2.edges)
edges = set(G.edges)
if edges1.union(edges2) == edges and len(edges1.intersection(edges2)) == 0:
# found splitting vertex
division1, source_targets1 = [subgraph1.copy()], [[source, node]]
# division1, source_targets1, _ = self.get_division(subgraph1.copy(), source, node, division_type='linear')
division2, source_targets2, _ = self.get_division(subgraph2.copy(), node, target, division_type='linear')
return division1 + division2, source_targets1 + source_targets2, 'linear'
# no splitting vertex found, check whether it has branches
source_target_edge = G.get_edge_data(source, target)
if source_target_edge is not None:
subgraph1 = G.subgraph([source, target]).copy()
subgraph2 = G.copy()
for id in G.get_edge_data(source, target):
subgraph2.remove_edge(source, target, id)
division2, source_targets2, _ = self.get_division(subgraph2, source, target, division_type='branch')
return [subgraph1] + division2, [[source, target]] + source_targets2, 'branch'
else:
random_node = nodes[np.random.choice(len(nodes))]
ancestors = nx.ancestors(G, random_node)
descendants = nx.descendants(G, random_node)
subgraph1_nodes = list(ancestors) + [random_node] + list(descendants)
queue = Queue()
for node in subgraph1_nodes:
if node != source and node != target:
for p in G.predecessors(node):
if p not in subgraph1_nodes:
queue.put(p)
subgraph1_nodes.append(p)
for s in G.successors(node):
if s not in subgraph1_nodes:
queue.put(s)
subgraph1_nodes.append(s)
while not queue.empty():
n = queue.get()
for p in G.predecessors(n):
if p not in subgraph1_nodes:
queue.put(p)
subgraph1_nodes.append(p)
for s in G.successors(n):
if s not in subgraph1_nodes:
queue.put(s)
subgraph1_nodes.append(s)
subgraph1 = G.subgraph(subgraph1_nodes).copy()
edges1 = set(subgraph1.edges)
edges = set(G.edges)
if edges1 == edges:
# division type is complicate
return self.get_division(G, source, target, division_type='complicate')
else:
subgraph2_nodes = set(nodes).difference(set(subgraph1_nodes))
subgraph2_nodes = list(subgraph2_nodes) + [source, target]
subgraph2 = G.subgraph(subgraph2_nodes).copy()
division2, source_targets2, divi_type = self.get_division(subgraph2, source, target, division_type='branch')
return [subgraph1] + division2, [[source, target]] + source_targets2, 'branch'
elif division_type == 'linear':
nodes = list(nx.topological_sort(G))
if len(nodes) == 2:
return [G], [[source, target]], division_type
for node in nodes:
if node == source or node == target:
continue
ancestors = nx.ancestors(G, node)
descendants = nx.descendants(G, node)
subgraph1 = G.subgraph(list(ancestors) + [node])
subgraph2 = G.subgraph(list(descendants) + [node])
edges1 = set(subgraph1.edges)
edges2 = set(subgraph2.edges)
edges = set(G.edges)
if edges1.union(edges2) == edges and len(edges1.intersection(edges2)) == 0:
# found splitting vertex
# division1, source_targets1, divi_type = self.get_division(subgraph1.copy(), source, node,
# division_type='linear')
division1, source_targets1 = [subgraph1.copy()], [[source, node]]
division2, source_targets2, divi_type = self.get_division(subgraph2.copy(), node, target,
division_type='linear')
return division1 + division2, source_targets1 + source_targets2, division_type
return [G], [[source, target]], division_type
elif division_type == 'branch':
nodes = list(nx.topological_sort(G))
nodes.remove(source)
nodes.remove(target)
random_node = nodes[np.random.choice(len(nodes))]
ancestors = nx.ancestors(G, random_node)
descendants = nx.descendants(G, random_node)
subgraph1_nodes = list(ancestors) + [random_node] + list(descendants)
queue = Queue()
for node in subgraph1_nodes:
if node != source and node != target:
for p in G.predecessors(node):
if p not in subgraph1_nodes:
queue.put(p)
subgraph1_nodes.append(p)
for s in G.successors(node):
if s not in subgraph1_nodes:
queue.put(s)
subgraph1_nodes.append(s)
while not queue.empty():
n = queue.get()
for p in G.predecessors(n):
if p not in subgraph1_nodes:
queue.put(p)
subgraph1_nodes.append(p)
for s in G.successors(n):
if s not in subgraph1_nodes:
queue.put(s)
subgraph1_nodes.append(s)
subgraph1 = G.subgraph(subgraph1_nodes).copy()
edges1 = set(subgraph1.edges)
edges = set(G.edges)
if edges1 == edges:
# division type is complicate
return [G], [[source, target]], division_type
else:
subgraph2_nodes = set(nodes).difference(set(subgraph1_nodes))
subgraph2_nodes = list(subgraph2_nodes) + [source, target]
subgraph2 = G.subgraph(subgraph2_nodes).copy()
division2, source_targets2, divi_type = self.get_division(subgraph2, source, target, division_type='branch')
return [subgraph1] + division2, [[source, target]] + source_targets2, division_type
elif division_type == 'complicate':
nodes = list(nx.topological_sort(G))
adjacency_matrix = np.array(nx.linalg.graphmatrix.adjacency_matrix(G, nodelist=nodes).todense())
reverse_mapping = {node: i for i, node in enumerate(nodes)}
# counting undirected adjacency
adjacency_matrix = adjacency_matrix + adjacency_matrix.T
path_adjacency_matrix = adjacency_matrix.copy()
descendants_all = {node: nx.descendants(G, node) for node in nodes}
ancestors_all = {node: nx.ancestors(G, node) for node in nodes}
for node in nodes:
idx = reverse_mapping[node]
for n in descendants_all[node]:
path_adjacency_matrix[idx, reverse_mapping[n]] += 1
for n in ancestors_all[node]:
path_adjacency_matrix[idx, reverse_mapping[n]] += 1
candidate_nodes = [node for i, node in enumerate(nodes) if np.sum(adjacency_matrix[i, :]) > 2 or node == source or node == target]
subgraphs = []
source_targets_all = []
for i in (range(len(candidate_nodes) - 1)):
# for j in range(i + 1, len(nodes)):
for j in range(len(candidate_nodes) - 1, i, -1):
node1, node2 = candidate_nodes[i], candidate_nodes[j]
if node1 == source and node2 == target:
continue
subgraph = self.get_largest_IS(G, node1, node2, descendants_all, ancestors_all, adjacency_matrix, path_adjacency_matrix, np.array(nodes), reverse_mapping)
if subgraph is not None:
subgraphs.append(subgraph)
source_targets_all.append([node1, node2])
subgraphs_source_targets = [[subgraph, source_target] for subgraph, source_target in zip(subgraphs, source_targets_all)]
subgraphs_source_targets = sorted(subgraphs_source_targets, key=lambda x:-len(list(x[0].edges)))
subgraphs = [subgraphs_source_target[0] for subgraphs_source_target in subgraphs_source_targets]
source_targets_all = [subgraphs_source_target[1] for subgraphs_source_target in subgraphs_source_targets]
divisions = []
source_targets = []
occupied_edges = set()
for subgraph, source_target in zip(subgraphs, source_targets_all):
subgraph_edges = set(subgraph.edges)
if len(subgraph_edges.intersection(occupied_edges)) == 0:
divisions.append(subgraph)
source_targets.append(source_target)
occupied_edges = occupied_edges.union(subgraph_edges)
if len(occupied_edges) != len(G.edges):
print('Something wrong with Complicate IS')
raise ValueError
return divisions, source_targets, division_type
else:
raise KeyError
# todo: this need to be speed up, the while loop is time consuming
def get_largest_IS(self, G, node1, node2, descendants_all, ancestors_all, adjacency_matrix, path_adjacency_matrix, adjacency_nodes_mapping, reverse_mapping):
ancestors = ancestors_all[node2]
descendants = descendants_all[node1]
subgraph_nodes = set(ancestors).intersection(set(descendants))
subgraph_nodes = subgraph_nodes.union({node1, node2})
other_nodes = set(G.nodes).difference(subgraph_nodes)
subgraph_inside_nodes_idxs = np.array([reverse_mapping[n] for n in subgraph_nodes.difference({node1, node2})])
if len(subgraph_inside_nodes_idxs) == 0:
subgraph = G.subgraph(subgraph_nodes)
else:
other_nodes_idxs = np.array([reverse_mapping[n] for n in other_nodes])
adjacency = np.sum(adjacency_matrix[other_nodes_idxs, :][:, subgraph_inside_nodes_idxs], axis=0)
removed_inside_nodes_idxs = subgraph_inside_nodes_idxs[adjacency > 0]
if len(removed_inside_nodes_idxs) > 0:
prev_remove_num = len(removed_inside_nodes_idxs)
while True:
remaining_subgraph_inside_nodes_idx = set(subgraph_inside_nodes_idxs).difference(set(removed_inside_nodes_idxs))
if len(remaining_subgraph_inside_nodes_idx) <= 0:
break
remaining_subgraph_inside_nodes_idx = np.array(list(remaining_subgraph_inside_nodes_idx))
path_adjacency_submat = path_adjacency_matrix[removed_inside_nodes_idxs, :][:, remaining_subgraph_inside_nodes_idx]
path_adjacency = np.sum(path_adjacency_submat, axis=0)
extra_removed_inside_nodes_idxs = remaining_subgraph_inside_nodes_idx[path_adjacency > 0]
removed_inside_nodes_idxs = set(removed_inside_nodes_idxs).union(set(extra_removed_inside_nodes_idxs))
cur_remove_num = len(removed_inside_nodes_idxs)
removed_inside_nodes_idxs = np.array(list(removed_inside_nodes_idxs))
if cur_remove_num == prev_remove_num:
break
prev_remove_num = cur_remove_num
removed_nodes = set(adjacency_nodes_mapping[removed_inside_nodes_idxs])
remaining_subgraph_nodes = subgraph_nodes.difference(removed_nodes)
else:
remaining_subgraph_nodes = subgraph_nodes
subgraph = G.subgraph(remaining_subgraph_nodes)
if len(subgraph.edges) == 0 or (not nx.is_connected(subgraph.to_undirected(as_view=True))):
return None
else:
return subgraph.copy()
class LinearSolver():
def __init__(self):
pass
def solve(self, G, source, target):
'''
solve optimal checkpoints for linear computation graph G
assuming G is linear DAG
:param G: networkx DI graph
:return:
'''
node_num = len(G.nodes)
max_terms, edge_costs, vertex_id_mapping = self.get_max_terms_and_costs(G, source)
best_GCPs = None
best_cost = np.inf
for max_term in max_terms:
GCPs, total_cost = self.solve_with_max(G, source, target, edge_costs, vertex_id_mapping, max_term)
total_cost += max_term
if total_cost < best_cost:
best_GCPs = GCPs
best_cost = total_cost
run_graph = nx.MultiDiGraph()
for i in range(len(best_GCPs) - 1):
s = best_GCPs[i]
t = best_GCPs[i + 1]
subgraph_nodes = [s]
vertex_id = s
while vertex_id != t:
vertex_id = [n for n in G.successors(vertex_id)][0]
subgraph_nodes.append(vertex_id)
subgraph = G.subgraph(subgraph_nodes).copy()
if s not in run_graph.nodes:
run_graph.add_node(s)
if t not in run_graph.nodes:
run_graph.add_node(t)
run_graph.add_edge(s, t, module=Segment(subgraph, s, t))
return run_graph, best_cost
def solve_with_max(self, G, source, target, edge_costs, vertex_id_mapping, max_term):
# construct accessibility graph
node_num = len(G.nodes)
AG = G.copy()
for i in range(node_num):
for j in range(i + 1, node_num):
if edge_costs[i, j] <= max_term:
# add accessibility edge
AG.add_edge(vertex_id_mapping[i], vertex_id_mapping[j], cost=0)
# move vertex cost to edge cost to use networkx shortest path algorithm
AG = add_vertex_cost_to_edge(AG)
GCPs = nx.shortest_path(AG, source=source, target=target, weight='weight')
total_cost = 0
for vertex_id in GCPs:
total_cost += G.nodes[vertex_id]['cost']
return GCPs, total_cost
def get_max_terms_and_costs(self, G, source):
node_num = len(G.nodes)
max_terms = set()
cost_sums = np.zeros([node_num])
edge_costs = np.zeros([node_num, node_num])
vertex_id = source
vertex_id_mapping = [source]
cost_sums[0] = G.nodes[vertex_id]['cost']
for i in range(1, node_num):
vertex_id = [n for n in G.successors(vertex_id)][0]
vertex_id_mapping.append(vertex_id)
cost_sums[i] = cost_sums[i - 1] + G.nodes[vertex_id]['cost']
for i in range(node_num - 1):
for j in range(i + 2, node_num):
cost = cost_sums[j - 1] - cost_sums[i]
edge_costs[i, j] = cost
max_terms.add(cost)
return max_terms, edge_costs, vertex_id_mapping