def compute(instance):
"""Return the Kou et al 2-approx algorithm"""
dists, paths = all_shp_between_sources(instance.g, instance.terms,
instance.weights)
gc = UndirectedGraph()
nodes = {x: gc.add_node() for x in instance.terms}
nodesback = {v: x for x, v in nodes.items()}
weights = {}
for i, x1 in enumerate(instance.terms):
v1 = nodes[x1]
for x2 in instance.terms[i + 1:]:
v2 = nodes[x2]
e = gc.add_edge(v1, v2)
try:
weights[e] = dists[x1][x2]
except KeyError:
weights[e] = dists[x2][x1]
treec = kruskal(gc, weights)
tree = set([])
for ec in treec:
v1, v2 = ec.extremities
x1 = nodesback[v1]
x2 = nodesback[v2]
try:
path = paths[x1][x2]
except KeyError:
path = paths[x2][x1]
for e in path:
tree.add(e)
return tree
def test_remove_node_raise_NodeMembershipError_with_not_contained_node(
self):
u = self.g.add_node()
v = self.g.add_node()
self.g.add_arc(u, v)
g2 = DirectedGraph()
v3 = g2.add_node()
v4 = g2.add_node()
g2.add_arc(v3, v4)
g3 = UndirectedGraph()
v5 = g3.add_node()
v6 = g3.add_node()
g3.add_edge(v5, v6)
with self.assertRaises(NodeMembershipError):
self.g.remove_node(v3)
with self.assertRaises(NodeMembershipError):
self.g.remove_node(v5)
def read_input_from_file(f):
"""Read the input from file"""
f.readline()
size = int(f.readline().split()[-1])
nb_edges = int(f.readline().split()[-1])
g = UndirectedGraph()
if parameters.DEBUG:
print('Build nodes')
nodes = [g.add_node() for _ in range(size)]
if parameters.DEBUG:
print('Build edges')
edges = []
weights = {}
i = 0
for i in range(nb_edges):
if parameters.DEBUG:
i += 1
if i % 1000 == 0:
print('Edge %d / %d' % (i, nb_edges))
line = f.readline()
_, u, v, w = line.split()
e = g.add_edge(nodes[int(u) - 1], nodes[int(v) - 1])
weights[e] = int(w)
edges.append((int(u), int(v), int(w)))
line = f.readline()
while 'Terminals' not in line:
line = f.readline()
if 'SECTION' in line:
line = f.readline()
while 'Terminals' not in line:
line = f.readline()
nb_terms = int(line.split()[-1])
terms = []
for i in range(nb_terms):
line = f.readline()
_, t = line.split()
terms.append(nodes[int(t) - 1])
return instances.SteinerInstance(g, terms, weights)
def test_remove_arc_raise_LinkMembershipError_with_not_contained_arc(self):
v1 = self.g.add_node()
v2 = self.g.add_node()
self.g.add_arc(v1, v2)
g2 = DirectedGraph()
v3 = g2.add_node()
v4 = g2.add_node()
e2 = g2.add_arc(v3, v4)
g3 = UndirectedGraph()
v5 = g3.add_node()
v6 = g3.add_node()
e3 = g3.add_edge(v5, v6)
with self.assertRaises(LinkMembershipError):
self.g.remove_arc(e2)
with self.assertRaises(LinkMembershipError):
self.g.remove_arc(e3)
class TestEdge(unittest.TestCase):
def setUp(self):
self.g = UndirectedGraph()
v1 = self.g.add_node()
v2 = self.g.add_node()
v3 = self.g.add_node()
v4 = self.g.add_node()
v5 = self.g.add_node()
v6 = self.g.add_node()
v7 = self.g.add_node()
v8 = self.g.add_node()
self.nodes = [v1, v2, v3, v4, v5, v6, v7, v8]
self.couples = [(v1, v5), (v2, v6), (v3, v7), (v4, v8), (v1, v2),
(v2, v3), (v3, v4), (v4, v1)]
self.edges = [self.g.add_edge(u, v) for u, v in self.couples]
def test_edge_is_undirected(self):
for e in self.edges:
self.assertFalse(e.directed)
def test_edge_extremities_are_nodes_defined_by_add_edge(self):
for e, couple in zip(self.edges, self.couples):
c1 = set(e.extremities)
c2 = set(couple)
self.assertEqual(c1, c2)
def test_neighbor_return_other_extremity_of_edge(self):
for e, couple in zip(self.edges, self.couples):
u, v = couple
self.assertEqual(u, e.neighbor(v))
self.assertEqual(v, e.neighbor(u))
def test_neighbor_raise_TypeError_with_not_node(self):
u = self.g.add_node()
v = self.g.add_node()
e = self.g.add_edge(u, v)
e1 = self.edges[0]
with self.assertRaises(TypeError):
e1.neighbor(1)
with self.assertRaises(TypeError):
e1.neighbor('abc')
with self.assertRaises(TypeError):
e1.neighbor((u, v))
with self.assertRaises(TypeError):
e1.neighbor(None)
with self.assertRaises(TypeError):
e1.neighbor(e)
def test_neighbor_raise_LinkError_with_not_extremity(self):
v1, v2, v3, v4, v5, v6, v7, v8 = self.nodes
u = self.g.add_node()
g2 = DirectedGraph()
w = g2.add_node()
e1 = self.edges[0]
for v in self.nodes:
if v == v1 or v == v5:
continue
with self.assertRaises(LinkError):
e1.neighbor(v)
with self.assertRaises(LinkError):
e1.neighbor(u)
with self.assertRaises(LinkError):
e1.neighbor(w)
class MelhornTwoApprox:
def __init__(self, instance):
dists, paths, closest_sources, limits = voronoi(
instance.g, instance.terms, instance.weights)
self.instance = instance
self.dists = dists
self.paths = paths
self.closest_sources = closest_sources
self.limits = limits
self.gc = UndirectedGraph()
self.treec = None
self.nontree_edges = SortedListWithKey(key=lambda ec: self.weights[ec])
self.sources = set(self.instance.terms)
self.nodes = {x: self.gc.add_node() for x in self.instance.terms}
self.nodesback = {v: x for x, v in self.nodes.items()}
self.weights = {}
self.pathslinks = {}
for x, limit_nodes in self.limits.items():
xc = self.nodes[x]
for u, edges in limit_nodes.items():
for e in edges:
v = e.neighbor(u)
y = self.closest_sources[v]
yc = self.nodes[y]
wc = self.dists[x][u] + self.instance.weights[
e] + self.dists[y][v]
try:
ec = xc.get_incident_edge(yc)
if self.weights[ec] > wc:
self.weights[ec] = wc
self.pathslinks[ec] = (u, v, e)
except NodeError:
ec = self.gc.add_edge(xc, yc)
self.weights[ec] = wc
self.pathslinks[ec] = (u, v, e)
self.treec = kruskal(self.gc, self.weights)
edges = set(self.treec)
for ec in self.gc.edges:
if ec not in edges:
self.nontree_edges.add(ec)
def compute(self):
return self.current_tree()
def current_tree(self):
tree = Tree(self.instance.g, self.instance.weights)
for ec in self.treec:
vuc, vvc = ec.extremities
xu = self.nodesback[vuc]
xv = self.nodesback[vvc]
u, v, e = self.pathslinks[ec]
try:
pathu = self.paths[xu][u]
pathv = self.paths[xv][v]
except KeyError:
pathu = self.paths[xu][v]
pathv = self.paths[xv][u]
for f in pathu:
tree.add_edge(f)
for f in pathv:
tree.add_edge(f)
tree.add_edge(e)
tree.simplify(self.instance.terms)
return tree
def current_cost(self):
return self.treec.cost
def _add_edge(self, ec):
fc = self.treec.add_edge(ec)
if fc is not None:
self.nontree_edges.add(fc)
def _decrease_edge_key(self, ec, wc, pathlink):
if ec in self.nontree_edges:
self.nontree_edges.remove(ec)
self.weights[ec] = wc
self.nontree_edges.add(ec)
else:
self.weights[ec] = wc
self.pathslinks[ec] = pathlink
def _remove_edges(self, rem_edges):
for ec in rem_edges:
try:
self.nontree_edges.remove(ec)
except ValueError:
self.treec.remove_edge(ec)
del self.weights[ec]
del self.pathslinks[ec]
to_remove_from_non_tree = []
for ec in self.nontree_edges:
remove_edge = self.treec.add_edge(ec, handle_conflict=False)
if remove_edge is None:
to_remove_from_non_tree.append(ec)
if len(self.treec) == len(self.gc) - 1:
break
for ec in to_remove_from_non_tree:
self.nontree_edges.remove(ec)
def add_sources(self, new_terms):
incremental_voronoi(self.instance.g, self.sources,
self.instance.weights, self.dists, self.paths,
self.closest_sources, self.limits, new_terms)
rem_edges = []
for ec in self.gc.edges:
vuc, vvc = ec.extremities
xu = self.nodesback[vuc]
xv = self.nodesback[vvc]
u, v, e = self.pathslinks[ec]
if self.closest_sources[u] != xu and self.closest_sources[u] != xv or \
self.closest_sources[v] != xu and self.closest_sources[v] != xv:
rem_edges.append(ec)
self.sources |= set(new_terms)
nodes = {x: self.gc.add_node() for x in new_terms}
self.nodes.update(nodes)
self.nodesback.update({v: x for x, v in nodes.items()})
add_edges = []
for x in new_terms:
xc = self.nodes[x]
limit_nodes = self.limits[x]
for u, edges in limit_nodes.items():
for e in edges:
v = e.neighbor(u)
y = self.closest_sources[v]
yc = self.nodes[y]
wc = self.dists[x][u] + self.instance.weights[
e] + self.dists[y][v]
try:
ec = xc.get_incident_edge(yc)
if self.weights[ec] > wc:
self.weights[ec] = wc
self.pathslinks[ec] = (u, v, e)
except NodeError:
ec = self.gc.add_edge(xc, yc)
self.weights[ec] = wc
self.pathslinks[ec] = (u, v, e)
add_edges.append(ec)
for ec in add_edges:
self._add_edge(ec)
self._remove_edges(rem_edges)
for ec in rem_edges:
self.gc.remove_edge(ec)
def rem_sources(self, rem_terms):
neighbor_sources = decremental_voronoi(self.instance.g, self.sources,
self.instance.weights,
self.dists, self.paths,
self.closest_sources,
self.limits, rem_terms)
self.sources -= set(rem_terms)
rem_nodes = set()
for x in rem_terms:
xc = self.nodes.pop(x)
del self.nodesback[xc]
rem_nodes.add(xc)
add_edges = set()
decrease_key_edges = {}
for x in neighbor_sources:
xc = self.nodes[x]
limit_nodes = self.limits[x]
for u, edges in limit_nodes.items():
for e in edges:
v = e.neighbor(u)
y = self.closest_sources[v]
if y not in neighbor_sources:
continue
yc = self.nodes[y]
wc = self.dists[x][u] + self.instance.weights[
e] + self.dists[y][v]
try:
ec = xc.get_incident_edge(yc)
if self.weights[ec] > wc:
if ec not in add_edges:
decrease_key_edges[ec] = (wc, (u, v, e))
else:
self.weights[ec] = wc
self.pathslinks[ec] = (u, v, e)
except NodeError:
ec = self.gc.add_edge(xc, yc)
self.weights[ec] = wc
self.pathslinks[ec] = (u, v, e)
add_edges.add(ec)
for ec, val in decrease_key_edges.items():
wc, pathlink = val
self._decrease_edge_key(ec, wc, pathlink)
for ec in add_edges:
self._add_edge(ec)
for xc in rem_nodes:
self._remove_edges(set(xc.incident_edges))
self.gc.remove_node(xc)
class TestUndirectedNode(unittest.TestCase):
def setUp(self):
self.g = UndirectedGraph()
v1 = self.g.add_node()
v2 = self.g.add_node()
v3 = self.g.add_node()
v4 = self.g.add_node()
v5 = self.g.add_node()
v6 = self.g.add_node()
v7 = self.g.add_node()
v8 = self.g.add_node()
self.nodes = [v1, v2, v3, v4, v5, v6, v7, v8]
self.couples = [(v1, v5), (v2, v6), (v3, v7), (v4, v8), (v1, v2),
(v2, v3), (v3, v4), (v4, v1)]
self.edges = [self.g.add_edge(u, v) for u, v in self.couples]
def test_add_node_increase_index(self):
v1, v2, v3, v4, v5, v6, v7, v8 = self.nodes
self.assertEqual(v1.index + 1, v2.index)
self.assertEqual(v2.index + 1, v3.index)
self.assertEqual(v3.index + 1, v4.index)
self.assertEqual(v4.index + 1, v5.index)
self.assertEqual(v5.index + 1, v6.index)
self.assertEqual(v6.index + 1, v7.index)
self.assertEqual(v7.index + 1, v8.index)
def test_add_edge_increase_nb_neighbors(self):
sizes = [3, 3, 3, 3, 1, 1, 1, 1]
for v, size in zip(self.nodes, sizes):
self.assertEqual(len(v), size)
self.assertEqual(v.nb_neighbors, size)
def test_add_node_do_not_increase_nb_neighbors(self):
self.g.add_node()
self.g.add_node()
self.g.add_node()
sizes = [3, 3, 3, 3, 1, 1, 1, 1]
for v, size in zip(self.nodes, sizes):
self.assertEqual(len(v), size)
self.assertEqual(v.nb_neighbors, size)
def test_remove_node_decrease_nb_neighbors_of_neighbors(self):
self.g.remove_node(self.nodes[0])
sizes = [2, 3, 2, 0, 1, 1, 1]
for v, size in zip(self.nodes[1:], sizes):
self.assertEqual(len(v), size)
self.assertEqual(v.nb_neighbors, size)
self.g.remove_node(self.nodes[5])
sizes = [1, 3, 2, 0, 1, 1]
for v, size in zip(self.nodes[1:5] + self.nodes[6:], sizes):
self.assertEqual(len(v), size)
self.assertEqual(v.nb_neighbors, size)
def test_remove_edge_decrease_nb_neighbors_of_extremities(self):
self.g.remove_edge(self.edges[0])
sizes = [2, 3, 3, 3, 0, 1, 1, 1]
for v, size in zip(self.nodes, sizes):
self.assertEqual(len(v), size)
self.assertEqual(v.nb_neighbors, size)
self.g.remove_edge(self.edges[4])
sizes = [1, 2, 3, 3, 0, 1, 1, 1]
for v, size in zip(self.nodes, sizes):
self.assertEqual(len(v), size)
self.assertEqual(v.nb_neighbors, size)
for i, e in enumerate(self.edges):
if i != 0 and i != 4:
self.g.remove_edge(e)
for v in self.nodes:
self.assertEqual(len(v), 0)
self.assertEqual(v.nb_neighbors, 0)
def test_add_edge_add_neighbors(self):
for u in self.g:
for v in self.g:
if u == v:
continue
if (u, v) in self.couples:
self.assertTrue(v.is_neighbor_of(u))
self.assertTrue(u.is_neighbor_of(v))
elif (v, u) not in self.couples:
self.assertFalse(v.is_neighbor_of(u))
self.assertFalse(u.is_neighbor_of(v))
def test_add_node_do_not_add_neighbors(self):
u = self.g.add_node()
v = self.g.add_node()
w = self.g.add_node()
for v2 in self.nodes:
self.assertFalse(u.is_neighbor_of(v2))
self.assertFalse(v.is_neighbor_of(v2))
self.assertFalse(w.is_neighbor_of(v2))
self.assertFalse(v2.is_neighbor_of(u))
self.assertFalse(v2.is_neighbor_of(v))
self.assertFalse(v2.is_neighbor_of(w))
def test_remove_node_remove_neighbor_of_neighbors(self):
v1, v2, v3, v4, v5, v6, v7, v8 = self.nodes
self.g.remove_node(v1)
self.assertFalse(v2.is_neighbor_of(v1))
self.assertFalse(v4.is_neighbor_of(v1))
self.assertFalse(v5.is_neighbor_of(v1))
self.assertFalse(v1.is_neighbor_of(v2))
self.assertFalse(v1.is_neighbor_of(v4))
self.assertFalse(v1.is_neighbor_of(v5))
def test_remove_edge_remove_neighbors_of_extremities(self):
v1, v2, v3, v4, v5, v6, v7, v8 = self.nodes
self.g.remove_edge(self.edges[0])
self.assertFalse(v1.is_neighbor_of(v5))
self.assertFalse(v5.is_neighbor_of(v1))
self.g.remove_edge(self.edges[7])
self.assertFalse(v1.is_neighbor_of(v4))
self.assertFalse(v4.is_neighbor_of(v1))
def test_add_edge_add_neighbors_2(self):
for u in self.g:
for v in self.g:
if u == v:
continue
if (u, v) in self.couples:
self.assertIn(u, list(v.neighbors))
self.assertIn(v, list(u.neighbors))
elif (v, u) not in self.couples:
self.assertNotIn(u, list(v.neighbors))
self.assertNotIn(v, list(u.neighbors))
def test_add_node_do_not_add_neighbors_2(self):
u = self.g.add_node()
v = self.g.add_node()
w = self.g.add_node()
for v2 in self.nodes:
self.assertNotIn(v2, list(u.neighbors))
self.assertNotIn(v2, list(v.neighbors))
self.assertNotIn(v2, list(w.neighbors))
self.assertNotIn(u, list(v2.neighbors))
self.assertNotIn(v, list(v2.neighbors))
self.assertNotIn(w, list(v2.neighbors))
def test_remove_node_remove_neighbor_of_neighbors_2(self):
v1, v2, v3, v4, v5, v6, v7, v8 = self.nodes
self.g.remove_node(v1)
self.assertNotIn(v1, list(v2.neighbors))
self.assertNotIn(v1, list(v4.neighbors))
self.assertNotIn(v1, list(v5.neighbors))
self.assertNotIn(v2, list(v1.neighbors))
self.assertNotIn(v4, list(v1.neighbors))
self.assertNotIn(v5, list(v1.neighbors))
def test_remove_edge_remove_neighbors_of_extremities_2(self):
v1, v2, v3, v4, v5, v6, v7, v8 = self.nodes
self.g.remove_edge(self.edges[0])
self.assertNotIn(v5, list(v1.neighbors))
self.assertNotIn(v1, list(v5.neighbors))
self.g.remove_edge(self.edges[7])
self.assertNotIn(v4, list(v1.neighbors))
self.assertNotIn(v1, list(v4.neighbors))
def test_nb_neighbors_equal_nb_incident_edges(self):
for v in self.nodes:
self.assertEqual(v.nb_neighbors, len(list(v.incident_edges)))
def test_add_edge_add_incident_edge(self):
for e, couple in zip(self.edges, self.couples):
u, v = couple
for w in self.nodes:
if w != u and w != v:
self.assertFalse(w.is_incident_to(e))
else:
self.assertTrue(w.is_incident_to(e))
def test_new_node_are_not_incident_to_previous_edges(self):
u = self.g.add_node()
v = self.g.add_node()
w = self.g.add_node()
for e in self.edges:
self.assertFalse(u.is_incident_to(e))
self.assertFalse(v.is_incident_to(e))
self.assertFalse(w.is_incident_to(e))
def test_remove_node_remove_incident_edges_of_neighbors(self):
v1, v2, v3, v4, v5, v6, v7, v8 = self.nodes
e1, e2, e3, e4, e5, e6, e7, e8 = self.edges
self.g.remove_node(v1)
self.assertFalse(v2.is_incident_to(e5))
self.assertFalse(v5.is_incident_to(e1))
self.assertFalse(v4.is_incident_to(e8))
def test_remove_edge_remove_incident_edges_extremities(self):
v1, v2, v3, v4, v5, v6, v7, v8 = self.nodes
e1, e2, e3, e4, e5, e6, e7, e8 = self.edges
self.g.remove_edge(e1)
self.assertFalse(v1.is_incident_to(e1))
self.assertFalse(v5.is_incident_to(e1))
self.g.remove_edge(e8)
self.assertFalse(v1.is_incident_to(e8))
self.assertFalse(v4.is_incident_to(e8))
def test_add_edge_add_incident_edge_2(self):
for e, couple in zip(self.edges, self.couples):
u, v = couple
for w in self.nodes:
if w != u and w != v:
self.assertNotIn(e, list(w.incident_edges))
else:
self.assertIn(e, list(w.incident_edges))
def test_new_node_are_not_incident_to_previous_edges_2(self):
u = self.g.add_node()
v = self.g.add_node()
w = self.g.add_node()
for e in self.edges:
self.assertNotIn(e, list(u.incident_edges))
self.assertNotIn(e, list(v.incident_edges))
self.assertNotIn(e, list(w.incident_edges))
def test_remove_node_remove_incident_edges_of_neighbors_2(self):
v1, v2, v3, v4, v5, v6, v7, v8 = self.nodes
e1, e2, e3, e4, e5, e6, e7, e8 = self.edges
self.g.remove_node(v1)
self.assertNotIn(e5, list(v2.incident_edges))
self.assertNotIn(e1, list(v5.incident_edges))
self.assertNotIn(e8, list(v4.incident_edges))
def test_remove_edge_remove_incident_edges_extremities_2(self):
v1, v2, v3, v4, v5, v6, v7, v8 = self.nodes
e1, e2, e3, e4, e5, e6, e7, e8 = self.edges
self.g.remove_edge(e1)
self.assertNotIn(e1, list(v1.incident_edges))
self.assertNotIn(e1, list(v5.incident_edges))
self.g.remove_edge(e8)
self.assertNotIn(e8, list(v1.incident_edges))
self.assertNotIn(e8, list(v4.incident_edges))
def test_add_edge_add_incident_edge_3(self):
for e, couple in zip(self.edges, self.couples):
u, v = couple
self.assertEqual(e, u.get_incident_edge(v))
self.assertEqual(e, v.get_incident_edge(u))
def test_get_incident_edge_raise_TypeError_with_not_node(self):
v1, v2, v3, v4, v5, v6, v7, v8 = self.nodes
e1, e2, e3, e4, e5, e6, e7, e8 = self.edges
g2 = DirectedGraph()
v = g2.add_node()
with self.assertRaises(TypeError):
v1.get_incident_edge(1)
with self.assertRaises(TypeError):
v1.get_incident_edge('abc')
with self.assertRaises(TypeError):
v1.get_incident_edge((v1, v2))
with self.assertRaises(TypeError):
v1.get_incident_edge(e5)
with self.assertRaises(TypeError):
v1.get_incident_edge(None)
with self.assertRaises(TypeError):
v1.get_incident_edge(v)
def test_get_incident_edge_raise_NodeError_with_not_neighbor(self):
v1, v2, v3, v4, v5, v6, v7, v8 = self.nodes
g2 = UndirectedGraph()
v = g2.add_node()
with self.assertRaises(NodeError):
v1.get_incident_edge(v3)
with self.assertRaises(NodeError):
v1.get_incident_edge(v)
def test_get_incident_edge_raise_NodeError_with_not_neighbor_due_to_remove_edge(
self):
v1, v2, v3, v4, v5, v6, v7, v8 = self.nodes
e1, e2, e3, e4, e5, e6, e7, e8 = self.edges
self.g.remove_edge(e1)
with self.assertRaises(NodeError):
v1.get_incident_edge(v5)
with self.assertRaises(NodeError):
v5.get_incident_edge(v1)
def test_get_incident_edge_raise_NodeError_with_not_neighbor_due_to_remove_node(
self):
v1, v2, v3, v4, v5, v6, v7, v8 = self.nodes
self.g.remove_node(v5)
with self.assertRaises(NodeError):
v1.get_incident_edge(v5)
},
v5: {
v6: 10
},
v6: {
v7: 9
},
v7: {
v8: 4
},
})
waux = defaultdict(dict)
for u in weights:
for v in weights[u]:
g.add_edge(u, v)
waux[v][u] = weights[u][v]
for v in waux:
weights[v].update(waux[v])
gd = GraphDrawer(g)
gd.move_node(v1, 100, 100)
gd.move_node(v2, 300, 100)
gd.move_node(v3, 500, 100)
gd.move_node(v4, 700, 100)
gd.move_node(v5, 100, 200)
gd.move_node(v6, 300, 200)
gd.move_node(v7, 500, 200)
gd.move_node(v8, 700, 200)
for e in g.edges:
def is_root(self):
return self.father is None
def __str__(self):
return str(self.node)
def __repr__(self):
return str(self)
if __name__ == '__main__':
from dynamicgraphviz.graph.undirectedgraph import UndirectedGraph
g = UndirectedGraph()
v1, v2, v3, v4, v5, v6, v7, v8 = [g.add_node() for _ in range(8)]
e0 = g.add_edge(v8, v1)
e1 = g.add_edge(v1, v2)
e2 = g.add_edge(v1, v3)
e3 = g.add_edge(v3, v4)
e4 = g.add_edge(v3, v5)
e5 = g.add_edge(v4, v6)
e6 = g.add_edge(v4, v7)
e7 = g.add_edge(v2, v6)
weights = {
e0: 1,
e1: 1,
e2: 1,
e3: 3,
e4: 1,
e5: 1,
e6: 1,