第六章课后习题6.1、6.3、6.4、6.5、6.7和例题6.10、6.11
习题6.1
a图
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import networkx as nx
import matplotlib.pyplot as plt
G = nx.Graph()
nodes = ['v1', 'v2', 'v3', 'v4', 'v5', 'v6']
G.add_nodes_from(nodes)
edges = [
('v1', 'v2'), ('v1', 'v3'), ('v1', 'v4'),
('v2', 'v3'), ('v2', 'v6'),
('v3', 'v4'),
('v4', 'v5'),
('v5', 'v6')
]
G.add_edges_from(edges)
pos = nx.circular_layout(G)
center = (0, 0)
pos['v1'] = center
plt.figure(figsize=(8, 8))
nx.draw(G, pos, with_labels=True, node_color='skyblue', node_size=700, font_size=15, font_weight='bold')
plt.title("Undirected Graph as Described")
plt.axis('equal')
plt.show()
print("学号：3001")
plt. show( )
b图
点击查看代码
import networkx as nx
import matplotlib.pyplot as plt
G = nx.Graph()
nodes = ['v1', 'v2', 'v3', 'v4', 'v5', 'v6']
G.add_nodes_from(nodes)
edges = [
('v1', 'v2', 7),
('v1', 'v3', 3),
('v1', 'v4', 12),
('v2', 'v3', 1),
('v2', 'v6', 1),
('v3', 'v4', 8),
('v4', 'v5', 9),
('v5', 'v6', 3)
]
G.add_weighted_edges_from(edges)
pos = nx.circular_layout(G)
center = (0, 0)
pos['v1'] = center
def draw_edges_with_weights(G, pos):
edge_labels = {(u, v): d['weight'] for u, v, d in G.edges(data=True)}
nx.draw_networkx_edge_labels(G, pos, edge_labels=edge_labels)
plt.figure(figsize=(10, 10))
nx.draw(G, pos, with_labels=True, node_color='lightblue', node_size=700, font_size=15, font_weight='bold')
draw_edges_with_weights(G, pos)
plt.title("Undirected Graph with Weights")
plt.axis('equal')
plt.show()
print("学号：3001")
c图
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import networkx as nx
import matplotlib.pyplot as plt
G = nx.DiGraph()
nodes = ['v1', 'v2', 'v3', 'v4', 'v5', 'v6']
G.add_nodes_from(nodes)
edges = [
('v2', 'v1', 7),
('v1', 'v3', 3),
('v4', 'v1', 12),
('v2', 'v3', 1),
('v6', 'v2', 1),
('v3', 'v4', 8),
('v5', 'v4', 9),
('v5', 'v6', 3)
]
G.add_weighted_edges_from(edges)
pos = nx.circular_layout(G)
def draw_edges_with_weights(G, pos):
edge_labels = {(u, v): d['weight'] for u, v, d in G.edges(data=True)}
nx.draw_networkx_edge_labels(G, pos, edge_labels=edge_labels, font_color='red')
plt.figure(figsize=(10, 10))
nx.draw(G, pos, with_labels=True, node_color='lightblue', node_size=700, font_size=15, font_weight='bold', arrows=True)
draw_edges_with_weights(G, pos)
plt.title("Directed Graph with Weights")
plt.axis('equal')
plt.show()
print("学号：3001")
习题6.3
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import heapq
def prim(graph, start):
num_nodes = len(graph)
visited = [False] * num_nodes
min_heap = [(0, start, -1)]
mst_cost = 0
mst_edges = []
while min_heap:
weight, u, parent = heapq.heappop(min_heap)
if visited[u]:
continue
visited[u] = True
mst_cost += weight
if parent != -1:
mst_edges.append((parent, u, weight))
for v in range(num_nodes):
if not visited[v] and graph[u][v] != 0:
heapq.heappush(min_heap, (graph[u][v], v, u))
return mst_cost, mst_edges
graph = [
[0,20,0,0,15,0],
[20,0,20,60,25,0],
[0,20,0,30,18,0],
[0,60,30,0,35,10],
[0,0,0,10,15,0]
]
mst_cost, mst_edges = prim(graph, 0)
print("Prim's MST Cost:", mst_cost)
print("Prim's MST Edges:", mst_edges)
print("学号：3001")
习题6.4
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initial_costs = [2.5, 2.6, 2.8, 3.1]
salvage_values = [2.0, 1.6, 1.3, 1.1]
maintenance_costs = [0.3, 0.8, 1.5, 2.0]
dp = [[float('inf')] * 2 for _ in range(4)]
dp[0][1] = initial_costs[0] + maintenance_costs[0]
for i in range(1, 4):
dp[i][1] = min(dp[i-1][1] + maintenance_costs[i],
initial_costs[i] + maintenance_costs[i])
if i > 0:
dp[i][0] = dp[i-1][1] + salvage_values[i-1]
min_cost = min(dp[3][1],
min(dp[i][0] for i in range(3)))
print(f"最优更新策略下的4年内最小总费用是：{min_cost}万元")
print("学号：3001")
习题6.5
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import numpy as np
distances = np.array([
[0, 2, 7, np.inf, np.inf, np.inf],
[2, 0, 4, 6, 8, np.inf],
[7, 4, 0, 1, 3, np.inf],
[np.inf, 6, 1, 0, 1, 6],
[np.inf, 8, 3, 1, 0, 3],
[np.inf, np.inf, np.inf, 6, 3, 0]
], dtype=float)
students = np.array([50, 40, 60, 20, 70, 90])
hospital_distances_sum = np.zeros(6)
for i in range(6):
connected_distances = distances[i, :i+1].copy()
connected_distances = connected_distances[connected_distances != np.inf]
hospital_distances_sum[i] = np.sum(connected_distances)
hospital_location = np.argmin(hospital_distances_sum)
print(f"医院应该建在村庄 {chr(65 + hospital_location)} 处，使得最远村庄的人到医院看病所走的路最短。")
school_total_distances = np.zeros(6)
for i in range(6):
weighted_distances = 0
for j in range(6):
if distances[j, i] != np.inf:
weighted_distances += students[j] * distances[j, i]
school_total_distances[i] = weighted_distances
school_location = np.argmin(school_total_distances)
print(f"小学应该建在村庄 {chr(65 + school_location)} 处，使得所有学生上学走的总路程最短。")
print("学号：3001")
习题6.7
(1)
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import numpy as np
import pandas as pd
import networkx as nx
import matplotlib.pyplot as plt
from matplotlib.patches import RegularPolygon # 用于绘制五角星
# 读取数据
a = pd.read_excel('ti6_7.xlsx')
b = a.values
# 提取数据
s = list(b[:, 0]) # 节点名称
x = b[:, 1] # x坐标
y = b[:, 2] # y坐标
num = b[:, 3].astype(float) # 节点类型
ad = b[:, 4:].astype(str) # 邻接矩阵（字符串形式）
# 转换节点类型
in1 = np.where(num == 1)[0] # 一类重要点
in2 = np.where(num == 2)[0] # 二类重要点
in3 = np.where(np.isnan(num))[0] # 一般点
# 绘制节点
plt.figure()
plt.plot(x[in1], y[in1], 'Pk')
for i in range(len(in1)):
plt.text(x[in1[i]] + 10, y[in1[i]], s[in1[i]], fontsize=10)
plt.plot(x[in2], y[in2], '*k') # 二类重要点
for i in range(len(in2)):
plt.text(x[in2[i]] + 10, y[in2[i]], s[in2[i]], fontsize=10)
plt.plot(x[in3], y[in3], '.k') # 一般点
for i in range(len(in3)):
plt.text(x[in3[i]] + 10, y[in3[i]], s[in3[i]], fontsize=10)
# 构建距离矩阵
c = np.zeros((len(s), len(s)))
for i in range(len(s)):
tt = list(ad[i])
tt = [t for t in tt if t != 'nan' and t in s] # 过滤掉非节点名称和NaN
for k in range(len(tt)):
j = s.index(tt[k])
c[i, j] = np.sqrt((x[i] - x[j]) ** 2 + (y[i] - y[j]) ** 2)
# 只保留非零元素作为图的边
i, j = np.nonzero(c)
c = c[i, j]
edges = list(zip(i, j, c))
# 创建图并添加边
G = nx.Graph()
G.add_nodes_from(s)
G.add_weighted_edges_from(edges)
plt.show()
print("学号:3001")
（2）和（3）
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import numpy as np
import pandas as pd
import networkx as nx
import matplotlib.pyplot as plt
# 读取数据
a = pd.read_excel('ti6_7.xlsx')
b = a.values
# 提取数据
s = list(b[:, 0])
x = b[:, 1]
y = b[:, 2]
num = b[:, 3].astype(float)
ad = b[:, 4:].astype(str)
# 分类索引
in1 = np.where(num == 1)[0]
in2 = np.where(num == 2)[0]
in3 = np.where(np.isnan(num))[0]
# 绘图
plt.plot(x[in1], y[in1], 'Pk') # 修正了变量名
for i in range(len(in1)):
plt.text(x[in1[i]] + 10, y[in1[i]], s[in1[i]])
plt.plot(x[in2], y[in2], '*k')
for i in range(len(in2)):
plt.text(x[in2[i]] + 10, y[in2[i]], s[in2[i]])
plt.plot(x[in3], y[in3], '.k')
for i in range(len(in3)):
plt.text(x[in3[i]] + 10, y[in3[i]], s[in3[i]])
plt.show()
# 创建距离矩阵
c = np.zeros((len(s), len(s)))
for i in range(len(s)):
tt = list(ad[i])
tt = [t for t in tt if t != 'nan' and t in s] # 确保t是有效的节点名
for k in range(len(tt)):
j = s.index(tt[k])
c[i, j] = np.sqrt((x[i] - x[j]) ** 2 + (y[i] - y[j]) ** 2) # 修正了距离计算
# 创建图并添加边
G = nx.Graph()
G.add_nodes_from(s)
edges = []
for i in range(len(s)):
for j in range(len(s)):
if c[i, j] > 0: # 只添加有实际距离的边
edges.append((s[i], s[j], c[i, j]))
G.add_weighted_edges_from(edges)
# 计算最小生成树
T = nx.minimum_spanning_tree(G)
w = nx.get_edge_attributes(T, 'weight')
total_weight = sum(w.values()) # 计算最小生成树的总权重
print('最小生成树的长度为：', round(total_weight, 4))
# 绘制最小生成树
pos = dict(zip(s, b[:, [1, 2]]))
plt.figure()
nx.draw_networkx(T, pos, node_size=180, font_weight='bold', with_labels=True, node_color='w')
plt.show()
# 计算最短路径和最短距离（确保起点和终点在图中）
if 'I' in G.nodes() and 'R3' in G.nodes():
p = nx.shortest_path(G, 'I', 'R3', weight='weight')
d = nx.shortest_path_length(G, 'I', 'R3', weight='weight')
print('最短路径为：', p)
print('最短距离为：', round(d, 4))
# 绘制原图并突出显示最短路径
plt.figure()
nx.draw_networkx(G, pos, node_size=180, font_weight='bold', with_labels=True, node_color='w')
path_edges = list(zip(p, p[1:]))
nx.draw_networkx_edges(G, pos, edgelist=path_edges, edge_color='r', style='dashed', width=4)
plt.show()
else:
print("起点或终点不在图中。")
print("学号:3001")
例题6.10
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import numpy as np
import networkx as nx
L=[(1,2,20),(1,5,15),(2,3,20),(2,4,60),(2,5,25),(3,4,30),(3,5,18),(5,6,15)]
G=nx.Graph();G.add_nodes_from(np.arange(1,7))
G.add_weighted_edges_from(L)
d=nx.floyd_warshall_numpy(G)
md=np.max(d,axis=1)
mmd=min(md)
ind=np.argmin(md)+1
print(d);print("最小值为:",mmd)
print("最小值的地址为:",ind)
print("学号:3001")
例题6.11
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import numpy as np
import cvxpy as cp
L=[(1,2,18),(1,5,15),(2,3,20),(2,4,60),(2,5,12),
(3,4,30),(3,5,18),(4,6,10),(5,6,15)]
a=np.ones((6,6)) * 100000
for i in range(len(L)):
a[L[i][0]-1,L[i][1]-1]=L[i][2]
a[L[i][1]-1,L[i][0]-1]=L[i][2]
x=cp.Variable((6,6),integer=True)
obj=cp.Minimize(cp.sum(cp.multiply(a,x)))
con=[sum(x[1,:])==1,sum(x[:,1])==0,
sum(x[:,3])==1,x>=0,x<=1]
for i in set(range(6))-{1,3}:
con.append(sum(x[i,:])==sum(x[:,i]))
prob=cp.Problem(obj,con)
prob.solve(solver='GLPK_MI')
print("最优值为:",prob.value)
print("最优解为:\n",x.value)
i,j=np.nonzero(x.value)
print("最短路径的起点:",i+1)
print("最短路径的终点:",j+1)
print("学号:3001")
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