6

6.1

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#学号17
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()

6.2

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#学号17

edges = [  
    ("Pe", "T", 13),  
    ("Pe", "N", 68),  
    ("Pe", "M", 78),  
    ("Pe", "L", 51),  
    ("Pe", "Pa", 51),  
    ("T", "N", 68),  
    ("T", "M", 70),  
    ("T", "L", 60),  
    ("T", "Pa", 61),  
    ("N", "M", 21),  
    ("N", "L", 35),  
    ("N", "Pa",36),  
    ("M", "L", 56),  
    ("M", "Pa", 57),  
    ("L", "Pa", 21),  
]
class UnionFind:  
    def __init__(self, n):  
        self.parent = list(range(n))  
        self.rank = [0] * n  
  
    def find(self, u):  
        if self.parent[u] != u:  
            self.parent[u] = self.find(self.parent[u])  
        return self.parent[u]  
  
    def union(self, u, v):  
        root_u = self.find(u)  
        root_v = self.find(v)  
        if root_u != root_v:  
            if self.rank[root_u] > self.rank[root_v]:  
                self.parent[root_v] = root_u  
            elif self.rank[root_u] < self.rank[root_v]:  
                self.parent[root_u] = root_v  
            else:  
                self.parent[root_v] = root_u  
                self.rank[root_u] += 1  
  
def kruskal(edges):  
    edges.sort(key=lambda edge: edge[2]) 
    uf = UnionFind(len(set(city for edge in edges for city in edge[:2])))  
    mst = []  
      
    for u, v, weight in edges:  
        if uf.find(u) != uf.find(v):  
            uf.union(u, v)  
            mst.append((u, v, weight))  
      
    return mst  
 
city_map = {  
    "Pe": 0, "T": 1, "N": 2, "M": 3, "L": 4, "Pa": 5  
}  
 
edges_with_indices = [(city_map[u], city_map[v], weight) for u, v, weight in edges]  
 
mst_edges = [(list(city_map.keys())[u], list(city_map.keys())[v], weight) for u, v, weight in kruskal(edges_with_indices)]  
  
print("最小生成树的边:")  
for u, v, weight in mst_edges:  
    print(f"{u} - {v}: {weight}")

6.3

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#学号17
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)

6.4

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 #学号17
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年内最小总费用为：{min_cost}万元")  

6.5

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#学号17
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。")

6.6

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#学号17
import numpy as np  
matches = np.array([  
    [0, 1, 0, 1, 1, 1],   
    [0, 0, 0, 1, 1, 1],  
    [1, 1, 0, 1, 0, 0],  
    [0, 0, 0, 0, 1, 1],   
    [0, 0, 1, 0, 0, 1],  
    [0, 0, 1, 0, 0, 0]     
], dtype=int)  
  
n = matches.shape[0]  
closure = matches.copy()  
for k in range(n):  
    for i in range(n):  
        for j in range(n):  
            closure[i, j] = closure[i, j] or (closure[i, k] and closure[k, j])  
  
strength = closure.sum(axis=1)  
   
ranking = np.argsort(-strength) 
  
for i, rank in enumerate(ranking):  
    print(f"{chr(65 + rank)}队 排名 {i + 1}")
    
    
import numpy as np  
from scipy.sparse import csr_matrix  
  
edges = [  
    (0, 1), (0, 3), (0, 4), (0, 5),    
    (1, 3), (1, 4), (1, 5),           
    (2, 0), (2, 1), (2, 3),          
    (3, 4), (3, 5),                  
    (4, 2), (4, 5),                 
    (5, 2)                          
]  
  
 
num_teams = 6  
  
 
row_ind = []  
col_ind = []  
data = []  
for u, v in edges:  
    row_ind.append(u)  
    col_ind.append(v)  
    data.append(1)  
adj_matrix = csr_matrix((data, (row_ind, col_ind)), shape=(num_teams, num_teams))  
  
 
adj_matrix_T = adj_matrix.T