第六章习题

学号后四位:3018
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("xuehao3018")

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年内最小总费用为：{min_cost}万元")  
print("xuehao3018")

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("xuehao3018")

6.6:

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import numpy as np  
  
matches = np.array([  
    [0, 1, 0, 1, 1, 1],  # 1队  
    [0, 0, 0, 1, 1, 1],  # 2队  
    [1, 1, 0, 1, 0, 0],  # 3队  
    [0, 0, 0, 0, 1, 1],  # 4队  
    [0, 0, 1, 0, 0, 1],  # 5队  
    [0, 0, 1, 0, 0, 0]   # 6队  
], 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  
  
 
d = 0.85  
out_degree = np.array(adj_matrix_T.sum(axis=1)).flatten()  
out_degree[out_degree == 0] = 1  
M = adj_matrix_T.multiply(1.0 / out_degree).tocsr()  
M = M + (1 - d) / num_teams * csr_matrix(np.ones((num_teams, num_teams)))  
  
 
R = np.ones(num_teams) / num_teams  
  
 
num_iterations = 100  
for _ in range(num_iterations):  
    R = R.dot(M.toarray())  
  
 
pagerank_ranking = np.argsort(-R) 
  
 
for i, rank in enumerate(pagerank_ranking):  
    print(f"{chr(65 + rank)}队 PageRank排名 {i + 1}")
print("xuehao3018")