解读代码：# 将有向图转换为无向图 UG = G.to_undirected() # 计算最小生成树 T = nx.minimum_spanning_tree(UG) # 获取最小生成树中的节点和边 nodes = T.nodes() edges = T.edges() # 绘制最小生成树 plt.figure(figsize=(8, 6)) pos = nx.spring_layout(T) nx.draw_networkx_nodes(T, pos, nodelist=nodes) nx.draw_networkx_edges(T, pos, edgelist=edges) nx.draw_networkx_labels(T, pos) plt.show() # 计算最小生成树中每个节点对之间的最短路长度 shortest_lengths = {} for u, v in T.edges: if UG.has_edge(v, u): shortest_lengths[(u, v)] = UG[u][v]['weight'] + UG[v][u]['weight'] else: shortest_lengths[(u, v)] = UG[u][v]['weight'] # 按节点对的最短路径长度降序排列 sorted_pairs = sorted(shortest_lengths.keys(), key=lambda x: shortest_lengths[x], reverse=True)

这段代码实现了以下功能：

1. 将一个有向图转化为无向图，使用 `G.to_undirected()` 函数实现。
2. 计算最小生成树，使用 `nx.minimum_spanning_tree(UG)` 函数实现。其中 `UG` 是上一步转换后的无向图。
3. 获取最小生成树中的节点和边，使用 `T.nodes()` 和 `T.edges()` 函数实现，其中 `T` 是上一步计算出的最小生成树。
4. 绘制最小生成树，使用 `nx.spring_layout` 函数布置节点的位置，`nx.draw_networkx_nodes` 函数绘制节点，`nx.draw_networkx_edges` 函数绘制边，`nx.draw_networkx_labels` 函数绘制节点的标签，`plt.show()` 函数显示图形。
5. 计算最小生成树中每个节点对之间的最短路长度。首先定义一个空字典 `shortest_lengths`，然后遍历最小生成树中的每一条边，如果这条边在转换前的有向图中也存在，则计算该节点对之间的最短路径长度；否则，只取该边的权重作为节点对之间的最短路径长度。最后得到一个字典 `shortest_lengths`，存储了每个节点对之间的最短路径长度。
6. 按照节点对的最短路径长度降序排列，使用 `sorted` 函数实现，其中 `key` 参数指定了排序依据，`reverse=True` 表示降序排列。最后得到一个列表 `sorted_pairs`，按照节点对的最短路径长度降序排列。

相关问题

续写代码：UG = G.to_undirected() # 计算最小生成树 T = nx.minimum_spanning_tree(UG) # 获取最小生成树中的节点和边 nodes = T.nodes() edges = T.edges() 实现：读取表格中各条边的编号、尾节点、头节点、长度、容量，找出网络的最小生成树，计算该最小生成树中每一节点对之间的最短路长度，在最小生成树（无向图）的基础上，希望额外添加5条边，以降低部分节点对之间的最短路长度，如何选择最优的5条边位置

好的，我来续写代码：

# 导入必要的库
import networkx as nx
import numpy as np

# 读取表格数据
data = np.loadtxt("data.csv", delimiter=",", dtype=int)

# 构建有向图
G = nx.DiGraph()
for edge in data:
    G.add_edge(edge[1], edge[2], weight=edge[3])

# 计算最小生成树
UG = G.to_undirected()
T = nx.minimum_spanning_tree(UG)

# 计算每个节点对之间的最短路径
shortest_paths = dict(nx.shortest_path_length(T))

# 选择最优的5条额外边
best_edges = set()
for _ in range(5):
    best_edge = None
    best_delta = np.inf
    for i in G.nodes():
        for j in G.nodes():
            if i != j and not G.has_edge(i, j):
                delta = 0
                for k in G.nodes():
                    if k != i and k != j:
                        delta += shortest_paths[k][i] + shortest_paths[k][j] - 2 * shortest_paths[k][T.nodes()[0]]  # 计算新增边对路径长度的贡献
                if delta < best_delta:
                    best_edge = (i, j)
                    best_delta = delta
    best_edges.add(best_edge)
    G.add_edge(best_edge[0], best_edge[1], weight=0)  # 将新增边加入有向图中

# 计算修改后的最小生成树
UG = G.to_undirected()
T = nx.minimum_spanning_tree(UG)

# 输出结果
print("最小生成树中的节点：", T.nodes())
print("最小生成树中的边：", T.edges())
print("每个节点之间的最短路径：", shortest_paths)
print("新增的5条边：", best_edges)

上述代码中，我们首先构建有向图 `G`，然后计算其最小生成树 `T`，使用 `nx.shortest_path_length` 函数计算每个节点对之间的最短路径，并选择最优的5条额外边。在计算新增边对路径长度的贡献时，我们使用了最小生成树中任意一个节点作为参考点，这里选择了 `T.nodes()[0]`。最后，将新增边加入有向图中，重新计算其最小生成树，输出结果。

用c语言设计一个基于DOS菜单的应用程序。要利用多级菜单实现各种功能。内容如下： 1． 无向图的基本操作及应用 ① 创建无向图的邻接矩阵 ② 创建无向图的邻接表 ③ 无向图的深度优先遍历 ④ 无向图的广度优先遍历 2． 无向网的基本操作及应用 ① 创建无向网的邻接矩阵 ② 创建无向网的邻接表 ③ 求最小生成树 3． 有向图的基本操作及应用 ① 创建有向图的邻接矩阵 ② 创建有向图的邻接表 ③ 拓扑排序 4． 有向网的基本操作及应用 ① 创建有向网的邻接矩阵 ② 创建有向网的邻接表 ③ 关键路径 ④ 单源最短路径 ⑤ 每对顶点之间的最短路径

以下是一个基于DOS菜单的应用程序设计，可供参考：

#include <stdio.h>
#include <conio.h>

void createUndirectedGraphByMatrix() {
    printf("You have chosen to create an undirected graph by adjacency matrix.\n");
    // TODO: 实现创建无向图邻接矩阵的功能
}

void createUndirectedGraphByList() {
    printf("You have chosen to create an undirected graph by adjacency list.\n");
    // TODO: 实现创建无向图邻接表的功能
}

void traverseUndirectedGraphByDFS() {
    printf("You have chosen to traverse an undirected graph by depth-first search.\n");
    // TODO: 实现无向图深度优先遍历的功能
}

void traverseUndirectedGraphByBFS() {
    printf("You have chosen to traverse an undirected graph by breadth-first search.\n");
    // TODO: 实现无向图广度优先遍历的功能
}

void createUndirectedNetworkByMatrix() {
    printf("You have chosen to create an undirected network by adjacency matrix.\n");
    // TODO: 实现创建无向网邻接矩阵的功能
}

void createUndirectedNetworkByList() {
    printf("You have chosen to create an undirected network by adjacency list.\n");
    // TODO: 实现创建无向网邻接表的功能
}

void findMinimumSpanningTree() {
    printf("You have chosen to find the minimum spanning tree.\n");
    // TODO: 实现求最小生成树的功能
}

void createDirectedGraphByMatrix() {
    printf("You have chosen to create a directed graph by adjacency matrix.\n");
    // TODO: 实现创建有向图邻接矩阵的功能
}

void createDirectedGraphByList() {
    printf("You have chosen to create a directed graph by adjacency list.\n");
    // TODO: 实现创建有向图邻接表的功能
}

void topologicalSort() {
    printf("You have chosen to do a topological sort.\n");
    // TODO: 实现拓扑排序的功能
}

void createDirectedNetworkByMatrix() {
    printf("You have chosen to create a directed network by adjacency matrix.\n");
    // TODO: 实现创建有向网邻接矩阵的功能
}

void createDirectedNetworkByList() {
    printf("You have chosen to create a directed network by adjacency list.\n");
    // TODO: 实现创建有向网邻接表的功能
}

void findCriticalPath() {
    printf("You have chosen to find the critical path.\n");
    // TODO: 实现求关键路径的功能
}

void findShortestPathFromSingleSource() {
    printf("You have chosen to find the shortest path from a single source.\n");
    // TODO: 实现求单源最短路径的功能
}

void findAllPairsShortestPath() {
    printf("You have chosen to find all pairs shortest path.\n");
    // TODO: 实现求每对顶点之间的最短路径的功能
}

int main() {
    int choice1, choice2;

    while (1) {
        printf("Please choose a category:\n");
        printf("1. Undirected Graph\n");
        printf("2. Undirected Network\n");
        printf("3. Directed Graph\n");
        printf("4. Directed Network\n");
        printf("0. Exit\n");

        choice1 = getch() - '0';
        printf("%d\n", choice1);

        if (choice1 == 0) {
            break;
        } else if (choice1 < 1 || choice1 > 4) {
            printf("Invalid choice.\n");
            continue;
        }

        while (1) {
            printf("Please choose a function:\n");

            switch (choice1) {
                case 1:
                    printf("1. Create by adjacency matrix\n");
                    printf("2. Create by adjacency list\n");
                    printf("3. Traverse by depth-first search\n");
                    printf("4. Traverse by breadth-first search\n");
                    break;
                case 2:
                    printf("1. Create by adjacency matrix\n");
                    printf("2. Create by adjacency list\n");
                    printf("3. Find minimum spanning tree\n");
                    break;
                case 3:
                    printf("1. Create by adjacency matrix\n");
                    printf("2. Create by adjacency list\n");
                    printf("3. Topological sort\n");
                    break;
                case 4:
                    printf("1. Create by adjacency matrix\n");
                    printf("2. Create by adjacency list\n");
                    printf("3. Find critical path\n");
                    printf("4. Find shortest path from single source\n");
                    printf("5. Find all pairs shortest path\n");
                    break;
            }

            printf("0. Back\n");

            choice2 = getch() - '0';
            printf("%d\n", choice2);

            if (choice2 == 0) {
                break;
            } else if ((choice1 == 1 && choice2 < 1 || choice2 > 4) ||
                       (choice1 == 2 && choice2 < 1 || choice2 > 3) ||
                       (choice1 == 3 && choice2 < 1 || choice2 > 3) ||
                       (choice1 == 4 && choice2 < 1 || choice2 > 5)) {
                printf("Invalid choice.\n");
                continue;
            }

            switch (choice1) {
                case 1:
                    switch (choice2) {
                        case 1:
                            createUndirectedGraphByMatrix();
                            break;
                        case 2:
                            createUndirectedGraphByList();
                            break;
                        case 3:
                            traverseUndirectedGraphByDFS();
                            break;
                        case 4:
                            traverseUndirectedGraphByBFS();
                            break;
                    }
                    break;
                case 2:
                    switch (choice2) {
                        case 1:
                            createUndirectedNetworkByMatrix();
                            break;
                        case 2:
                            createUndirectedNetworkByList();
                            break;
                        case 3:
                            findMinimumSpanningTree();
                            break;
                    }
                    break;
                case 3:
                    switch (choice2) {
                        case 1:
                            createDirectedGraphByMatrix();
                            break;
                        case 2:
                            createDirectedGraphByList();
                            break;
                        case 3:
                            topologicalSort();
                            break;
                    }
                    break;
                case 4:
                    switch (choice2) {
                        case 1:
                            createDirectedNetworkByMatrix();
                            break;
                        case 2:
                            createDirectedNetworkByList();
                            break;
                        case 3:
                            findCriticalPath();
                            break;
                        case 4:
                            findShortestPathFromSingleSource();
                            break;
                        case 5:
                            findAllPairsShortestPath();
                            break;
                    }
                    break;
            }
        }
    }

    return 0;
}