用邻接矩阵表示无向带权图prim c++

无向带权图可以用邻接矩阵来表示，而Prim算法是一种用来寻找最小生成树的算法。首先，我们需要构建一个二维数组作为邻接矩阵来表示图的边的权重。假设该无向带权图有n个顶点，我们可以构建一个n*n的矩阵来表示图中顶点之间的边的权重。

接着，我们可以利用Prim算法来找出该图的最小生成树。具体做法是从任意一个顶点开始，将其加入到最小生成树中，并标记为已访问。然后依次找出与已访问的顶点相连且权重最小的边所连接的顶点，将其加入到最小生成树中，并标记为已访问。

重复上述步骤直到所有的顶点都被访问过，即可得到该图的最小生成树。最小生成树的邻接矩阵就是Prim算法中每一步都选取的顶点与其连接的边所构成的矩阵。

总的来说，用邻接矩阵表示无向带权图prim c 就是利用二维数组来表示图的边的权重，然后利用Prim算法来寻找最小生成树，最后得到最小生成树的邻接矩阵来表示整个图的结构。

相关问题

用c++ 给出无向网的Prim算法和Kruskal算法的代码

在C++中，Prim算法和Kruskal算法用于求解最小生成树的问题。这里我会提供简化的伪代码以及基本的C++实现，实际项目中可能需要优化处理大规模数据。

### Prim算法（基于邻接矩阵）

#include <vector>
#include <set>

// 边结构体
struct Edge {
    int src, dest, weight;
};

class Graph {
public:
    // 添加边和获取邻接矩阵
    void addEdge(int src, int dest, int weight);
    std::vector<int> prim();
private:
    std::vector<std::vector<int>> adjMatrix; // 邻接矩阵存储权重
    std::set<int> visited; // 记录已访问节点
};

void Graph::addEdge(int src, int dest, int weight) {
    adjMatrix[src][dest] = weight;
    adjMatrix[dest][src] = weight;
}

std::vector<int> Graph::prim() {
    // 初始化最小生成树和集合
    std::vector<int> minSet(adjMatrix.size(), INT_MAX), MST = {0};
    std::set<int> MST_set;
    
    // 将第一顶点加入MST
    minSet[0] = 0;
    MST_set.insert(0);

    for (int i = 1; i < adjMatrix.size(); ++i) {
        int u = -1;
        for (int v : MST_set) {
            if (adjMatrix[v][i] && minSet[u] > adjMatrix[v][i]) {
                u = v;
            }
        }

        if (u != -1) {
            minSet[i] = adjMatrix[u][i];
            MST.push_back(i);
            MST_set.insert(i);
            visited.insert(u);
        }
    }

    return MST;
}

### Kruskal算法（基于并查集）

#include <vector>
#include <set>
#include <algorithm>

struct Edge {
    int src, dest, weight;
};

class UnionFind {
public:
    void makeSet(int node);
    int findSet(int node);
    void unionSets(int x, int y);
private:
    std::vector<int> parent;
    std::vector<int> rank;
};

// ... 实现UnionFind类的其他方法

std::vector<Edge> kruskal(Graph& graph) {
    std::vector<Edge> edges(graph.getEdges()); // 获取所有边
    sort(edges.begin(), edges.end(), [](const Edge& a, const Edge& b) { return a.weight < b.weight; });

    UnionFind uf(graph.getNumVertices());
    std::vector<Edge> MST;

    for (const auto& edge : edges) {
        int u = uf.findSet(edge.src);
        int v = uf.findSet(edge.dest);

        if (u != v) {
            MST.push_back(edge);
            uf.unionSets(u, v);
        }

        if (uf.getNumSets() == graph.getNumVertices() - 1) break; // 找到最小生成树
    }

    return MST;
}

利用邻接矩阵表示无向图，请实现prim算法

Prim's algorithm, also known as the Prim's minimum spanning tree (MST) algorithm, is a popular algorithm for finding the minimum spanning tree of an undirected weighted graph represented using an adjacency matrix. Here's how you can implement it in C++:

#include <iostream>
#include <vector>
#include <climits>

using namespace std;

// Function to find the minimum edge weight
int min_edge(vector<vector<int>>& adj, vector<bool>& visited, int node, int& min_weight) {
    int min_val = INT_MAX;
    int min_index = -1;
    for (int i = 0; i < adj[node].size(); i++) {
        if (!visited[i] && adj[node][i] < min_val) {
            min_val = adj[node][i];
            min_index = i;
        }
    }
    min_weight = min_weight ? min_weight : min_val;
    return min_index;
}

// Function to perform Prim's Algorithm
vector<int> prim_mst(vector<vector<int>>& adj, int V) {
    vector<bool> visited(V, false);
    vector<int> parent(V, -1); // Parent array
    priority_queue<pair<int, int>, vector<pair<int, int>>, greater<pair<int, int>>> pq; // Min heap for edges

    // Add the first vertex with weight 0
    visited[0] = true;
    pq.push({0, 0});

    while (!pq.empty()) {
        int u = pq.top().second;
        pq.pop();

        // Find the minimum edge from the current node
        int min_weight = INT_MAX;
        int v = min_edge(adj, visited, u, min_weight);

        if (min_weight != INT_MAX) {
            // If found, mark the node as visited and update the parent
            visited[v] = true;
            parent[v] = u;

            // Relax the adjacent vertices
            for (int i = 0; i < adj[v].size(); i++) {
                if (!visited[i] && adj[u][i] < INT_MAX) {
                    pq.push({adj[u][i], i});
                }
            }
        }
    }

    return parent;
}

int main() {
    int V, E;
    cout << "Enter number of vertices: ";
    cin >> V;
    cout << "Enter number of edges: ";
    cin >> E;
    
    vector<vector<int>> adj(V, vector<int>(V, INT_MAX)); // Initialize adjacency matrix with infinite weights
    // Read the edges and their weights here

    prim_mst(adj, V); // Call the function
    // Print or use the parent array to reconstruct the MST
    return 0;
}

In this code, we create a `prim_mst` function that takes an adjacency matrix (`adj`) and the number of vertices (`V`). It uses a priority queue to keep track of the smallest edge weights and keeps a boolean array `visited` to mark which nodes are already included in the MST.

To use this implementation, you'll need to input the edges and their weights into the adjacency matrix before calling the `prim_mst` function. The returned `parent` array will then contain the parent-child relationships for each node in the minimum spanning tree.