c语言实现prim算法与克鲁斯卡尔算法

### C语言实现Prim算法和Kruskal算法

#### 1. Prim算法的C语言实现

Prim算法用于寻找加权连通图中的最小生成树。该算法通过逐步扩展一棵树来进行工作，每次迭代都会找到一条权重最小的新边并将其加入到现有的树中。

#include <stdio.h>
#include <limits.h>

#define V 10

int minKey(int key[], int mstSet[]) {
    int min = INT_MAX, min_index;
    
    for (int v = 0; v < V; v++)
        if (!mstSet[v] && key[v] < min)
            min = key[v], min_index = v;

    return min_index;
}

void printMST(int parent[], int graph[V][V]) {
    printf("Edge \tWeight\n");
    for (int i = 1; i < V; i++)
        printf("%d - %d \t%d \n", parent[i], i, graph[i][parent[i]]);
}

void primMST(int graph[V][V]) {
    int parent[V];
    int key[V];
    int mstSet[V];

    for (int i = 0; i < V; i++) {
        key[i] = INT_MAX;
        mstSet[i] = 0;
    }

    key[0] = 0;
    parent[0] = -1;

    for (int count = 0; count < V - 1; count++) {
        int u = minKey(key, mstSet);
        
        mstSet[u] = 1;

        for (int v = 0; v < V; v++)

            if (graph[u][v] && !mstSet[v] && graph[u][v] < key[v])
                parent[v] = u, key[v] = graph[u][v];
    }
    
    printMST(parent, graph);
}

这段代码展示了如何利用数组`key[]`保存最短路径估计值以及使用布尔型数组`mstSet[]`跟踪节点是否已处理完毕[^1]。

#### 2. Kruskal算法的C语言实现

Kruskal算法同样用来构建最小生成树，不过它采取了一种不同的策略——按照边的权重升序排列，并依次尝试添加每条边至最终的结果集中，前提是这样做不会形成任何循环。

为了有效地管理这些集合之间的关系，在此引入了不相交集数据结构（Disjoint Set Data Structure），也称为Union-Find结构体。

#include <stdio.h>
#include <stdlib.h>

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

struct Graph {
    int V, E;
    struct Edge* edge;
};

struct subset {
    int parent;
    int rank;
};

int find(struct subset subsets[], int i) {
    if (subsets[i].parent != i)
        subsets[i].parent = find(subsets, subsets[i].parent);

    return subsets[i].parent;
}

void Union(struct subset subsets[], int x, int y) {
    int xroot = find(subsets, x);
    int yroot = find(subsets, y);

    if (subsets[xroot].rank < subsets[yroot].rank)
        subsets[xroot].parent = yroot;
    else if (subsets[xroot].rank > subsets[yroot].rank)
        subsets[yroot].parent = xroot;
    else {
        subsets[yroot].parent = xroot;
        subsets[xroot].rank++;
    }
}

int myComp(const void *a, const void *b) {
    struct Edge *a1 = (struct Edge *) a;
    struct Edge *b1 = (struct Edge *) b;
    return a1->weight > b1->weight;
}

void kruskalMST(struct Graph* graph) {
    int V = graph->V;
    struct Edge result[V];  
    int e = 0; 
    int i = 0; 

    qsort(graph->edge, graph->E, sizeof(graph->edge[0]), myComp);

    struct subset *subsets =
           (struct subset*) malloc(V * sizeof(struct subset));

    for (int v = 0; v < V; ++v) {
        subsets[v].parent = v;
        subsets[v].rank = 0;
    }

    while (e < V - 1 && i < graph->E) {

        struct Edge next_edge = graph->edge[i++];

        int x = find(subsets, next_edge.src);
        int y = find(subsets, next_edge.dest);

        if (x != y) {
            result[e++] = next_edge;
            Union(subsets, x, y);
        }
    }

    printf("Following are the edges in the constructed MST\n");
    for (i = 0; i < e; ++i)
        printf("%d -- %d == %d\n", result[i].src,
               result[i].dest, result[i].weight);
    return;
}

上述程序实现了Kruskal算法的核心逻辑，包括快速排序函数`qsort()`的应用、查找根结点的方法`find()`以及合并子集的操作`Union()`. 此外还定义了一个辅助性的比较器`myComp()`以便于对所有的边按其权重进行排序[^2].