最小生成树

在一给定的无向图G = (V, E) 中,(u, v) 代表连接顶点 u 与顶点 v 的边,而 w(u, v) 代表此边的权重,若存在 T 为 E 的子集,使得 w(T) 最小,则此 T 为 G 的最小生成树。最小生成树可以用kruskal（克鲁斯卡尔）算法或prim（普里姆）算法求出。

Prim算法

普利姆算法（Prim）求的是最小生成树,也就是在包含n个顶点的无向连通图中找到（n-1）条边包含所有n个顶点的连通子图。

代码

/*无向连通图*/
public class Graph {
    int vertexs;
    char[] data;
    int[][] weight;

    public Graph(int vertexs,char[] data,int[][] weight){
        this.vertexs = vertexs;
        this.data = new char[vertexs];
        this.weight = new int[vertexs][vertexs];

        for (int i = 0; i < vertexs; i++) {
            this.data[i] = data[i];
            for (int j = 0; j < vertexs; j++) {
                this.weight[i][j] = weight[i][j];
            }
        }
    }
}

public class Prim {
    public static void main(String[] args) {
        char[] data = {'A','B','C','D','E','F','G'};
        /*0表示两个顶点之间不能直接到达*/
        int[][] weight = {{0,5,7,0,0,0,2},
                          {5,0,0,9,0,0,3},
                          {7,0,0,0,8,0,0},
                          {0,9,0,0,0,4,0},
                          {0,0,8,0,0,5,4},
                          {0,0,0,4,5,0,6},
                          {2,3,0,0,4,6,0}};
		/*初始化图*/
        Graph graph = new Graph(data.length,data,weight);
        prim(graph,0);
    }

    /**
     * prim算法求最小生成树
     * @param graph 带权连通图
     * @param v 从哪个顶点开始生成
     */
    public static void prim(Graph graph,int v){
        /*标记顶点是否被访问*/
        boolean[] isVisited = new boolean[graph.vertexs];
        /*标记当前顶点已经被访问*/
        isVisited[v] = true;
        /*h1,h2记录两个顶点下标*/
        int h1 = -1;
        int h2 = -1;
        int miniWeight = 10000;

        for (int i = 1; i < graph.vertexs; i++) { //因为有vertexs个顶点 根据prim算法会生成vertexs-1条边 所以i从1开始

            for (int j = 0; j < graph.vertexs; j++) { //遍历已经访问过的顶点  例如图中的<A,G>
                for (int k = 0; k < graph.vertexs; k++) { //遍历未访问过的顶点  例如图中的A-C[7],A-B[5],G-B[3],G-E[4],G-F[6]
                    if (isVisited[j] && !isVisited[k]  &&graph.weight[j][k] > 0 && graph.weight[j][k] < miniWeight){
                        miniWeight = graph.weight[j][k];
                        h1 = j;
                        h2 = k;
                    }
                }
            }
            System.out.println(graph.data[h1]+"->"+graph.data[h2]+"权值："+miniWeight);
            isVisited[h2] = true;
            miniWeight = 10000;
        }
    }
}

运行结果

A->G权值：2
G->B权值：3
G->E权值：4
E->F权值：5
F->D权值：4
A->C权值：7

Process finished with exit code 0