Minimum Spanning Tree Prim Algorithm Kruskal Algorithm

Features of the minimum spanning tree Features of the
tree:
1. No loops
2. V vertices have V-1 edges

Features of the spanning tree:
1. Contains all vertices of the tree
2. V-1 edges are in the graph

Minimum: the weight and minimum of the edge

Prim algorithm

Find the node with the smallest edge weight outside the tree each time;
add the node to the tree;
key points When
finding the node, you must also meet
1. You can only connect the node with the existing edge in the graph
2. Only just Use the V-1 edge
3. After adding a new node, no loop appears.
Solution
How to determine whether the node is included in the tree: the distance between the collected node and the tree is 0, and update the distance of the surrounding nodes
Every time we grab the node outside the tree, so there will be no loop;
if the V-1 edge is not used, it means that the graph is not connected and there is no minimum spanning tree

Build code
Adjacency matrix storage
1. Find the node with the smallest edge weight connected to the tree

Vertex FindMinDist( MGraph Graph, WeightType dist[] )
{ /* 返回未被收录顶点中dist最小者 */
    Vertex MinV, V;
    WeightType MinDist = INFINITY;
 
    for (V=0; V<Graph->Nv; V++) {
        if ( dist[V]!=0 && dist[V]<MinDist) {
            /* 若V未被收录，且dist[V]更小 */
            MinDist = dist[V]; /* 更新最小距离 */
            MinV = V; /* 更新对应顶点 */
        }
    }
    if (MinDist < INFINITY) /* 若找到最小dist */
        return MinV; /* 返回对应的顶点下标 */
    else return ERROR;  /* 若这样的顶点不存在，返回-1作为标记 */
}

2.prim function

int Prim( MGraph Graph, LGraph MST )
{ /* 将最小生成树保存为邻接表存储的图MST，返回最小权重和 */
    WeightType dist[MaxVertexNum], TotalWeight;
    Vertex parent[MaxVertexNum], V, W;
    int VCount;
    Edge E;
     
    /* 初始化。默认初始点下标是0 */
       for (V=0; V<Graph->Nv; V++) {
        /* 这里假设若V到W没有直接的边，则Graph->G[V][W]定义为INFINITY */
           dist[V] = Graph->G[0][V];
           parent[V] = 0; /* 暂且定义所有顶点的父结点都是初始点0 */ 
    }
    TotalWeight = 0; /* 初始化权重和     */
    VCount = 0;      /* 初始化收录的顶点数 */
    /* 创建包含所有顶点但没有边的图。注意用邻接表版本 */
    MST = CreateGraph(Graph->Nv);
    E = (Edge)malloc( sizeof(struct ENode) ); /* 建立空的边结点 */
            
    /* 将初始点0收录进MST */
    dist[0] = 0;
    VCount ++;
    parent[0] = -1; /* 当前树根是0 */
 
    while (1) {
        V = FindMinDist( Graph, dist );
        /* V = 未被收录顶点中dist最小者 */
        if ( V==ERROR ) /* 若这样的V不存在 */
            break;   /* 算法结束 */
             
        /* 将V及相应的边<parent[V], V>收录进MST */
        E->V1 = parent[V];
        E->V2 = V;
        E->Weight = dist[V];
        InsertEdge( MST, E );
        TotalWeight += dist[V];
        dist[V] = 0;
        VCount++;
         
        for( W=0; W<Graph->Nv; W++ ) /* 对图中的每个顶点W */
            if ( dist[W]!=0 && Graph->G[V][W]<INFINITY ) {
            /* 若W是V的邻接点并且未被收录 */
                if ( Graph->G[V][W] < dist[W] ) {
                /* 若收录V使得dist[W]变小 */
                    dist[W] = Graph->G[V][W]; /* 更新dist[W] */
                    parent[W] = V; /* 更新树 */
                }
            }
    } /* while结束*/
    if ( VCount < Graph->Nv ) /* MST中收的顶点不到|V|个 */
       TotalWeight = ERROR;
    return TotalWeight;   /* 算法执行完毕，返回最小权重和或错误标记 */
}