08-图7 公路村村通 (30分)

题目描述

现有村落间道路的统计数据表中，列出了有可能建设成标准公路的若干条道路的成本，求使每个村落都有公路连通所需要的最低成本。

输入格式:

输入数据包括城镇数目正整数N（≤1000）和候选道路数目M（≤3N）；随后的M行对应M条道路，每行给出3个正整数，分别是该条道路直接连通的两个城镇的编号以及该道路改建的预算成本。为简单起见，城镇从1到N编号。

输出格式:

输出村村通需要的最低成本。如果输入数据不足以保证畅通，则输出−1，表示需要建设更多公路。

输入样例:

6 15
1 2 5
1 3 3
1 4 7
1 5 4
1 6 2
2 3 4
2 4 6
2 5 2
2 6 6
3 4 6
3 5 1
3 6 1
4 5 10
4 6 8
5 6 3

输出样例:

12

解题思路

很明显的最小生成树问题，直接使用Prim算法即可。

代码

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

#define MAXSIZE 1001
#define INFINITY 65535

struct Graph {
    int vertexCount;
    int edgeCount;
    int matrix[MAXSIZE][MAXSIZE];
};
typedef struct Graph *MGraph;

MGraph createGraph();
void Prim(MGraph graph);
int findMinDis(MGraph graph, int dis[]);

int main() {
    MGraph graph = createGraph();
    Prim(graph);
    return 0;
}

MGraph createGraph() {
    int N, M;
    scanf("%d %d", &N, &M);

    MGraph graph = (MGraph) malloc(sizeof(struct Graph));
    graph->vertexCount = N; 
    graph->edgeCount = M;

    for (int i = 1; i <= N; i++) {
        for (int j = 1; j <= N; j++) {
            graph->matrix[i][j] = INFINITY;
            graph->matrix[j][i] = INFINITY;
        }
    }
        
    for (int i = 0; i < M; i++) {
        int x, y, weight;
        scanf("%d %d %d", &x, &y, &weight);
        graph->matrix[x][y] = weight;
        graph->matrix[y][x] = weight;
    }

    return graph;
}

void Prim(MGraph graph) {
    //vCount用于判断是否成功得到MST，totalWeight用于输出结果
    int vCount = 1, totalWeight = 0;

    int dis[MAXSIZE] = {0};     //dis是图中结点到MST的距离，为0表示在MST中
    for (int i = 2; i <= graph->vertexCount; i++)
        dis[i] = graph->matrix[1][i];   //从第一个顶点开始Prim算法

    while (1) {
        int vertex = findMinDis(graph, dis);    //找到距离最近且不在MST中的顶点
        if (vertex == -1) break;                //没找到就返回

        vCount++;                               //顶点数加一
        totalWeight += dis[vertex];             //权重增加
        dis[vertex] = 0;                        //将顶点放入MST中

        for (int i = 1; i <= graph->vertexCount; i++)
            if (dis[i] != 0 && graph->matrix[vertex][i] < INFINITY)
                if (dis[i] > graph->matrix[vertex][i])
                    dis[i] = graph->matrix[vertex][i];  //更新dis数组
    }

    if (vCount < graph->vertexCount) printf("-1\n");    //顶点数不够就错误
    else printf("%d\n", totalWeight);                   //成功生成MST
}

int findMinDis(MGraph graph, int dis[]) {
    int minDis = INFINITY, minVertex = 1;
    for (int i = 1; i <= graph->vertexCount; i++) {
        if (dis[i] != 0 && dis[i] < minDis) {
            minDis = dis[i];
            minVertex = i;
        }
    }
    if (minDis == INFINITY) return -1;
    return minVertex;
}