There is No Alternative~最小生成树变形

Posted 2020-11-03 Fitz

Description

ICPC (Isles of Coral Park City) consist of several beautiful islands.

The citizens requested construction of bridges between islands to resolve inconveniences of using boats between islands, and they demand that all the islands should be reachable from any other islands via one or more bridges.

The city mayor selected a number of pairs of islands, and ordered a building company to estimate the costs to build bridges between the pairs. With this estimate, the mayor has to decide the set of bridges to build, minimizing the total construction cost.

However, it is difficult for him to select the most cost-efficient set of bridges among those connecting all the islands. For example, three sets of bridges connect all the islands for the Sample Input 1. The bridges in each set are expressed by bold edges in Figure F.1.

Figure F.1. Three sets of bridges connecting all the islands for Sample Input 1

As the first step, he decided to build only those bridges which are contained in all the sets of bridges to connect all the islands and minimize the cost. We refer to such bridges as no alternative bridges. In Figure F.2, no alternative bridges are drawn as thick edges for the Sample Input 1, 2 and 3.

Write a program that advises the mayor which bridges are no alternative bridges for the given input.

Input

The input consists of several tests case.

Figure F.2. No alternative bridges for Sample Input 1, 2 and 3

 

N MS1 D1 C1?SM DM CMN MS1 D1 C1?SM DM CM

For each test, the first line contains two positive integers N and M . N represents the number of islands and each island is identified by an integer 1 through N. M represents the number of the pairs of islands between which a bridge may be built.

 

Each line of the next M lines contains three integers Si, Di and Ci (1?≤?i?≤?M) which represent that it will cost Ci to build the bridge between islands Si and Di. You may assume 3?≤?N?≤?500, N???1?≤?M?≤?min(50000,?N(N???1)/2), 1?≤?Si?<?Di?≤?N, and 1?≤?Ci?≤?10000. No two bridges connect the same pair of two islands, that is, if i?≠?j and Si?=?Sj , then Di?≠?Dj. If all the candidate bridges are built, all the islands are reachable from any other islands via one or more bridges.

Output

Output two integers, which mean the number of no alternative bridges and the sum of their construction cost, separated by a space.

Sample Input

4 4
1 2 3
1 3 3
2 3 3
2 4 3

4 4
1 2 3
1 3 5
2 3 3
2 4 3

4 4
1 2 3
1 3 1
2 3 3
2 4 3

3 3
1 2 1
2 3 1
1 3 1

Sample Output

1 3
3 9
2 4
0 0

可以组成多种最小生成树，求他们的公共边，和权值和；
这个n ，可以直接暴力枚举；
暴力出奇迹
暴力枚举一下就好了；
先求出一个最小生成树，记录边；
依次删边，看新的最小生成树的权值是否相等
不相等则证明，必须有的边，

 1 #include <cstdio>
 2 #include <algorithm>
 3 #include <vector>
 4 #include <queue>
 5 #include <cstring>
 6 #include <string>
 7 #include <cmath>
 8 #include <map>
 9 using namespace std;
10 const int maxn = 5e4 + 10;
11 const int INF = 1e9 + 7;
12 int fa[510], vis[maxn];
13 struct node {
14     int u, v, w;
15 } qu[maxn];
16 int cmp(node a, node b) {
17     return  a.w < b.w;
18 }
19 int find(int x) {
20     return fa[x] == x ? x : fa[x] = find(fa[x]);
21 }
22 int combine(int x, int y) {
23     int nx = find(x);
24     int ny = find(y);
25     if(nx != ny) {
26         fa[nx] = ny;
27         return 1;
28     }
29     return 0;
30 }
31 int kruskal(int num, int flag, int x) {
32     int sum = 0, k = 0;
33     for(int i = 0; i < num; i++) {
34         if(x == i)  continue;
35         if(combine(qu[i].u, qu[i].v)) {
36             sum += qu[i].w;
37             if(flag)  vis[k++] = i;
38         }
39     }
40     return sum;
41 }
42 int main() {
43  //   freopen("DATA.txt", "r", stdin);
44     int n, m;
45     while(scanf("%d%d", &n, &m) != EOF) {
46         for (int i = 0 ; i < m ; i++) {
47             scanf("%d%d%d", &qu[i].v, &qu[i].u, &qu[i].w);
48         }
49         sort(qu, qu + m, cmp);
50         int temp = kruskal(m, 1, -1);
51         int ans1 = 0, ans2 = 0;
52         for (int i = 1 ; i <= n ; i++) fa[i] = i;
53         for (int i = 0 ; i < n - 1 ; i++ ) {
54             for (int j = 0 ; j <= n ; j++) fa[j] = j;
55             int sum = kruskal(m, 0, vis[i]);
56             if (sum != temp) {
57                 ans1++;
58                 ans2 += qu[vis[i]].w;
59             }
60         }
61         printf("%d %d\n", ans1, ans2 );
62     }
63     return 0;
64 }