poj 1679 The Unique MST(次小生成树)

题意：判断最小生成树是否唯一。

分析：只需要判断最小生成树与次小生成树的总权值是否相等。

判断次小生成树的方法：

kruskal O(n^3):先求一次最小生成树，然后枚举去掉生成树中的边，再求n-1次最小生成树，去最小的一棵。

prim O(n^2):先求一次最小生成树，记录树上的边，并且保存每个环中的最大边，然后用一次DP枚举掉树上的边，更新生成树的权和，取最小值。

/*
Author:Zhaofa Fang
Lang:C++
*/
#include <cstdio>
#include <cstdlib>
#include <sstream>
#include <iostream>
#include <cmath>
#include <cstring>
#include <algorithm>
#include <string>
#include <utility>
#include <vector>
#include <queue>
#include <stack>
#include <map>
#include <set>
using namespace std;

typedef long long ll;
#define DEBUG(x) cout<< #x << ':' << x << endl
#define REP(i,n) for(int i=0;i < (n);i++)
#define FOR(i,s,t) for(int i = (s);i <= (t);i++)
#define FORD(i,s,t) for(int i = (s);i >= (t);i--)
#define PII pair<int,int>
#define PB push_back
#define MP make_pair
#define ft first
#define sd second
#define lowbit(x) (x&(-x))
#define INF (1<<30)


const int maxn = 5050;
int pa[111],a[111];
struct Edge
{
    int u,v,w;
}edge[maxn];

bool cmp(Edge x,Edge y)
{
    return x.w < y.w;
}
void init(int n)
{
    FOR(i,1,n)pa[i] = i;
}
int findset(int x)
{
    return x == pa[x] ? x : pa[x] = findset(pa[x]);
}
int kruskal(int n,int m,int key)
{
    init(n);
    int ans = 0,cnt=0;
    REP(i,m)
    {
        if(i == key)continue;
        int x = findset(edge[i].u),y=findset(edge[i].v);
        if(x != y)
        {
            pa[x] = y;ans += edge[i].w;cnt++;
        }
        if(cnt == n-1)return ans;
    }
    return -INF;
}
int main()
{
    //freopen("in","r",stdin);
    int T;
    scanf("%d",&T);
    while(T--)
    {
        int n,m;
        scanf("%d%d",&n,&m);
        REP(i,m)scanf("%d%d%d",&edge[i].u,&edge[i].v,&edge[i].w);
        sort(edge,edge+m,cmp);
        int ans = 0,cnt=0;
        init(n);
        REP(i,m)//最小生成树
        {
            int x = findset(edge[i].u),y=findset(edge[i].v);
            if(x != y)
            {
                pa[x] = y;ans += edge[i].w;a[cnt++]=i;
            }
            if(cnt == n-1)break;
        }
        REP(i,n-1)
        {
            int tmp = kruskal(n,m,a[i]);
            if(ans == tmp)
            {
                ans = -INF;break;
            }
        }
        if(ans == -INF)puts("Not Unique!");
        else printf("%d\n",ans);
    }
    return 0;
}

/*
Author:Zhaofa Fang
Lang:C++
*/
#include <cstdio>
#include <cstdlib>
#include <sstream>
#include <iostream>
#include <cmath>
#include <cstring>
#include <algorithm>
#include <string>
#include <utility>
#include <vector>
#include <queue>
#include <stack>
#include <map>
#include <set>
using namespace std;

typedef long long ll;
#define DEBUG(x) cout<< #x << ':' << x << endl
#define REP(i,n) for(int i=0;i < (n);i++)
#define FOR(i,s,t) for(int i = (s);i <= (t);i++)
#define FORD(i,s,t) for(int i = (s);i >= (t);i--)
#define PII pair<int,int>
#define PB push_back
#define MP make_pair
#define ft first
#define sd second
#define lowbit(x) (x&(-x))
#define INF (1<<30)


const int maxn = 5050;
int maz[maxn][maxn],path[maxn][maxn];
bool vist[maxn];
int dist[maxn],pre[maxn],Stack[maxn];
int n,m;
void init(void){
    FOR(i,1,n)FOR(j,1,n)maz[i][j] = INF, path[i][j] = 0;
}
int prim(int s){
    FOR(i,1,n)dist[i] = maz[s][i],vist[i] = 0,pre[i] = s;
    int top =  0,sum = 0;
    vist[s] = 1,dist[s] = 0, Stack[top++] = s;
    while(1){
        int u = -1;
        FOR(i,1,n)
            if(!vist[i] && (u == - 1 || dist[u] > dist[i]))
                u = i;
        if(u == -1)break;
        sum += dist[u],Stack[top++] =  u,vist[u] = 1;
        REP(i,top){
            if(Stack[i] == u)continue;
            path[Stack[i]][u] = max(path[pre[u]][u],dist[u]);
            path[u][Stack[i]] = path[Stack[i]][u];
        }
        FOR(i,1,n)
            if(!vist[i] && (maz[u][i]<dist[i]))
                dist[i] = maz[u][i],pre[i] = u;
    }
    return sum;
}
void SMT(void){
    int Min = INF,sum = prim(1);
    FOR(i,1,n-1)
        FOR(j,i+1,n)
            if(pre[i] != j && pre[j] != i && maz[i][j] < INF)
                Min = min(Min, sum  - path[i][j] + maz[i][j]);
    if(Min == sum)puts("Not Unique!");
    else printf("%d\n",sum);
}
int main(){
    //freopen("in","r",stdin);
    int T;
    scanf("%d",&T);
    while(T--){
        scanf("%d%d",&n,&m);
        init();
        int u,v,w;
        while(m--)
        {
            scanf("%d%d%d",&u,&v,&w);
            maz[u][v] = maz[v][u] = w;
        }
        SMT();
    }
    return 0;
}