SRM691 Sunnygraphs2

Problem Statement

Hero has just constructed a very specific graph. He started with n isolated vertices, labeled 0 through n-1. For each vertex i Hero then chose a vertex a[i] (other than i) and he added an edge that connected i and a[i]. This way he created a graph with n vertices and n edges. Note that if a[x]=y and a[y]=x, the vertices x and y were connected by two different edges. Hero now wants to perform the following procedure:

1. Add a new isolated vertex number n.
2. Choose a subset M of the original vertices.
3. For each x in M, erase an edge between vertices x and a[x].
4. For each x in M, add a new edge between vertices x and n.

Hero‘s goal is to create a final graph in which the vertices 0 through n-1 are all in the same connected component. (I.e., there must be a way to reach any of these vertices from any other of them by following one or more consecutive edges, possibly visiting vertex n along the way.) Note that Hero does not care whether vertex n is in the same component as the other vertices: both possibilities are fine. In step 2 of the above procedure Hero has 2^n possible subsets to choose from. A choice of M is good if it produces a graph with the desired property. Count how many of the 2^n possibilities are good choices. Return that count as a long.

Definition

• ClassSunnygraphs2
• Methodcount
• Parametersvector<int>
• Returnslong long
• Method signaturelong long count(vector<int> a)

(be sure your method is public)

Limits

• Time limit (s)2.000
• Memory limit (MB)256

Constraints

• a will contain n elements.
• n will be between 2 and 50, inclusive.
• Each element in a will be between 0 and n - 1, inclusive.
• For each i between 0 and n - 1 holds a[i] != i.

Test cases

1. • a{ 1, 0 }

   Returns4

   The original graph contained the vertices 0 and 1. This pair of vertices was connected by two edges. Next, Hero added a new vertex 2. Then he had to choose one of four possible subsets M:

   • If he chose M = {}, the resulting graph contained the edges 0-1 and 0-1. The vertices 0 and 1 were in the same component.
   • If he chose M = {0}, the resulting graph contained the edges 0-1 and 0-2. The vertices 0 and 1 were in the same component.
   • If he chose M = {1}, the resulting graph contained the edges 0-1 and 1-2. The vertices 0 and 1 were in the same component.
   • Finally, if he chose M = {0, 1}, the resulting graph contained the edges 0-2 and 1-2. And again, the vertices 0 and 1 were in the same component. (In the resulting graph we can still go from vertex 0 to vertex 1, even though we have to go via vertex 2.)

   As all four choices of M are good, the correct answer is 4.

2.  
   • a{ 1, 0, 0 }

   Returns7

   Here, M = {2} is not a good choice. This choice produces a graph with edges 0-1, 0-1, and 2-3. In this graph vertex 2 is not in the same component as vertices 0 and 1. The other seven possible choices of M are all good.

3.  
   • a{ 2, 3, 0, 1 }

   Returns9

4.  
   • a{ 2, 3, 0, 1, 0 }

   Returns18

5.  
   • a{ 2, 3, 0, 1, 0, 4, 5, 2, 3 }

   Returns288

6.  
   • a{ 29, 34, 40, 17, 16, 12, 0, 40, 20, 35, 5, 13, 27, 7, 29, 13, 14, 39, 42, 9, 30, 38, 27, 40, 34, 33, 42, 20, 29, 42, 12, 29, 30, 21, 4, 5, 7, 25, 24, 17, 39, 32, 9 }

   Returns6184752906240

   "Watch out for integer overflow."

7.  
   • a{ 9, 2, 0, 43, 12, 14, 39, 25, 24, 3, 16, 17, 22, 0, 6, 21, 18, 29, 34, 35, 23, 43, 28, 28, 20, 11, 5, 12, 31, 24, 8, 13, 17, 10, 15, 9, 15, 26, 4, 13, 21, 27, 36, 39 }

   Returns17317308137473

题意。。看了很久。

其实就是用一个点n来连接其他联通分量，使得标号0~n-1这些点再一个联通分量中。

为了简化题目，假设原图中联通分量>=2，我们可以先找出原图中的环，因为按照题目规则，只有两个环分别拓展出一条边连接点n，使得所有联通分量和为1个联通分量。

下面解释一下样例3.

如图0，2是一个环   1，3是一个环。现在借助5把{4，0，2}和{1，3}这两个联通分量连接起来。先不考虑点4.那么{0，2}中有3种方案可以选择，{1，3}中有3中方案可以选则所以一共有9中方案。接下来考虑点4，那么挂上点4后，答案也是9.  所以种答案是18.

如果0~n-1这些点原来在一个联通分量中。那么 我们还要加上空集这种情况。

代码：

#include <cstdio>
#include <cmath>
#include <cstring>
#include <ctime>
#include <iostream>
#include <algorithm>
#include <set>
#include <vector>
#include <sstream>
#include <typeinfo>
#include <fstream>
#define ll long long
using namespace std;

class Sunnygraphs2 {
    public:
    int vis[60]={0};
    int used[60]={0};
    vector<int>edge[60];
    int num=0;
    void dfs(int u){
        num++;used[u]=1;
        for(int i:edge[u])if(!used[i])dfs(i);
    }
    long long count(vector<int> a) {
        int n=a.size();
        int m=n;
        for(int i=0;i<n;i++){
            edge[i].push_back(a[i]);
            edge[a[i]].push_back(i);
        }
        dfs(0);
        ll ans=1;
        int cnt=0,mark=0,cur;
        for(int i=0;i<n;i++){//找环
            if(!vis[i]){
                cur=i;cnt=mark=0;
                for(int j=1;j<=50;j++){
                    cur=a[cur];cnt++;//记录环中节点数
                    if(cur==i){
                        mark=1;break;
                    }
                }
                if(mark){
                    cur=i;
                    for(int j=1;j<=50;j++){
                        vis[cur]=1;
                        cur=a[cur];
                    }
                    ans*=(ll)pow(2ll,cnt)-1;
                    m-=cnt;
                }
            }
        }
        ans*=(ll)pow(2ll,m);
        if(num==n)ans++;//包含空集
        return ans;
    }
};