Tree Cutting 题解

最新推荐文章于 2022-10-09 23:28:27 发布

原创最新推荐文章于 2022-10-09 23:28:27 发布 · 453 阅读

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本文介绍了一个有趣的问题，即如何通过移除特定节点来确保网络分裂后的各部分规模不超过原网络的一半。文章详细阐述了解决方案，包括使用邻接表表示网络结构、采用深度优先搜索算法确定关键节点等。

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After Farmer John realized that Bessie had installed a "tree-shaped" network among his N (1 <= N <= 10,000) barns at an incredible cost, he sued Bessie to mitigate his losses.
Bessie, feeling vindictive, decided to sabotage Farmer John's network by cutting power to one of the barns (thereby disrupting all the connections involving that barn). When Bessie does this, it breaks the network into smaller pieces, each of which retains full connectivity within itself. In order to be as disruptive as possible, Bessie wants to make sure that each of these pieces connects together no more than half the barns on FJ.
Please help Bessie determine all of the barns that would be suitable to disconnect.

Input

* Line 1: A single integer, N. The barns are numbered 1..N.
* Lines 2..N: Each line contains two integers X and Y and represents a connection between barns X and Y.

Output

* Lines 1..?: Each line contains a single integer, the number (from 1..N) of a barn whose removal splits the network into pieces each having at most half the original number of barns. Output the barns in increasing numerical order. If there are no suitable barns, the output should be a single line containing the word "NONE".

Sample Input

Sample Output

3
8

Hint

INPUT DETAILS:
The set of connections in the input describes a "tree": it connects all the barns together and contains no cycles.

OUTPUT DETAILS:
If barn 3 or barn 8 is removed, then the remaining network will have one piece consisting of 5 barns and two pieces containing 2 barns. If any other barn is removed then at least one of the remaining pieces has size at least 6 (which is more than half of the original number of barns, 5).

题意：给了一棵节点数为n的树的对应关系，让判断去掉哪些节点能使子数的大小小于等于n/2，并将符合条件的节点从小到大输出，如果没有符合条件的节点，则输出 “NONE”。

思路：既然是让求哪些ji节点符合条件，可以用邻接表构建一个图，然后用dfs找出每一个节点下连的图的da'x大小些节点符合条件，将节点存入ans数组中。

代码如下：

#include<cstdio>
#include<cstring>
#include<algorithm>
#define N 10010
using namespace std;
int n,s,e,first[N],book[N],ans[N];
struct node
{
    int x,y;
}que[2*N];
void add(int x,int y) //构建邻接表的函数
{
    que[e].x=x;
    que[e].y=first[y];
    first[y]=e++;
}
int dfs(int x)
{

    int num=0,sum=0;
    book[x]=1;
    int k=first[x],flag=0;
    while(k!=-1)
    {
        if(!book[que[k].x])
        {
            num=dfs(que[k].x);
            sum+=num;
            if(num>n/2)   //x节点下的图的大小超过了n/2，不符合条件
                flag=1;
        }
        k=que[k].y;
    }
    if(n-sum-1>n/2)    //另一半是否小于等于n/2
        flag=1;
    if(!flag)
        ans[s++]=x;
    return sum+1;     //加上自己
}
int main()
{
    while(~scanf("%d",&n))
    {
        memset(first,-1,sizeof(first));//初始化
        memset(book,0,sizeof(book));
        s=e=0;   //有s个节点符合条件，e用邻接表表示无向图的大小
        int a,b;
        for(int i=1;i<n;i++)
        {
            scanf("%d%d",&a,&b);
            add(a,b);       //构建邻接表
            add(b,a);       //无向图
        }
        a=dfs(1);        //从第一个点开始尝试
        if(s)      //有s个节点符合条件
        {
            sort(ans,ans+s);    //从小到大排序，输出
            for(int i=0;i<s;i++)
                printf("%d\n",ans[i]);
        }
        else    //没有符合条件的节点
            printf("NONE\n");
    }
}