maximum clique 1

maximum clique 1

时间限制：C/C++ 1秒，其他语言2秒
空间限制：C/C++ 262144K，其他语言524288K
Special Judge, 64bit IO Format: %lld

题目描述

You are given a set S containing n distinct positive integers a1,a2,…,an.

Please find a subset of S which has the maximum size while satisfying the following constraint:

The binary representations of any two numbers in this subset must have at least two different bits.

If there are multiple valid subsets, please output any one of them.

输入描述:

The input contains only two lines.

The first line contains an integer N denoting the size of S.
The second line contains N positive integers a1,a2,…,aN​ denoting the members of set S.

* 1≤N≤5000

* 1≤ai≤1e9

* all ai are distinct

输出描述:

You should output two lines.

The first line contains one integer m denoting the size of the subset you found.

The second line contains m positive integers denoting the member of this subset.

输入

5
3 1 4 2 5

输出

3
4 1 2

说明

3 (11₂) and 1 (01₂) has only 1 different bit so they can not be in the set together. 4 (100₂) and 1 (001₂) has 2 different bits so they can be in the set together.

Following unordered pairs are allowed to be in the set together: <1, 2>, <1, 4>, <2, 4>, <2, 5>, <3, 4>, <3, 5>. Thus the maximum set is of size 3 ({1, 2, 4}).

链接：https://ac.nowcoder.com/acm/contest/885/F?&headNav=acm
来源：牛客网

---

题意：在一个集合中求一个最大的子集，使得子集里任意两个数之间至少有两个二进制位不同。

思路：两个相异的数至少两个bit不一样的否命题是：恰一个bit不一样。然后考虑到A和B一个bit不一样，A和C一个bit不一样，A，B，C各不相同，那么B和C一定不会只有一个bit不一样，所以若是两个数只有一个bit不一样，我们就在它们之间连一条边，可以发现这个图是个二分图。答案就是这个二分图的最大点独立集。

#include<bits/stdc++.h>
#define N 5050
using namespace std;

typedef struct
{
    int to,next;
    long long flow;
    bool is_match;
}ss;

ss edg[64*N];
int now_edge=0,s,t;
int head[N];

void addedge(int u,int v)
{
    edg[now_edge]=(ss){v,head[u],0};
    head[u]=now_edge++;
}

void addedge(int u,int v,long long flow)
{
   // printf("%d %d
",u,v);

    edg[now_edge]=(ss){v,head[u],flow};
    head[u]=now_edge++;

    edg[now_edge]=(ss){u,head[v],0};
    head[v]=now_edge++;
}

int dis[N];

bool bfs()
{
    memset(dis,0,sizeof(dis));
    queue<int>q;
    q.push(s);
    dis[s]=1;

    while(!q.empty())
    {
        int now=q.front();
        q.pop();
        for(int i=head[now];i!=-1;i=edg[i].next)
        {
            ss &e=edg[i];
            if(e.flow>0&&dis[e.to]==0)
            {
                dis[e.to]=dis[now]+1;
                q.push(e.to);
            }
        }
    }

    if(dis[t]==0)return 0;
    return 1;
}

int current[N];
long long dfs(int x,long long maxflow)
{
    if(x==t)return maxflow;
    for(int i=current[x];i!=-1;i=edg[i].next)
    {
        current[x]=i;
        ss &e=edg[i];
        if(e.flow>0&&dis[e.to]==dis[x]+1)
        {
            long long flow=dfs(e.to,min(maxflow,e.flow));
            if(flow!=0)
            {
                e.flow-=flow;
                edg[i^1].flow+=flow;
                return flow;
            }
        }
    }
    return 0;
}

long long dinic()//最大流
{
    long long ans=0,flow;
    while(bfs())
    {
        for(int i=0;i<N;i++)current[i]=head[i];
        while(flow=dfs(s,LLONG_MAX/2))ans+=flow;
    }
    return ans;
}

void init()
{
    for(int i=0;i<N;i++)head[i]=-1;
    now_edge=0;
}

int arr[N];
int color[N]={0};

void Bipartite_graph_coloring(int x,int now_color)//二分图染色
{
    if(color[x])return;
    color[x]=now_color;
    for(int i=head[x];i!=-1;i=edg[i].next)Bipartite_graph_coloring(edg[i].to,now_color*-1);
}

bool is_match_vertex[N]={0};
int is_ans[N]={0};
void dfs(int x,int type)//寻找最小点覆盖集中的点，不在该集合中的点就是最大点独立集的点
{
    if(is_ans[x])return;
    is_ans[x]=1;
    for(int i=head[x];i!=-1;i=edg[i].next)
    {
        int v=edg[i].to;
        if(v==s||v==t)continue;

        if(edg[i].is_match==(bool)type)dfs(v,1-type);
    }
}

int main()
{
    int n;
    scanf("%d",&n);
    for(int i=1;i<=n;i++)scanf("%d",&arr[i]);

    init();
    for(int i=1;i<=n;i++)
    {
        for(int j=i+1;j<=n;j++)
        {
            int now=arr[i]^arr[j];
            if(now==(now&-now))
            {
                addedge(i,j);
                addedge(j,i);
              //  printf("%d %d
",arr[i],arr[j]);
            }
        }
    }

    for(int i=1;i<=n;i++)
        if(!color[i])Bipartite_graph_coloring(i,1);//先将图划分成二分图

   /* for(int i=1;i<=n;i++)if(color[i]==1)printf("%d ",arr[i]);
    printf("
");
    for(int i=1;i<=n;i++)if(color[i]==-1)printf("%d ",arr[i]);
    printf("
");*/

    init();

    s=n+1,t=s+1;
    for(int i=1;i<=n;i++)
    {
        if(color[i]==1)addedge(s,i,1);
        else
            addedge(i,t,1);

        if(color[i]==1)
        {
            for(int j=1;j<=n;j++)
            {
                int now=arr[i]^arr[j];
                if(now==(now&-now))
                {
                    addedge(i,j,1);//建边准备跑二分图匹配
                }
            }
        }
    }
    int ans=n-dinic();
    printf("%d
",ans);

    for(int i=1;i<=n;i++)
    {
        if(color[i]==1)
        {
            for(int j=head[i];j!=-1;j=edg[j].next)
            {
                if(edg[j].to!=s&&edg[j^1].flow)
                {
                    edg[j].is_match=edg[j^1].is_match=true;
                    is_match_vertex[i]=is_match_vertex[edg[j].to]=true;
                }
                else
                {
                    edg[j].is_match=edg[j^1].is_match=false;
                }
            }
        }
    }

    for(int i=1;i<=n;i++)
    {
        if(color[i]==-1&&is_match_vertex[i]==false)
        {
            dfs(i,0);
        }
    }

    for(int i=1;i<=n;i++)
    {
        if(color[i]==1&&!is_ans[i]||color[i]==-1&&is_ans[i])printf("%d%c",arr[i],--ans==0 ? '
': ' ');
    }
    return 0;
}