CodeForces 665D Simple Subset

题意:给你n个数,你需要找到一个最大的子集,使得这个子集中的任何两个数加起来都是质数。

思路:建边的话,跑dfs一波最大团


#include<bits/stdc++.h>
using namespace std;
const int maxn = 1005;
int pri[2000005];
bool flag[maxn], a[maxn][maxn];
int ans, cnt[maxn], group[maxn], n, vis[maxn];
// 最大团: V中取K个顶点,两点间相互连接
// 最大独立集: V中取K个顶点,两点间不连接
// 最大团数量 = 补图中最大独立集数

bool dfs( int u, int pos ){
    int i, j;
    for( i = u+1; i <= n; i++){
        if( cnt[i]+pos <= ans ) return 0;
        if( a[u][i] ){
             // 与目前团中元素比较,取 Non-N(i)
            for( j = 0; j < pos; j++ ) if( !a[i][ vis[j] ] ) break;
            if( j == pos ){     // 若为空,则皆与 i 相邻,则此时将i加入到 最大团中
                vis[pos] = i;
                if( dfs( i, pos+1 ) ) return 1;
            }
        }
    }
    if( pos > ans ){
            for( i = 0; i < pos; i++ )
                group[i] = vis[i]; // 最大团 元素
            ans = pos;
            return 1;
    }
    return 0;
}
void maxclique()
{
    ans=-1;
    for(int i=n;i>0;i--)
    {
        vis[0]=i;
        dfs(i,1);
        cnt[i]=ans;
    }
}
void pre()
{
    pri[1]=1;
    pri[0]=1;
    for(int i=2;i<2000005;i++)
    {
        if(pri[i])continue;
        for(int j=i+i;j<2000005;j+=i)
            pri[j]=1;
    }
}
int aa[maxn];
int main()
{
    pre();
    scanf("%d",&n);
    for(int i=1;i<=n;i++)scanf("%d",&aa[i]);
    for(int i=1;i<=n;i++)
        for(int j=1;j<i;j++)
            if(!pri[aa[i]+aa[j]])
                a[i][j]=1,a[j][i]=1;
    maxclique();
    cout<<ans<<endl;
    for(int i=0;i<ans;i++)
        cout<<aa[group[i]]<<" ";
    cout<<endl;
}


Description

A tuple of positive integers {x1, x2, ..., xk} is called simple if for all pairs of positive integers (i,  j) (1  ≤ i  <  j ≤ k), xi  +  xj is a prime.

You are given an array a with n positive integers a1,  a2,  ...,  an (not necessary distinct). You want to find a simple subset of the arraya with the maximum size.

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Let's define a subset of the array a as a tuple that can be obtained from a by removing some (possibly all) elements of it.

Input

The first line contains integer n (1 ≤ n ≤ 1000) — the number of integers in the array a.

The second line contains n integers ai (1 ≤ ai ≤ 106) — the elements of the array a.

Output

On the first line print integer m — the maximum possible size of simple subset of a.

On the second line print m integers bl — the elements of the simple subset of the array a with the maximum size.

If there is more than one solution you can print any of them. You can print the elements of the subset in any order.

Sample Input

Input
2
2 3
Output
2
3 2
Input
2
2 2
Output
1
2
Input
3
2 1 1
Output
3
1 1 2
Input
2
83 14
Output
2
14 83



### Codeforces 1487D Problem Solution The problem described involves determining the maximum amount of a product that can be created from given quantities of ingredients under an idealized production process. For this specific case on Codeforces with problem number 1487D, while direct details about this exact question are not provided here, similar problems often involve resource allocation or limiting reagent type calculations. For instance, when faced with such constraints-based questions where multiple resources contribute to producing one unit of output but at different ratios, finding the bottleneck becomes crucial. In another context related to crafting items using various materials, it was determined that the formula `min(a[0],a[1],a[2]/2,a[3]/7,a[4]/4)` could represent how these limits interact[^1]. However, applying this directly without knowing specifics like what each array element represents in relation to the actual requirements for creating "philosophical stones" as mentioned would require adjustments based upon the precise conditions outlined within 1487D itself. To solve or discuss solutions effectively regarding Codeforces' challenge numbered 1487D: - Carefully read through all aspects presented by the contest organizers. - Identify which ingredient or component acts as the primary constraint towards achieving full capacity utilization. - Implement logic reflecting those relationships accurately; typically involving loops, conditionals, and possibly dynamic programming depending on complexity level required beyond simple minimum value determination across adjusted inputs. ```cpp #include <iostream> #include <vector> using namespace std; int main() { int n; cin >> n; vector<long long> a(n); for(int i=0;i<n;++i){ cin>>a[i]; } // Assuming indices correspond appropriately per problem statement's ratio requirement cout << min({a[0], a[1], a[2]/2LL, a[3]/7LL, a[4]/4LL}) << endl; } ``` --related questions-- 1. How does identifying bottlenecks help optimize algorithms solving constrained optimization problems? 2. What strategies should contestants adopt when translating mathematical formulas into code during competitive coding events? 3. Can you explain why understanding input-output relations is critical before implementing any algorithmic approach? 4. In what ways do prefix-suffix-middle frameworks enhance model training efficiency outside of just tokenization improvements? 5. Why might adjusting sample proportions specifically benefit models designed for tasks requiring both strong linguistic comprehension alongside logical reasoning skills?
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