CodeForce 981 D. Bookshelves(贪心+dp)

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本文介绍了一个关于书架分配的问题，需要将一定数量的书籍分配到多个书架上以最大化所有书架的美观度，美观度定义为各书架上书籍价格之和的按位与结果。文章提供了一种高效的解决方案，通过从高位到低位考虑来确定最优解。

Mr Keks is a typical white-collar in Byteland.

He has a bookshelf in his office with some books on it, each book has an integer positive price.

Mr Keks defines the value of a shelf as the sum of books prices on it.

Miraculously, Mr Keks was promoted and now he is moving into a new office.

He learned that in the new office he will have not a single bookshelf, but exactly $k$ bookshelves. He decided that the beauty of the $k$ shelves is the bitwise AND of the values of all the shelves.

He also decided that he won't spend time on reordering the books, so he will place several first books on the first shelf, several next books on the next shelf and so on. Of course, he will place at least one book on each shelf. This way he will put all his books on $k$ shelves in such a way that the beauty of the shelves is as large as possible. Compute this maximum possible beauty.

Input

The first line contains two integers $n$ and $k$ ( $1 \leq k \leq n \leq 50$ ) — the number of books and the number of shelves in the new office.

The second line contains $n$ integers $a_{1}, a_{2}, \dots a_{n}$ , ( $0 < a_{i} < 2^{50}$ ) — the prices of the books in the order they stand on the old shelf.

Output

Print the maximum possible beauty of $k$ shelves in the new office.

  Examples
 
    input
    
     Copy
    
10 4
9 14 28 1 7 13 15 29 2 31

    output
    
     Copy
    
24

    input
    
     Copy
    
7 3
3 14 15 92 65 35 89

    output
    
     Copy
    
64

题解：
从高位到低位考虑，若高位满足条件则先满足高位。
#include<bits/stdc++.h>
using namespace std;
const int maxn=55;
#define ll long long
ll a[maxn],sum[maxn];
bool dp[maxn][maxn];
int n,K;
bool check(ll x)//判断x是否满足条件
{
    memset(dp,0,sizeof(dp));
    dp[0][0]=1;
    for(int i=1;i<=K;i++)
    {
        for(int j=1;j<=n;j++)
        {
            for(int k=1;k<=j;k++)
            {
                if((((sum[j]-sum[k-1])&x)==x)&&dp[i-1][k-1])//若满足条件则每个区间和&x都等于x
                    dp[i][j]=1;
            }
        }
    }
    return dp[K][n];
}
int main()
{
    scanf("%d%d",&n,&K);
    for(int i=1;i<=n;i++)
    {
        scanf("%lld",&a[i]);
        sum[i]=sum[i-1]+a[i];
    }
    ll ans=0;
    for(ll i=62;i>=0;i--)
        if(check(ans|(1LL<<i))) ans=ans|(1LL<<i);
    printf("%lld\n",ans);
    return 0;
}