区间dp——异或推公式

For an array b of length m we define the function f

as
f(b)={b[1]f(b[1]⊕b[2],b[2]⊕b[3],…,b[m−1]⊕b[m])if m=1otherwise,

where ⊕

is bitwise exclusive OR.

For example, f(1,2,4,8)=f(1⊕2,2⊕4,4⊕8)=f(3,6,12)=f(3⊕6,6⊕12)=f(5,10)=f(5⊕10)=f(15)=15

You are given an array a
and a few queries. Each query is represented as two integers l and r. The answer is the maximum value of f on all continuous subsegments of the array al,al+1,…,ar

    .
Input

    The first line contains a single integer n(1≤n≤5000) — the length of a.

The second line contains n integers a1,a2,…,an (0≤ai≤230−1) — the elements of the array.

The third line contains a single integer q(1≤q≤100000) — the number of queries.

Each of the next q lines contains a query represented as two integers l, r (1≤l≤r≤n).
Output

    Print q lines — the answers for the queries.
Examples
    Input

    3
    8 4 1
    2
    2 3
    1 2

    Output

    5
    12

    Input

    6
    1 2 4 8 16 32
    4
    1 6
    2 5
    3 4
    1 2

    Output

    60
    30
    12
    3

Note

    In first sample in both queries the maximum value of the function is reached on the subsegment that is equal to the whole segment.

    In second sample, optimal segment for first query are [3,6], for second query — [2,5], for third — [3,4], for fourth — [1,2].

这道题就是推出来一段区间具体是怎么算出来的。

dp[l][r]=dp[l+1][r]^dp[l][r-1];

就是这个公式。当时比赛的时候公式推错了。

然后就很显然用区间dp来做了。

第二个代码是错的。因为我立即更新了。而事实上立即更新会让计算出来的i-j的值发生差异。

AC代码：

#include <bits/stdc++.h>
using namespace std;
const int maxn=5500;
int a[maxn];
int dp[maxn][maxn];
int sum[maxn];
int n;
int main()
{
    scanf("%d",&n);
    sum[0]=0;
    memset(dp,0,sizeof(dp));
    for(int i=1;i<=n;i++)
    {
        scanf("%d",&a[i]);
        dp[i][i]=a[i];
    }
    int i,j;
    for(int i=1;i<n;i++)
    {
        for(int l=1;l<=n-i;l++)
        {
            int r=l+i;
            dp[l][r]=dp[l+1][r]^dp[l][r-1];
        }
    }
    for(int i=1;i<n;i++)///区间长度
    {
        for(int l=1;l<=n-i;l++)///起点
        {
            int r=l+i;
            dp[l][r]=max(dp[l][r],max(dp[l+1][r],dp[l][r-1]));
           /// cout<<"l: "<<l<<"  r:"<<r<<"   "<<dp[l][r]<<endl;
        }
    }
    int m;
    int l,r;
    scanf("%d",&m);
    for(int i=1;i<=m;i++)
    {
        scanf("%d%d",&l,&r);
        cout<<dp[l][r]<<endl;
    }
    return 0;
}

错误代码：

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

long long dp[5010][5010];
int main()
{
    long long a[5005];
    int n;
    scanf("%d",&n);
    for(int i=1;i<=n;i++){
        scanf("%lld",&a[i]);
        dp[i][i]=a[i];
    }
    for(int i=1;i<n;i++){///区间长度
        for(int shou=1;shou+i<=n;shou++){
            int end=shou+i;
            ///dp[shou][end]=dp[shou+1][end]^dp[shou][end-1];
            dp[shou][end]=max(max(dp[shou+1][end],dp[shou][end-1]),dp[shou][end]);
            cout<<"shou:"<<shou<<"   wei:"<<end<<"   "<<dp[shou][end]<<endl;
        }
    }
    int m,l,r;
    scanf("%d",&m);
    for(int i=1;i<=m;i++){
        scanf("%d%d",&l,&r);
        printf("%lld\n",dp[l][r]);
    }
    return 0;
}

ps：自己太菜了，昨天比赛居然爆0了。更要拼命努力了