UVA 10891 Game of Sum

題目鏈接：http://acm.hust.edu.cn/vjudge/problem/19461

題意：一個長度爲n的序列，兩個人輪流取數。每次玩家可以從左端或者右端取任意數量個數，但不能兩端都取。兩人都希望自己的得分儘量大。當所有數都取完後，求先手取數之和減去後手取數之和。

思路：dp[l][r]表示取區間[l,r]內的數時，先手之和減去後手之和的最大值。dp[l][r] = max( sum[l,k] - dp[k+1][r] , sum[k,r] - dp[l][k-1] )分別枚舉先手從左/右端取走的個數，取最大值，記憶化搜索計算答案。

還有一種方法是參考書上的遞推做法,dp[l][r]表示先手取區間[l,r]內的數獲得的最大值。dp[l][r] = sum[i,j] - min( dp[l+1][r] , dp[l+2][r] , dp[l+3][r]...dp[r][r] (取左端) ,   dp[l][l] , dp[l][l+1] , dp[l][l+2] , dp[l][l+3]....dp[l][r-1] (取右端)  , 0(全部取完)     );

f[l][r] = min( dp[l+1][r] , dp[l+2][r] , dp[l+3][r]...dp[r][r]  )

g[l][r] = min( dp[l][l] , dp[l][l+1] , dp[l][l+2] , dp[l][l+3]....dp[l][r-1] )

dp[l][r] = sum - min( 0 , min(f[l][r] , g[l][r]) )

f和g存在遞推式

#include <cstdio>
#include <cmath>
#include <cstring>
#include <string>
#include <cstdlib>
#include <iostream>
#include <algorithm>
#include <stack>
#include <map>
#include <set>
#include <vector>
#include <sstream>
#include <queue>
#include <utility>
using namespace std;

#define rep(i,j,k) for (int i=j;i<=k;i++)
#define Rrep(i,j,k) for (int i=j;i>=k;i--)

#define Clean(x,y) memset(x,y,sizeof(x))
#define LL long long
#define ULL unsigned long long
#define inf 0x7fffffff
#define mod 100000007

const int maxn = 109;
int n;
LL sum[maxn];
int a[maxn];
LL dp[maxn][maxn];
bool flag[maxn][maxn];

void init()
{
    sum[0] = 0;
    rep(i,1,n)
    {
        scanf("%d",&a[i]);
        sum[i] = sum[i-1] + a[i];
    }
    Clean(flag,false);
}

LL dfs( int l , int r )
{
    if ( flag[l][r] ) return dp[l][r];
    if ( l == r ) return a[l];
    if ( l > r ) return 0;
    LL ans = sum[r] - sum[l-1];
    rep(i,l,r-1)
    {
        ans = max( ans , sum[i]-sum[l-1] - dfs(i+1,r) );
        ans = max( ans , sum[r] - sum[i] - dfs(l,i) );
    }
    flag[l][r] = true;
    return dp[l][r] = ans;
}

int main()
{
    while( cin>>n )
    {
        if ( !n ) break;
        init();
        cout<<dfs(1,n)<<endl;
    }
    return 0;
}

#include <cstdio>
#include <cmath>
#include <cstring>
#include <string>
#include <cstdlib>
#include <iostream>
#include <algorithm>
#include <stack>
#include <map>
#include <set>
#include <vector>
#include <sstream>
#include <queue>
#include <utility>
using namespace std;

#define rep(i,j,k) for (int i=j;i<=k;i++)
#define Rrep(i,j,k) for (int i=j;i>=k;i--)

#define Clean(x,y) memset(x,y,sizeof(x))
#define LL long long
#define ULL unsigned long long
#define inf 0x7fffffff
#define mod 100000007

#define mp make_pair
#define fi first
#define se second

#define lson l,m,rt<<1
#define rson m+1,r,rt<<1|1
#define mid int m=(l+r)>>1
const int maxn = 109;
LL a[maxn];
LL sum[maxn];
LL dp[maxn][maxn]; //先手取最大值
LL f[maxn][maxn];
LL g[maxn][maxn];
int n;
void init()
{
    sum[0] = 0;
    rep(i,1,n)
    {
        cin>>a[i];
        sum[i] = sum[i-1] + a[i];
        dp[i][i] = a[i];
        f[i][i] = g[i][i] = inf;
    }
    rep(L,2,n)
    {
        rep(i,1,n-L+1)
        {
            int j = i + L - 1;
            f[i][j] = min( f[i+1][j] , dp[i+1][j] );
            g[i][j] = min( g[i][j-1] , dp[i][j-1] );
            dp[i][j] = sum[j] - sum[i-1] - min( 0LL , min( f[i][j] , g[i][j] ) );
        }
    }
    cout<<2*dp[1][n] - sum[n]<<endl;
}

int main()
{
    while(cin>>n)
    {
        if ( !n ) break;
        init();
    }
    return 0;
}