kuangbin专题十二 POJ3186 Treats for the Cows (区间dp)
Time Limit: 1000MS | Memory Limit: 65536K | |
Total Submissions: 7949 | Accepted: 4217 |
Description
The treats are interesting for many reasons:
- The treats are numbered 1..N and stored sequentially in single file in a long box that is open at both ends. On any day, FJ can retrieve one treat from either end of his stash of treats.
- Like fine wines and delicious cheeses, the treats improve with age and command greater prices.
- The treats are not uniform: some are better and have higher intrinsic value. Treat i has value v(i) (1 <= v(i) <= 1000).
- Cows pay more for treats that have aged longer: a cow will pay v(i)*a for a treat of age a.
Given the values v(i) of each of the treats lined up in order of the index i in their box, what is the greatest value FJ can receive for them if he orders their sale optimally?
The first treat is sold on day 1 and has age a=1. Each subsequent day increases the age by 1.
Input
Lines 2..N+1: Line i+1 contains the value of treat v(i)
Output
Sample Input
5
1
3
1
5
2
Sample Output
43
Hint
Five treats. On the first day FJ can sell either treat #1 (value 1) or treat #5 (value 2).
FJ sells the treats (values 1, 3, 1, 5, 2) in the following order of indices: 1, 5, 2, 3, 4, making 1x1 + 2x2 + 3x3 + 4x1 + 5x5 = 43.
题目大意:给一个长度为n的序列,每次只能从队首或队尾取一个数,第几次取 * f[i] 就是利润,求最大利润。
看到题目果断贪心,只能局部最优,因为是dp专题,但是丝毫不会dp,看了题解发现是区间dp,然后看着理解了一下,
dp[i][j] 表示 从第i个数到第j个数的最大利润,由于只能从dp[i+1][j] 和 dp[i][j-1]到达dp[i][j],所以状态转移方程可以表示为 dp[i][j] = max(dp[i+1][j] + f[i] * (n-j+i), dp[i][j-1] + f[j] * (n-j+i),
这里是由里向外递推的,逆向遍历i。用n-j+i表示第几次取(可以模拟一个简单的看看)
初始化条件要注意一下,dp[i][i] = f[i] * n 只有一个数时,它就是最后取的。
#include <iostream>
#include <stdio.h>
#include <math.h>
#include <string.h>
#include <stdlib.h>
#include <string>
#include <vector>
#include <set>
#include <map>
#include <queue>
#include <algorithm>
#include <sstream>
#include <stack>
using namespace std;
#define mem(a,b) memset((a),(b),sizeof(a))
#define mp make_pair
#define pb push_back
#define fi first
#define se second
#define sz(x) (int)x.size()
#define all(x) x.begin(),x.end()
typedef long long ll;
const int inf = 0x3f3f3f3f;
const ll INF =0x3f3f3f3f3f3f3f3f;
const double pi = acos(-1.0);
const double eps = 1e-;
const ll mod = 1e9+;
//head
const int maxn = + ;
int dp[maxn][maxn], f[maxn];//dp[i][j] 表示从i到j的 最大值 int main() {
int n;
while(~scanf("%d", &n)) {
for(int i = ; i <= n; i++)
scanf("%d", &f[i]);
mem(dp, );
for(int i = ; i <= n; i++)//初始化 每一个都是最后取的
dp[i][i] = f[i] * n;
for(int i = n - ; i >= ; i--) {//从里往外推
for(int j = i + ; j <= n; j++) {// n - j + i 很巧妙
dp[i][j] = max(dp[i+][j] + f[i] * (n - j + i), dp[i][j-] + f[j] * (n - j + i));//转移方程
}
}
printf("%d\n", dp[][n]);
}
}
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