poj3666 Making the Grade（基础dp + 离散化）

Description

A straight dirt road connects two fields on FJ’s farm, but it changes elevation more than FJ would like. His cows do not mind climbing up or down a single slope, but they are not fond of an alternating succession of hills and valleys. FJ would like to add and remove dirt from the road so that it becomes one monotonic slope (either sloping up or down).

You are given N integers A1, … , AN (1 ≤ N ≤ 2,000) describing the elevation (0 ≤ Ai ≤ 1,000,000,000) at each of N equally-spaced positions along the road, starting at the first field and ending at the other. FJ would like to adjust these elevations to a new sequence B1, . … , BN that is either nonincreasing or nondecreasing. Since it costs the same amount of money to add or remove dirt at any position along the road, the total cost of modifying the road is

|A1 – B1| + |A2 – B2| + … + |AN – BN |
Please compute the minimum cost of grading his road so it becomes a continuous slope. FJ happily informs you that signed 32-bit integers can certainly be used to compute the answer.

Input

* Line 1: A single integer: N
* Lines 2..N+1: Line i+1 contains a single integer elevation: Ai

Output

* Line 1: A single integer that is the minimum cost for FJ to grade his dirt road so it becomes nonincreasing or nondecreasing in elevation.

题意：给你一个序列，求最少的代价让其变成单调不递增序列或单调不递减序列。

看到这道题目应该会想到用dp处理，dp[i][max]表示处理到前i位的最大值位max，于是只要两个for就可以，i=1～n，j=1～max。
dp[i][j]=min(dp[i][1~j]) + abs(a[i]-j)，然后再从dp[n][1~max]中找最小的。
这题数据有点大如果用max会TLE，所以要将数据离散化一下。

#include <iostream>
#include <cstring>
#include <algorithm>
#include <cmath>
#define Abs(a) ((a)>0?(a):-(a))
using namespace std;
typedef long long ll;
const int M = 2e3 + 10;
ll dp[M][M];
ll a[M] , b[M] , c[M];
int main()
{
    int n;
    cin >> n;
    for(int i = 1 ; i <= n ; i++) {
        cin >> a[i];
        c[n + 1 - i] = a[i];
        b[i] = a[i];
    }
    sort(b + 1 , b + n + 1);
    for(int i = 1 ; i <= n ; i++) {
        ll MIN = dp[i - 1][1];
        for(int j = 1 ; j <= n ; j++) {
            MIN = min(MIN , dp[i - 1][j]);
            dp[i][j] = MIN + Abs((a[i] - b[j]));
        }
    }
    ll ans = dp[n][1];
    for(int i = 1 ; i <= n ; i++) {
        ans = min(ans , dp[n][i]);
    }
    memset(dp , 0 , sizeof(dp));
    for(int i = 1 ; i <= n ; i++) {
        ll MIN = dp[i - 1][1];
        for(int j = 1 ; j <= n ; j++) {
            MIN = min(MIN , dp[i - 1][j]);
            dp[i][j] = MIN + Abs((c[i] - b[i]));
        }
    }
    for(int i = 1 ; i <= n ; i++) {
        ans = min(ans , dp[n][i]);
    }
    cout << ans << endl;
    return 0;
}