HDOJ 3516 Tree Construction 四边形优化dp

原题链接：http://acm.hdu.edu.cn/showproblem.php?pid=3516

题意：

大概就是给你个下凸包的左侧，然后让你用平行于坐标轴的线段构造一棵树，并且这棵树的总曼哈顿距离最短

题解：

很容易得到转移方程：

$$dp[i][j]=min { dp[i][k-1]+dp[k][j] + dis(uni(i,k-1),uni(k,j))}$$

其中$dp[i][j]$表示从$i$到$j$的最优解，$dis(i,j)$表示$i$和$j$之间的曼哈顿距离，$uni(i,j)$表示将$i$和$j$用平行于坐标轴的线段连在一起时的拐角点。

然后可以证明这是满足四边形优化条件的。

然后四边形优化一下就好。

代码：

#include<iostream>
#include<cstring>
#include<queue>
#include<vector>
#include<algorithm>
#include<climits>
#include<cmath>
#define MAX_N 1234
#define INF LLONG_MAX/1234
using namespace std;

typedef long long LL;

LL dp[MAX_N][MAX_N];

int n;

struct point {
public:
    LL x, y;

    point(LL xx, LL yy) : x(xx), y(yy) { }

    point() { }
};

point P[MAX_N];

LL dis(point a,point b){
    return abs(a.x-b.x)+abs(a.y-b.y);
}

point uni(point a,point b) {
    return point(min(a.x, b.x), min(a.y, b.y));
}

int s[MAX_N][MAX_N];

int main() {
    cin.sync_with_stdio(false);
    while (cin >> n) {
        for (int i = 1; i <= n; i++)
            cin >> P[i].x >> P[i].y;
        for (int i = 0; i <= n; i++)
            for (int j = 0; j <= n; j++) {
                if (i == j)
                    dp[i][j] = 0;
                else
                    dp[i][j] = INF;
            }
        for (int i = 1; i < n; i++)
            dp[i][i + 1] = dis(P[i], P[i + 1]), s[i][i + 1] = i + 1;
        for (int t = 2; t <= n; t++)
            for (int i = 1; i + t <= n; i++) {
                int j = i + t;
                int pos = 0;
                for (int k = s[i][j - 1]; k <= s[i + 1][j]; k++)
                    if (dp[i][j] > dp[i][k - 1] + dp[k][j]+ dis(uni(P[i], P[k - 1]), uni(P[k], P[j])))
                        dp[i][j] = dp[i][k - 1] + dp[k][j] + dis(uni(P[i], P[k - 1]), uni(P[k], P[j])), pos = k;
                s[i][j] = pos;
            }
        cout << dp[1][n] << endl;
    }
    return 0;
}