Rhythm of Codes

[OpenJudge][DP]162 Post Office 题解

总时间限制:
1000ms
内存限制:
65536kB
描述
There is a straight highway with villages alongside the highway. The highway is represented as an integer axis, and the position of each village is identified with a single integer coordinate. There are no two villages in the same position. The distance between two positions is the absolute value of the difference of their integer coordinates.

Post offices will be built in some, but not necessarily all of the villages. A village and the post office in it have the same position. For building the post offices, their positions should be chosen so that the total sum of all distances between each village and its nearest post office is minimum.

You are to write a program which, given the positions of the villages and the number of post offices, computes the least possible sum of all distances between each village and its nearest post office.

输入
Your program is to read from standard input. The first line contains two integers: the first is the number of villages V, 1 <= V <= 300, and the second is the number of post offices P, 1 <= P <= 30, P <= V. The second line contains V integers in increasing order. These V integers are the positions of the villages. For each position X it holds that 1 <= X <= 10000.
输出
The first line contains one integer S, which is the sum of all distances between each village and its nearest post office.
样例输入

样例输出

来源
IOI 2000

题意:用数轴描述一条高速公路,有V个村庄,每一个村庄坐落在数轴的某个点上,需要选择P个村庄在其中建立邮局,要求每个村庄到最近邮局的距离和最小。

分析:典型的DP:

1、考虑在V个村庄中只建立一个邮局的情况,显然可以知道,将邮局建立在中间的那个村庄即可。也就是在a到b间建立

一个邮局,若使消耗最小,则应该将邮局建立在(a+b)/2这个村庄上。

2、下面考虑建立多个邮局的问题,可以这样将该问题拆分为若干子问题,在前i个村庄中建立j个邮局的最短距离,是

在前k个村庄中建立j-1个邮局的最短距离与 在k+1到第i个邮局建立一个邮局的最短距离的和。而建立一个邮局我们在

上面已经求出。

3、状态表示,由上面的讨论,可以开两个数组

f[i][j]:在前i个村庄中建立j个邮局的最小耗费

dst[i][j]:在第i个村庄到第j个村庄中建立1个邮局的最小耗费

那么就有转移方程:f[i][j] = min(f[i][j],f[k][j-1]+dst[k+1][i]) DP的边界状态即为f[i][1] = f[1][i];

所要求的结果即为f[n][m];

 

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