HDOJ1300 Pearls[DP（先证明出相互代替不会跳跃）]

Pearls

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 65536/32768 K (Java/Others)
Total Submission(s): 767    Accepted Submission(s): 325

Problem Description

In Pearlania everybody is fond of pearls. One company, called The Royal Pearl, produces a lot of jewelry with pearls in it. The Royal Pearl has its name because it delivers to the royal family of Pearlania. But it also produces bracelets and necklaces for ordinary people. Of course the quality of the pearls for these people is much lower then the quality of pearls for the royal family. In Pearlania pearls are separated into 100 different quality classes. A quality class is identified by the price for one single pearl in that quality class. This price is unique for that quality class and the price is always higher then the price for a pearl in a lower quality class.

Every month the stock manager of The Royal Pearl prepares a list with the number of pearls needed in each quality class. The pearls are bought on the local pearl market. Each quality class has its own price per pearl, but for every complete deal in a certain quality class one has to pay an extra amount of money equal to ten pearls in that class. This is to prevent tourists from buying just one pearl.

Also The Royal Pearl is suffering from the slow-down of the global economy. Therefore the company needs to be more efficient. The CFO (chief financial officer) has discovered that he can sometimes save money by buying pearls in a higher quality class than is actually needed. No customer will blame The Royal Pearl for putting better pearls in the bracelets, as long as the prices remain the same.

For example 5 pearls are needed in the 10 Euro category and 100 pearls are needed in the 20 Euro category. That will normally cost: (5+10)*10 + (100+10)*20 = 2350 Euro.

Buying all 105 pearls in the 20 Euro category only costs: (5+100+10)*20 = 2300 Euro.

The problem is that it requires a lot of computing work before the CFO knows how many pearls can best be bought in a higher quality class. You are asked to help The Royal Pearl with a computer program.

Given a list with the number of pearls and the price per pearl in different quality classes, give the lowest possible price needed to buy everything on the list. Pearls can be bought in the requested, or in a higher quality class, but not in a lower one.

 

Input

The first line of the input contains the number of test cases. Each test case starts with a line containing the number of categories c (1 <= c <= 100). Then, c lines follow, each with two numbers ai and pi. The first of these numbers is the number of pearls ai needed in a class (1 <= ai <= 1000). The second number is the price per pearl pi in that class (1 <= pi <= 1000). The qualities of the classes (and so the prices) are given in ascending order. All numbers in the input are integers.

 

Output

For each test case a single line containing a single number: the lowest possible price needed to buy everything on the list.

 

Sample Input

2 2 100 1 100 2 3 1 10 1 11 100 12

 

Sample Output

330 1344

 

Source

Northwestern Europe 2002

 

Recommend

Eddy

 

 

 

 

 

 

 

 

转自：http://blog.sina.com.cn/s/blog_63509b890100nvqx.html

题意解释：

有n个等级的珠宝，等级依次升高，等级越高价钱越高，每买一个等级的任何数量的珠宝必须多付10颗此种珠宝的价钱，可以用高等级的珠宝代替低等级的，问要买到若干规定的数量和等级珠宝的最少花费。例如买5颗价值为10的、100颗价值为20的珠宝，有两种方案：一种为分别买两种等级的珠宝价钱为(5+10)*10+(100+10)*20 = 2350；另一种是将等级低的（即价格低）的珠宝全部换为等级高的，此时价钱为(5+100+10)*20 = 2300，故第二种方案较优。

选题原因：

此题是比较典型的动态规划在实际问题中应用的问题，难度适中，特征为通过组合子问题的解来解决整个问题，且子问题不是互相独立，即各子问题包含公共的子子问题，这也是所有动态规划问题的共同特点。（摘自《算法导论》）

解题思路：

首先需要证明一点，用高一等级的代替低一等级的必须是连续代替。例如 3 5，90 7，100 12要想代替3个5价值的首先考虑用7价值的代替，因为3*7=21 （3+10）*5=65 显然比较合算，这样就不需要考虑用12价值的那个代替了，因为前面已经有了最优解，也就是说相互代替不会跳跃。

证明这一点后开始分析，用sum[i]表示前i中珠宝总数，对于第i等级珠宝，ans[i]表示其最优解，那么ans[i]可能是(sum[i]+10) *p[i] 即前面所有的珠宝用i的价值去买，或者(sum[i]-sum[1]+10)*p[i]+ans[1] 即从第二个开始前面的所有珠宝用i的价值去买，或者（sum[i]-sum[2] +10）*p[i] + ans[2] 即从第三个。。。依次计算，其中最小的一个就是i的解。注意，上面提到的跳跃是指不用i+1等级珠价值去计算i等级以及其以下的珠宝，但可以将i以下等级的珠宝归为等级为i的价格进行计算，这样就会保证当前价值最小。这样状态方程便有ans[i]=min(ans[i],(sum[i]-sum[j]+10)*p[i]+ans[j])，（1<=j<=i）。

在写状态方程时是整体到局部的思路去考虑，具体写代码特别是边界条件时则需要自底下向上推导。通常需要将状态数组（此题为ans[]）分开处理，即先对其边界赋值，处理特殊情况，再按状态方程进行计算。

 

 

 

 

 

 

PS：

比赛的时候没有做出来，后来去网上找了解题报告，发现这道题目的思想值得我去好好学习，DP弱爆了。

code:

#include <iostream>   
#include <iomanip>   
#include <fstream>   
#include <sstream>   
#include <algorithm>   
#include <string>   
#include <set>   
#include <utility>   
#include <queue>   
#include <stack>   
#include <list>   
#include <vector>   
#include <cstdio>   
#include <cstdlib>   
#include <cstring>   
#include <cmath>   
#include <ctime>   
#include <ctype.h> 
using namespace std;

#define MAXN 1100

typedef struct kind
{
    int num,price;
}Kind;
Kind kind[MAXN];

int sum[MAXN];
int ans[MAXN];

int main()
{
    int t;
    int i,j;
    int n;
    scanf("%d",&t);
    while(t--)
    {
        scanf("%d",&n);
        sum[0]=0;
        memset(ans,0,sizeof(ans));
        for(i=1;i<=n;i++)
        {
            scanf("%d%d",&kind[i].num,&kind[i].price);
            sum[i]=sum[i-1]+kind[i].num;
            ans[i]=(sum[i]+10)*kind[i].price;
        }
        for(i=1;i<=n;i++)                                
            for(j=1;j<=i;j++)                                    //枚举
                ans[i]=min(ans[i],(sum[i]-sum[j]+10)*kind[i].price+ans[j]);
        printf("%d\n",ans[n]);
    }
    return 0;
}