第二讲 完全背包问题 HD Piggy-Bank 1114

Piggy-Bank

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 65536/32768 K (Java/Others)
Total Submission(s): 17611    Accepted Submission(s): 8879

Problem Description

Before ACM can do anything, a budget must be prepared and the necessary financial support obtained. The main income for this action comes from Irreversibly Bound Money (IBM). The idea behind is simple. Whenever some ACM member has any small money, he takes all the coins and throws them into a piggy-bank. You know that this process is irreversible, the coins cannot be removed without breaking the pig. After a sufficiently long time, there should be enough cash in the piggy-bank to pay everything that needs to be paid. 

But there is a big problem with piggy-banks. It is not possible to determine how much money is inside. So we might break the pig into pieces only to find out that there is not enough money. Clearly, we want to avoid this unpleasant situation. The only possibility is to weigh the piggy-bank and try to guess how many coins are inside. Assume that we are able to determine the weight of the pig exactly and that we know the weights of all coins of a given currency. Then there is some minimum amount of money in the piggy-bank that we can guarantee. Your task is to find out this worst case and determine the minimum amount of cash inside the piggy-bank. We need your help. No more prematurely broken pigs! 

 

Input

The input consists of T test cases. The number of them (T) is given on the first line of the input file. Each test case begins with a line containing two integers E and F. They indicate the weight of an empty pig and of the pig filled with coins. Both weights are given in grams. No pig will weigh more than 10 kg, that means 1 <= E <= F <= 10000. On the second line of each test case, there is an integer number N (1 <= N <= 500) that gives the number of various coins used in the given currency. Following this are exactly N lines, each specifying one coin type. These lines contain two integers each, Pand W (1 <= P <= 50000, 1 <= W <=10000). P is the value of the coin in monetary units, W is it's weight in grams. 

 

Output

Print exactly one line of output for each test case. The line must contain the sentence "The minimum amount of money in the piggy-bank is X." where X is the minimum amount of money that can be achieved using coins with the given total weight. If the weight cannot be reached exactly, print a line "This is impossible.". 

 

Sample Input

   
   
   
   

    
    
    
    3
10 110
2
1 1
30 50
10 110
2
1 1
50 30
1 6
2
10 3
20 4
   
   
   
   

 

Sample Output

   
   
   
   

    
    
    
    The minimum amount of money in the piggy-bank is 60.
The minimum amount of money in the piggy-bank is 100.
This is impossible.
   
   
   
   

基本思路：

有N种物品和一个容量为V的背包，每种物品都有无限件可用。第i种物品的费用是c[i]，价值是w[i]。求解将哪些物品装入背包可使这些物品的费用总和不超过背包容量，且价值总和最大。
既然01背包问题是最基本的背包问题，那么我们可以考虑把完全背包问题转化为01背包问题来解。

这个算法使用一维数组，先看伪代码：

for i=1..N
for v=0..V
f[v]=max{f[v],f[v-cost]+weight}

你会发现，这个伪代码与P01的伪代码只有v的循环次序不同而已。 为什么这样一改就可行呢？首先想想为什么P01中要按照v=V..0的逆序来循环。这是因为要保证第i次循环中的状态f[i][v]是由状态f[i-1] [v-c[i]]递推而来。换句话说，这正是为了保证每件物品只选一次，保证在考虑“选入第i件物品”这件策略时，依据的是一个绝无已经选入第i件物品的 子结果f[i-1][v-c[i]]。而现在完全背包的特点恰是每种物品可选无限件，所以在考虑“加选一件第i种物品”这种策略时，却正需要一个可能已选入第i种物品的子结果f[i][v-c[i]]，所以就可以并且必须采用v=0..V的顺序循环。这就是这个简单的程序为何成立的道理。

本题是完全背包的巧用，每次都找质量为b时最少的钱

代码如下：

#include<iostream>//c++
#include<cmath>//数学公式
#include<cstdlib>//malloc
#include<cstring>
#include<string>
#include<cstdio>//输入输出
#include<algorithm>//快排
#include<queue>//队列
#include<functional>//优先队列
#include<stack>//栈
#include<vector>//容器
#include<map>//地图  if continue
typedef long long ll;
const int N=10005;
const int inf=0x3f3f3f3f;
using namespace std;
int dp[N];
int main()
{
    //freopen("C:\\Users\\ch\\Desktop\\1.txt","r",stdin);
	//freopen("C:\\Users\\lenovo\\Desktop\\2.txt","w",stdout);
	int i,j,k;
	int v,n,text;
	int a,b;
	cin>>text;
	while(text--)
	{
	    memset(dp,inf,sizeof(dp));
	    dp[0]=0;//初始化数组
	    cin>>a>>b;  v=b-a;
	    cin>>n;
	    for(i=0;i<n;i++)
        {
            cin>>a>>b;//价值a->w	     体积b->v
            for(j=b;j<=v;j++)
                dp[j]=min(dp[j-b]+a,dp[j]);
        }
        if(dp[v] == inf)
            printf("This is impossible.\n");
        else
            printf("The minimum amount of money in the piggy-bank is %d.\n",dp[v]);
	}
	return 0;
}