完全背包问题 一维数组就地滚动写法

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.

 

题意：

有一个存钱罐，求出恰好达到指定重量（装满钱后的重量-装钱前的重量）时，其中钱的最小数量，若达不到指定重量，输出impossible。

 

初始化的细节问题:我们看到的求最优解的背包问题题目中，事实上有两种不太相同的问法。有的题目要求“恰好装满背包”时的最优解，有的题目
则并没有要求必须把背包装满。一种区别这两种问法的实现方法是在初始化的时候有所不同。
如果是第一种问法，要求恰好装满背包，那么在初始化
时除了f[0]为0其它f[1..V]均设为-∞，这样就可以保证最终得到的f[N]是一种恰好装满背包的最优解。
如果并没有要求必须把背包装满，而是只希
望价格尽量大，初始化时应该将f[0..V]全部设为0。
为什么呢？可以这样理解：初始化的f数组事实上就是在没有任何物品可以放入背包时的合法状
态。如果要求背包恰好装满，那么此时只有容量为0的背包可能被价值为0的nothing“恰好装满”，其它容量的背包均没有合法的解，属于未定义的
状态，它们的值就都应该是−∞了。如果背包并非必须被装满，那么任何容量的背包都有一个合法解“什么都不装”，这个解的价值为0，所以初始时
状态的值也就全部为0了.

 

 1 #include <iostream>

 2 #include <cstring>

 3 #include <algorithm>

 4 #include <fstream>

 5 using namespace std;

 6 const int INF = 0x7ffffff;

 7 int main()

 8 {

 9     int T;

10     cin>>T;

11     while(T--)

12     {

13         int x1,x2;

14         cin>>x1>>x2;

15         int dp[10001];

16         int lose_w = x2 - x1;

17         int p[500+5],w[500+5];

18         int n;

19         cin>>n;

20         for(int i = 0; i < n; i++)

21         {

22             cin>>p[i]>>w[i];

23         }

24         /*初始化*/

25         for(int i=1; i<=lose_w; i++)

26             dp[i]=INF;

27         dp[0]=0;

28         for(int i = 0; i< n; i ++)

29             for(int j = w[i]; j <= lose_w; j++)

30             {

31                 dp[j] = min(dp[j],dp[j-w[i]]+p[i]);

32             }

33         if(dp[lose_w]==INF)

34             cout<<"This is impossible."<<endl;

35         else

36             cout<<"The minimum amount of money in the piggy-bank is "<<dp[lose_w]<<"."<<endl;

37 

38     }

39     return 0;

40 

41 }

View Code

 1 /*借鉴的代码*/

 2 #include<stdio.h>

 3 #include<string.h>

 4 #define INF 0x7ffffff

 5 #define MAXN 10000

 6 int dp[MAXN+10];//dp[i]表容量为i的时候所装东西的最小价值

 7 int main()

 8 {

 9     int w1,w2;

10     int P,W;

11     int T,n;

12     int i,j;

13     scanf("%d",&T);

14     while(T--)

15     {

16         scanf("%d%d",&w1,&w2);

17         scanf("%d",&n);

18         for(i=1;i<=w2-w1;i++)

19            dp[i]=INF;//初始化为无穷大

20         dp[0]=0;

21         while(n--)

22         {

23             scanf("%d%d",&P,&W);

24             for(i=W;i<=w2-w1;i++)  //直接在读入中处理时间空间效率更高

25                 if(dp[i]>dp[i-W]+P)

26                    dp[i]=dp[i-W]+P;

27         }

28         if(dp[w2-w1]==INF) printf("This is impossible.\n");

29         else

30            printf("The minimum amount of money in the piggy-bank is %d.\n",dp[w2-w1]);

31     }

32     return 0;

33 }

View Code