HDU1963 && POJ2063：Investment(完全背包)

Problem Description

John never knew he had a grand-uncle, until he received the notary’s letter. He learned that his late grand-uncle had gathered a lot of money, somewhere in South-America, and that John was the only inheritor.

John did not need that much money for the moment. But he realized that it would be a good idea to store this capital in a safe place, and have it grow until he decided to retire. The bank convinced him that a certain kind of bond was interesting for him.

This kind of bond has a fixed value, and gives a fixed amount of yearly interest, payed to the owner at the end of each year. The bond has no fixed term. Bonds are available in different sizes. The larger ones usually give a better interest. Soon John realized that the optimal set of bonds to buy was not trivial to figure out. Moreover, after a few years his capital would have grown, and the schedule had to be re-evaluated.

Assume the following bonds are available:

Value Annual interest

4000   400

3000   250

With a capital of $10 000 one could buy two bonds of $4 000, giving a yearly interest of $800. Buying two bonds of $3 000, and one of $4 000 is a better idea, as it gives a yearly interest of $900. After two years the capital has grown to $11 800, and it makes sense to sell a $3 000 one and buy a $4 000 one, so the annual interest grows to $1 050. This is where this story grows unlikely: the bank does not charge for buying and selling bonds. Next year the total sum is $12 850, which allows for three times $4 000, giving a yearly interest of $1 200.

Here is your problem: given an amount to begin with, a number of years, and a set of bonds with their values and interests, find out how big the amount may grow in the given period, using the best schedule for buying and selling bonds.

 

Input

The first line contains a single positive integer N which is the number of test cases. The test cases follow.

The first line of a test case contains two positive integers: the amount to start with (at most $1 000 000), and the number of years the capital may grow (at most 40).

The following line contains a single number: the number d (1 <= d <= 10) of available bonds.

The next d lines each contain the description of a bond. The description of a bond consists of two positive integers: the value of the bond, and the yearly interest for that bond. The value of a bond is always a multiple of $1 000. The interest of a bond is never more than 10% of its value.

 

Output

For each test case, output – on a separate line – the capital at the end of the period, after an optimal schedule of buying and selling.

 

Sample Input

1
10000 4
2
4000 400
3000 250

 

Sample Output

14050

 

题意：给出初始资金，还有年数，然后给出每个物品的购买价格与每年获得的利益，要求在给出的年份后所能得到的最大本利之和。

思路：因为每种物品可以多次购买，可以看做是完全背包的题目，但是要注意的是，由于本金可能会很大，所以我们要对背包的大小进行压缩，值得注意的是，题目已经说了本金与物品的购买价格都是1000的倍数，所以我们可以将他们都除以1000来进行压缩，然后就是一道完全背包模板题了。

#include <stdio.h>
#include <string.h>
#include <algorithm>
using namespace std;

struct node
{
    int v,w;
}a[20];

int dp[100000];

int main()
{
    int t,n,i,j,k,val,y;
    scanf("%d",&t);
    while(t--)
    {
        scanf("%d%d",&val,&y);
        scanf("%d",&n);
        for(i = 1;i<=n;i++)
        {
            scanf("%d%d",&a[i].v,&a[i].w);
            a[i].v/=1000;//进行压缩
        }
        for(i = 1;i<=y;i++)
        {
            int s = val/1000;//每年本金都是上一年本金与利息之和
            memset(dp,0,sizeof(dp));//每年都要重新存利息
            for(j = 1;j<=n;j++)//完全背包
            {
                for(k = a[j].v;k<=s;k++)
                {
                    dp[k]=max(dp[k],dp[k-a[j].v]+a[j].w);
                }
            }
            val+=dp[s];//每年的最大本利和
        }
        printf("%d\n",val);
    }

    return 0;
}