[poj 1276] Cash Machine 多重背包及优化

Description

A Bank plans to install a machine for cash withdrawal. The machine is able to deliver appropriate @ bills for a requested cash amount. The machine uses exactly N distinct bill denominations, say Dk, k=1,N, and for each denomination Dk the machine has a supply of nk bills. For example, 

N=3, n1=10, D1=100, n2=4, D2=50, n3=5, D3=10 

means the machine has a supply of 10 bills of @100 each, 4 bills of @50 each, and 5 bills of @10 each. 

Call cash the requested amount of cash the machine should deliver and write a program that computes the maximum amount of cash less than or equal to cash that can be effectively delivered according to the available bill supply of the machine. 

Notes: 
@ is the symbol of the currency delivered by the machine. For instance, @ may stand for dollar, euro, pound etc. 

Input

The program input is from standard input. Each data set in the input stands for a particular transaction and has the format: 

cash N n1 D1 n2 D2 ... nN DN 

where 0 <= cash <= 100000 is the amount of cash requested, 0 <=N <= 10 is the number of bill denominations and 0 <= nk <= 1000 is the number of available bills for the Dk denomination, 1 <= Dk <= 1000, k=1,N. White spaces can occur freely between the numbers in the input. The input data are correct. 

Output

For each set of data the program prints the result to the standard output on a separate line as shown in the examples below. 

Sample Input

735 3  4 125  6 5  3 350
633 4  500 30  6 100  1 5  0 1
735 0
0 3  10 100  10 50  10 10

Sample Output

735
630
0
0

Hint

The first data set designates a transaction where the amount of cash requested is @735. The machine contains 3 bill denominations: 4 bills of @125, 6 bills of @5, and 3 bills of @350. The machine can deliver the exact amount of requested cash. 

In the second case the bill supply of the machine does not fit the exact amount of cash requested. The maximum cash that can be delivered is @630. Notice that there can be several possibilities to combine the bills in the machine for matching the delivered cash. 

In the third case the machine is empty and no cash is delivered. In the fourth case the amount of cash requested is @0 and, therefore, the machine delivers no cash.

 

根据题意可转化为重量和价值相同的多重背包问题

最基本的dp方程为: DP(i, j) = max( DP(i-1, V - k*Ci) + k*Wi| k∈[0, Mi] )

通过把物品进行分解 ,可转化为0-1背包问题(如把M件相同物品当做M件独立的不同的物品), 如果采用二进制进行分解, 将把复杂度降到O(V∑logMi)

将其分解成二进制可表示0～M之间的任何数

#include <iostream>
#include <stdio.h>
#include <cstring>
#include <algorithm>
using namespace std;
const int Max = 100005;
int cash, N, cnt[15], d[15];
int dp[Max];

void ZeroOnePack(int d)
{
    for (int i = cash; i >= d; i--)
        dp[i] = max(dp[i], dp[i-d]+d);
}

void CompletePack(int d)
{
    for (int i = d; i <= cash; i++) 
        dp[i] = max(dp[i], dp[i-d]+d);
}

void MultiplePack(int m, int d)
{
    if (m*d >= cash) {
        CompletePack(d);
        return ;
    }
    int k = 1;
    while (k < m) {
        ZeroOnePack(k*d);
        m -= k;
        k *= 2;
    }
    ZeroOnePack(m*d);
}


int main()
{
    //freopen("1.txt", "r", stdin);
    while(cin >> cash) {
        cin >> N;
        for (int i = 1; i <= N; i++)
            cin >> cnt[i] >> d[i];
        memset(dp, 0, sizeof(dp));
        for (int i = 1; i <= N; i++)
            MultiplePack(cnt[i], d[i]);
        printf("%d
", dp[cash]);
    }



    return 0;
}

参考资料：背包九讲