ACM: 动态规划题 poj 3093 0-1背包

Margaritas on the River Walk

Description

One of the more popular activities in SanAntonio is to enjoy margaritas in the park along the river know asthe River Walk. Margaritas may be purchased at manyestablishments along the River Walk from fancy hotels to Joe’sTaco and Margarita stand. (The problem is not to find out howJoe got a liquor license. That involves Texas politics and thus ismuch too difficult for an ACM contest problem.) The prices of themargaritas vary depending on the amount and quality of theingredients and the ambience of the establishment. You haveallocated a certain amount of money to sampling differentmargaritas.

Given the price of a single margarita (including applicabletaxes and gratuities) at each of the various establishments and theamount allocated to sampling the margaritas, find out how manydifferent maximal combinations, choosing at most one margarita fromeach establishment, you can purchase. A valid combination must havea total price no more than the allocated amount and the unusedamount (allocated amount – total price) must be less thanthe price of any establishment that was not selected. (Otherwiseyou could add that establishment to the combination.)

For example, suppose you have $25 to spend and the prices (wholedollar amounts) are:

VendorABCDHJPrice8987165

Then possible combinations (with their prices) are:

ABC(25), ABD(24), ABJ(22), ACD(23), ACJ(21), ADJ( 20), AH(24),BCD(24), BCJ(22), BDJ(21), BH(25), CDJ(20), CH(24), DH(23) andHJ(21).

Thus the total number of combinations is 15.

Input

The input begins with a line containing aninteger value specifying the number of datasets that follow,N (1 ≤ N ≤ 1000). Each dataset startswith a line containing two integer values V andD representing the number of vendors (1 ≤V ≤ 30) and the dollar amount to spend (1 ≤D ≤ 1000) respectively. The two values will beseparated by one or more spaces. The remainder of each datasetconsists of one or more lines, each containing one or more integervalues representing the cost of a margarita for each vendor. Therewill be a total of V cost values specified. The costof a margarita is always at least one (1). Input values will bechosen so the result will fit in a 32 bit unsignedinteger.

Output

For each problem instance, the output will be a single linecontaining the dataset number, followed by a single space and thenthe number of combinations for that problem instance.

Sample Input

2
6 25
8 9 8 7 16 5
30 250
1 2 3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30

Sample Output

1 15
2 16509438

 

题意: 从n个物品选出一部分, 使它们的值累加起来Sum,如果Sum+a[i]任何一个不在Sum之内的值,

     将大于限制S. 问有多少种可能组合.

解题思路:

     1. 从物品中选取一部分出来, 先将全部按值从小到大排序. 进行背包问题求解.

        设sum[i]表示前i个排序后值的和, 并且假设a[i]是当前最小不能再放进去的值.

        即: 前i-1种物品都必须已经放进去了. dp[ sum[i-1] ] = 1

     2. 我们知道背包的临界值是S(最大值), 并且假设sum[i-1]物品已经在背包中了,

         之后枚举后面的物品即可.范围区间[ sum[i-1]+a[j] , S ]; (j> i)

 

代码：

#include <cstdio>
#include <iostream>
#include <cstring>
#include <algorithm>
using namespace std;
#define MAX 1005
#define MAXSIZE 32

int n, S;
int sum[MAXSIZE], a[MAXSIZE];
int dp[MAXSIZE*MAX];

inline int max(int a, int b)
{
 return a > b ? a : b;
}

int DP()
{
 int result = 0;
 for(int i = 1; i <= n; ++i)
 {
  if(sum[i-1] > S)break;
  memset(dp, 0,sizeof(dp));
  dp[ sum[i-1] ] = 1;
  for(int j = i+1; j<= n; ++j)
  {
   for(int k =S; k >= sum[i-1]+a[j]; --k)
    dp[k]+= dp[ k-a[j] ];
  }

  int temp = max(sum[i-1],S-a[i]+1);
  for(int p = temp; p<= S; ++p)
   result +=dp[p];
 }

 return result;
}

int main()
{
// freopen("input.txt", "r", stdin);
 int caseNum, num = 1, i;
 scanf("%d", &caseNum);
 while(caseNum--)
 {
  scanf("%d %d",&n, &S);
  for(i = 1; i <=n; ++i)
   scanf("%d",&a[i]);
  sort(a+1, a+n+1);

  if(a[1] >S)
  {
   printf("%d%d\n", num++, 0);
   continue;
  }
  sum[0] = 0;
  for(i = 1; i <=n; ++i)
   sum[i] =(sum[i-1]+a[i]);

  printf("%d %d\n", num++,DP());
 }
 return 0;
}