Solving the Integer Knapsack

问题:

I a new to dynamic programing and have tried the integer knapsack problem here at SPOJ (http://www.spoj.pl/problems/KNAPSACK/) . However, for the given test cases my solution is not giving the correct output. I'd be thankful to you if you could suggest if the following implementation by me is correct. Please note that the variable back is for backtracking, about which I'm not sure how to do. I hope to have your help in implementing the backtracking too. Thanks.

#include <cstdio>
#include <cstdlib>
#include <algorithm>
#include <vector>
#include <string>
#include <iostream>

using namespace std;

int knapsack(int value[], int weight[], int n, int C,
         vector < int >&back)
{
    int *M = new int[C + 1];
    int k;
    int i, j, tmp, pos;
    for (i = 1; i <= C; i++) {
        M[i] = M[i - 1];
        pos = i - 1;
        for (j = 0; j < n; j++) {
            k = i - weight[j];
            if (k >= 0)
                tmp = M[k] + value[j];
            if (tmp > M[i]) {
                M[i] = tmp;
            }
        }
        back.push_back(pos);
    }
    int ans = M[C];
    delete[]M;
    return ans;
}


int main()
{
    int C, N;
    cin >> C >> N;
    int* value = new int[N+1];
    int* weight = new int[N+1];
    for ( int i = 0; i <= N; i++) {
        cin>>value[i]>>weight[i];
    }
    vector < int >back(N, 0);
    cout<<knapsack(value,weight,N,C,back)<<endl;
    return 0;
}

Here are the correct input/output test cases:

Input:
4 5
1 8
2 4
3 0
2 5
2 3


Output:
13

However, the output that I am getting is 17.

回答1:

This is a version of the Knapsack problem known as the 0-1 knapsack.

You are making some silly mistakes in your code.

To begin with the first integer in input is the weight and the second is the value. While you are taking first as value and second as weight. Moreover you are taking n+1 values as input 0 to N inclusive.

Now in your algorithm, you are considering that you can take any number of copies of the items, this is not true this is a 0-1 knapsack. Read this http://en.wikipedia.org/wiki/Knapsack_problem .

I came up with this algorithm and I submitted and got accepted. So this should work fine.

int M[2000][2000];
int knapsack(int value[], int weight[], int C, int n)
{      
  for(int i = 1; i <= C; i++){
    for(int j = 0; j <n; j++){
      if(j > 0){
        M[j][i] = M[j-1][i];
        if (weight[j] <= i)
          M[j][i] = max(M[j][i], M[j-1][i-weight[j]]+value[j]);
      }
      else{
        M[j][i] = 0;
        if(weight[j] <= i)
          M[j][i] = max(M[j][i], value[j]);
      }
    }
    //    cout << M[i][n-1] << endl;
  }        
  return M[n-1][C];
}  

int main()
{
    int C, N;
    cin >> C >> N;
    //    cout << C << endl;
    int* value = new int[N+1];
    int* weight = new int[N+1];
    for ( int i = 0; i < N; i++) {
        cin>>weight[i]>>value[i];
    }
    //   vector < int >back(N, 0);
    cout<<knapsack(value,weight,C,N)<<endl;
    return 0;
}

BTW don't dynamically allocate arrays simply use vectors

vector <int> My_array(n);

回答2:

There is a version of the knapsack problem documented well at https://sites.google.com/site/mikescoderama/Home/0-1-knapsack-problem-in-p in Python.

EDIT: Nevermind, I skipped the part where the first line input defines C and N. That said, your input example doesn't seem to load with the code you provided (it is looking for one more pair than would be expected due to the <= N). I expect that loop should be < N, to get 0..n-1 as items.

Fixing that I get a result of 134516145 printed, which is nonsensical.