Problem statement

Given n items with size Ai and value Vi, and a backpack with size m. What's the maximum value can you put into the backpack?

Solution

0/1 knapsack problem is a classical dynamic programming model. There is a knapsack with the capacity of m, you should find the maximum volume can be filled in.

Still, we need:

  • DP memory and the representation
  • The initialization of DP memory
  • DP formula
  • Return value.

DP memory and the representation

Suppose, size is the number of elements in A.

A two dimension array: dp[size + 1][m + 1]

  • dp[i][j]: means the maximum volume formed by first i elements whose volume is at most j.

The key word is the first and at most.

  • The first means there are i + 1 elements.
  • At most means the total volume can not exceed j.

Initialization

For a two dimension DP memory, normally, we should initialize the first row and column, and start from i = 1 and j = 1. The initialization comes from general knowledge.

  • dp[0][i]: first 0 elements can form at most i volume. Obviously, the initialization is 0 since we can get nothing if there is no elements.
  • dp[i][0]: first i elements can form at most 0 volume. Obviously, the initialization is 0 since we can get 0 volume by any elements.

DP formula

For current element A[i], we need to know what is the maximum volume can get if we add it into the backpack.

  • dp[i][j] = dp[i - 1][j] if A[i - 1] is greater than j
  • dp[i][j] = max(dp[i - 1][j], dp[i - 1][j - A[i - 1]]) if j >= A[i - 1], we find the maximum value.

Return value.

Just return dp[size][m]

Time complexity is O(size * m)

class Solution {
public:
/**
* @param m: An integer m denotes the size of a backpack
* @param A & V: Given n items with size A[i] and value V[i]
* @return: The maximum value
*/
int backPackII(int m, vector<int> A, vector<int> V) {
// write your code here
// write your code here
int size = A.size();
//vector<vector<int>> dp(size + 1, vector<int>(m + 1, 0));
int dp[size + ][m + ] = {};
for(int i = ; i <= size; i++){
for(int j = ; j <= m; j++){
dp[i][j] = dp[i - ][j];
if(j >= A[i - ]){
dp[i][j] = max(dp[i][j], V[i - ] + dp[i - ][j - A[i - ]]);
}
}
}
return dp[size][m];
}
};

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