24. Maximum Sum Problem

The problem can be found at the following link: Question Link

My Approach

For this problem, I used dynamic programming to iteratively calculate the maximum sum for each number up to n.

• Initialize a vector dp with size n+1, initialized with zeros.

• Loop through numbers from 1 to n.

  • For each number i, determine dp[i] as the larger value between the sum of dp[i/2], dp[i/3], and dp[i/4], and i itself.

  • This process helps us in getting whether breaking i into i/2, i/3, and i/4 yields the maximum sum at each iteration.

• Return dp[n] as the maximum sum.

Time and Auxiliary Space Complexity

• Time Complexity: O(n), where n is the input number.

• Auxiliary Space Complexity: O(n), as we use a vector of size n+1 to store the dynamic programming values.

Code (C++)

class Solution {
public:
    int maxSum(int n) {
        vector<int> dp(n+1, 0);
        for(int i = 1; i <= n; ++i) {
            dp[i] = dp[i/2] + dp[i/3] + dp[i/4];
            dp[i] = max(dp[i], i);
        }
        return dp[n];
    }
};

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