Profitable Schemes

class Solution {
public:
    int modulo = 1000000007;
    int memo[101][101][101];
              
    int maxProfitPlans(int n, int minProfit, vector<int>& group, vector<int>& profit) {
        // Set initial conditions
        for (int count = 0; count <= n; count++) {
            memo[group.size()][count][minProfit] = 1;
        }
            
        for (int i = group.size() - 1; i >= 0; i--) {
            for (int count = 0; count <= n; count++) {
                for(int profitSoFar = 0; profitSoFar <= minProfit; profitSoFar++) {
                    // Carry over previous scenarios without including the current group
                    memo[i][count][profitSoFar] = memo[i + 1][count][profitSoFar];
                    if (count + group[i] <= n) {
                        // Include current group's contribution
                        memo[i][count][profitSoFar] 
                            = (memo[i][count][profitSoFar] + memo[i + 1][count + group[i]][min(minProfit, profitSoFar + profit[i])]) % modulo;
                    }
                }
            }
        }
            
        return memo[0][0][0];
    }
};

The provided C++ code defines a solution to the "Profitable Schemes" problem using dynamic programming. The class Solution includes the method maxProfitPlans which computes the number of ways to achieve at least minProfit profit using up to n members from various groups, each potentially contributing a different amount of profit.

Key elements of the solution include:

• An integer modulo set to 1000000007 to ensure results are returned modulo 10^9 + 7.
• A 3D memoization array memo[101][101][101] used to store intermediate results.
• Initialization of memoization values for base cases where the group size is zero but a certain profit needs to be achieved with a certain number of people.
• Nested loops to iterate through every group member, every possible worker count up to n, and every possible profit amount up to minProfit.
• Conditions to either skip including a group or to include it while adjusting the profit sum and member count accordingly.

The function finally returns the number of ways to achieve at least minProfit from no groups and no members.

Ensure to have vector<int> included in your packages, as this data structure is crucial for holding the group sizes and profit contributions. This solution provides an optimized approach to handle large inputs efficiently by utilizing memoization and iterative calculations, suitable for problems involving combinations and resource allocation under constraints.

class SchemeCalculator {
    int modulus = 1000000007;
    int[][][] memo = new int[101][101][101];
    
    public int calculateSchemes(int G, int targetProfit, int[] teams, int[] earnings) {
        // Initialize dp for base cases
        for (int g = 0; g <= G; g++) {
            memo[teams.length][g][targetProfit] = 1;
        }
    
        for (int i = teams.length - 1; i >= 0; i--) {
            for (int g = 0; g <= G; g++) {
                for (int p = 0; p <= targetProfit; p++) {
                    // Calculate schemes without using current team
                    memo[i][g][p] = memo[i + 1][g][p];
                    if (g + teams[i] <= G) {
                        // Incorporate current team into the schemes
                        memo[i][g][p] = (memo[i][g][p] + memo[i + 1][g + teams[i]][Math.min(targetProfit, p + earnings[i])]) % modulus;
                    }
                }
            }
        }
    
        return memo[0][0][0];
    }
}

The Profitable Schemes problem consists of counting the number of ways to achieve a target profit using a designated number of groups, each having a certain number of members and potential profit. This solution is implemented in Java using a dynamic programming approach.

The approach involves utilizing a three-dimensional dynamic programming (DP) table memo, where:

• The first dimension represents the team index.
• The second dimension corresponds to the number of people involved.
• The third dimension is used for the accumulated profit.

The solution initializes the base cases in the DP table where for all possible group sizes up to the limit G, if actions reach the target profit at the last job, they are considered viable, hence initialized to 1.

The core of the solution iteratively calculates the number of schemes:

• For each backward iteration from the last available team to the first (teams.length - 1 to 0).
• For each possible group size up to G.
• For each profit level from 0 to targetProfit.

In each iteration:

• The DP cell memo[i][g][p] is initially set to the value from the case where the current team is not chosen – memo[i + 1][g][p].
• If the current team can be used without breaching the group size limit (g + teams[i] <= G), the DP value is updated to add schemes that include the profits from using the current team.

Values are computed modulo 1000000007 to manage large numbers.

The result, which is the count of different ways to achieve at most the target profit with the available teams and members limit, is stored in memo[0][0][0]. The return from calculateSchemes method gives this result, making this a comprehensive solution utilizing memoization to ensure efficiency by avoiding redundant calculations.