How do you solve Profitable Schemes on LeetCode?

Profitable Schemes (LeetCode #879) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(m * n * minProfit) time complexity and O(minProfit) space complexity. Key insight: Dynamic programming is powerful for problems involving combinations and constraints.

What is the brute force solution for Profitable Schemes?

The brute force approach for Profitable Schemes is Brute Force, with O(2^m * m) time complexity and O(1) space complexity. The brute-force approach involves generating all possible subsets of crimes and checking each subset to see if it meets the criteria of total members and minimum profit. This is straightforward but inefficient as the number of subsets grows exponentially with the number of crimes. The optimal Optimal Solution approach improves this to O(m * n * minProfit).

What algorithm or data structure is used to solve Profitable Schemes?

Profitable Schemes uses the following concepts: Array, Dynamic Programming. The recommended patterns to study are: Dynamic Programming, Combinatorial Optimization.

What are the interview tips for Profitable Schemes?

Clearly explain your thought process and the reasoning behind each step. Practice writing both brute-force and optimal solutions to show a range of problem-solving skills. Be prepared to discuss trade-offs between time and space complexity.

What are common mistakes when solving Profitable Schemes?

Not considering the constraints on both members and profit simultaneously. Failing to account for large numbers in the final result, leading to overflow.

#879

Profitable Schemes

Hard

ArrayDynamic ProgrammingDynamic ProgrammingCombinatorial Optimization

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(2^m * m)	O(m * n * minProfit)
Space	O(1)	O(minProfit)

💡

Intuition

Time O(m * n * minProfit)Space O(minProfit)

The optimal solution uses dynamic programming to keep track of the number of ways to achieve different profits with a limited number of members. This approach avoids the need to generate all subsets and instead builds up the solution incrementally.

⚙️

Algorithm

3 steps

1Step 1: Initialize a DP array where dp[j] represents the number of ways to achieve profit j with the current number of members.
2Step 2: For each crime, iterate backwards through the DP array and update the number of ways to achieve profits based on the current crime's profit and required members.
3Step 3: Sum all the ways to achieve at least minProfit while ensuring the total members used do not exceed n.

solution.py13 lines

1# Full working Python code
2
3def profitableSchemes(n, minProfit, group, profit):
4    MOD = 10**9 + 7
5    dp = [0] * (minProfit + 1)
6    dp[0] = 1  # 1 way to achieve 0 profit
7    for i in range(len(group)):
8        g = group[i]
9        p = profit[i]
10        for j in range(minProfit, -1, -1):
11            for k in range(n, g - 1, -1):
12                dp[j] = (dp[j] + dp[max(0, j - p)]) % MOD
13    return sum(dp) % MOD

ℹ

Complexity note: The time complexity is O(m * n * minProfit) because we iterate through each crime (m), each possible number of members (n), and each profit level (minProfit). The space complexity is O(minProfit) as we only need to store the ways to achieve profits up to minProfit.

1Dynamic programming is powerful for problems involving combinations and constraints.
2Understanding how to build solutions incrementally can save time and resources.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.