879. Profitable Schemes

879. Profitable Schemes

Tag: Dynamic Programming

Difficulty: Medium

There is a group of n members, and a list of various crimes they could commit. The ith crime generates a profit[i] and requires group[i] members to participate in it. If a member participates in one crime, that member can't participate in another crime.

Let's call a profitable scheme any subset of these crimes that generates at least minProfit profit, and the total number of members participating in that subset of crimes is at most n.

Return the number of schemes that can be chosen. Since the answer may be very large, return it modulo 10⁹ + 7.

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Example 1:

Input: n = 5, minProfit = 3, group = [2,2], profit = [2,3]
Output: 2
Explanation: To make a profit of at least 3, the group could either commit crimes 0 and 1, or just crime 1.
In total, there are 2 schemes.

Example 2:

Input: n = 10, minProfit = 5, group = [2,3,5], profit = [6,7,8]
Output: 7
Explanation: To make a profit of at least 5, the group could commit any crimes, as long as they commit one.
There are 7 possible schemes: (0), (1), (2), (0,1), (0,2), (1,2), and (0,1,2).

Constraints:

• 1 <= n <= 100
• 0 <= minProfit <= 100
• 1 <= group.length <= 100
• 1 <= group[i] <= 100
• profit.length == group.length
• 0 <= profit[i] <= 100

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The Framework

Top-Down Dynamic Programming

class Solution:
    def profitableSchemes(self, n: int, minProfit: int, group: List[int], profit: List[int]) -> int:
        m = len(group)

        @lru_cache(None)
        def dp(curr, total_profit, remaining):
            # Base case
            if curr == m:
                if total_profit >= minProfit:
                    return 1
                return 0
            
            # Recurrence relation: to commit or to do nothing and skip the current crime

            # To do nothing: simply move to the next crime
            skip = dp(curr + 1, total_profit, remaining)

            # To commit the crime
            pick = 0
            # Check if there are enough members to do the current job
            if remaining >= group[curr]:
                # Move on the next crime, gather the profit as long as it is larger than minProfit, reduce the members
                pick = dp(curr + 1, min(minProfit, total_profit + profit[curr]), remaining - group[curr])
            
            return pick + skip

        return dp(0, 0, n) % (10**9 + 7)