Problem Statement
Profitable Schemes
There is a group of n members, and a list of various crimes they could commit. The i-th 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 10^9 + 7.
Examples
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
Problem Breakdown
To solve this problem, we need to:
- This problem can be solved using dynamic programming with three dimensions: the number of crimes considered, the number of members used, and the profit generated.
- For each crime, we have two options: include it in our scheme or exclude it.
- If we include a crime, we need to ensure that we have enough members to commit it.
- We need to count the number of schemes that generate at least minProfit profit, not exactly minProfit profit.
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Profitable Schemes
There is a group of n members, and a list of various crimes they could commit. The i-th 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 10^9 + 7.
Examples
Example 1:
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:
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).
Problem Breakdown
Constraints:
- 1 <= n <= 100
- 0 <= minProfit <= 100
- 1 <= group.length <= 100
- 1 <= group[i] <= 100
- profit.length == group.length
- 0 <= profit[i] <= 100
Key Insights:
This problem can be solved using dynamic programming with three dimensions: the number of crimes considered, the number of members used, and the profit generated.
For each crime, we have two options: include it in our scheme or exclude it.
If we include a crime, we need to ensure that we have enough members to commit it.
We need to count the number of schemes that generate at least minProfit profit, not exactly minProfit profit.
Real-World Application
This problem has several practical applications:
Resource Allocation
Optimizing the allocation of limited resources to maximize profit in business operations.
Team Assignment
Assigning team members to different projects to achieve specific goals with limited personnel.
Project Selection
Selecting a subset of projects to undertake with limited resources to meet profit targets.