Profitable Schemes

We can solve this problem using a dynamic programming approach. Let’s break down the solution into simple steps:

Understanding the Problem: We need to find the total number of profitable schemes where we have n members, a minimum profit of minProfit, and a list of crimes characterized by the number of members needed (group) and the profit they generate (profit).
Initialize a 3D DP Array: We will use a 3D dynamic programming table dp[i][j][k] to represent the number of ways to use the first i crimes, with j members, to make a profit of at least k.
Base Case: If we have 0 crimes, and we need 0 profit, there is 1 way to do it (i.e., do nothing).
Iterative Case: For each crime, we iterate through the number of members and the profit, and update the DP table accordingly.
Return Value: Finally, we sum up the possible ways for all profits greater than or equal to minProfit with n or fewer members.

Here is the Python code:

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class Solution:
    def profitableSchemes(self, n: int, minProfit: int, group: List[int], profit: List[int]) -> int:
        MOD = 10**9 + 7
        dp = [[[0] * (minProfit + 1) for _ in range(n + 1)] for _ in range(len(group) + 1)]
        dp[0][0][0] = 1

        for i in range(1, len(group) + 1):
            members = group[i - 1]
            gain = profit[i - 1]
            for j in range(n + 1):
                for k in range(minProfit + 1):
                    dp[i][j][k] = dp[i - 1][j][k] # Do not commit this crime
                    if j >= members:
                        dp[i][j][k] += dp[i - 1][j - members][max(0, k - gain)] # Commit this crime
                        dp[i][j][k] %= MOD

        return sum(dp[len(group)][j][minProfit] for j in range(n + 1)) % MOD

Key Takeaways

We use a 3D dynamic programming approach to represent the state of our problem.
The iterative case considers whether to commit a crime or not, and updates the DP table accordingly.
The solution considers both the number of members and the profit generated, ensuring that the constraints are met.
The code returns the final count modulo (10^9 + 7), as requested by the problem statement.

10 Prerequisite LeetCode Problems

“Profitable Schemes” involves using dynamic programming (DP) and understanding combinatorics. It requires understanding of concepts related to two-dimensional DP, handling large numbers, and problem decomposition. Here are 10 problems to build these foundational skills:

“Climbing Stairs” (LeetCode Problem #70): This problem introduces the basics of dynamic programming, which is a key concept for “Profitable Schemes”.
“Unique Paths” (LeetCode Problem #62): This problem helps with understanding the counting aspect of dynamic programming.
“Coin Change” (LeetCode Problem #322): This problem introduces the concept of dynamic programming with state transition, which is also used in “Profitable Schemes”.
“0/1 Knapsack” (LeetCode Problem #416): This problem gives you practice on solving a classic dynamic programming problem, which has a similar structure to “Profitable Schemes”.
“Target Sum” (LeetCode Problem #494): This problem helps with understanding dynamic programming in the context of finding all possible combinations to reach a target.
“Partition Equal Subset Sum” (LeetCode Problem #416): This problem gives you practice on 2D dynamic programming similar to “Profitable Schemes”.
“Combination Sum IV” (LeetCode Problem #377): This problem provides an understanding of dynamic programming in the context of counting combinations.
“Number of Longest Increasing Subsequence” (LeetCode Problem #673): This problem helps to understand the concept of counting and selecting optimal sequences, which is used in “Profitable Schemes”.
“Super Egg Drop” (LeetCode Problem #887): This problem introduces the concept of dynamic programming in minimizing worst-case scenarios.
“Best Time to Buy and Sell Stock with Cooldown” (LeetCode Problem #309): This problem practices the idea of maximizing profit with certain constraints, similar to “Profitable Schemes”.

Problem Classification

Problem Statement: 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.

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

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Alternatively, if you’re working on a specific problem, you might ask something like:

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.