Maximum Strength of a Group


10 Prerequisite LeetCode Problems
For this, the following are a good preparation:
“152. Maximum Product Subarray”  This problem helps understand how to calculate maximum product in a subset of elements.
“198. House Robber”  This dynamic programming problem deals with selecting nonadjacent elements to maximize the result, a related concept to forming the maximum strength group.
“238. Product of Array Except Self”  A useful problem to understand handling products in an array.
“628. Maximum Product of Three Numbers”  This problem also revolves around the concept of finding maximum product but from three numbers.
“53. Maximum Subarray”  Understanding this problem will help in understanding subarrays and their properties which can be useful.
“121. Best Time to Buy and Sell Stock”  This problem helps understand the concept of optimizing for a maximum outcome, similar to maximizing the group’s strength.
“322. Coin Change”  Although this problem deals with a different context (coins), it provides good practice for dynamic programming, which can be used to solve the given problem.
“416. Partition Equal Subset Sum”  This problem practices forming subsets from an array, which is a required skill for the given problem.
“494. Target Sum”  This problem, like the others, helps in understanding how to form subsets and their sums, similar to forming groups and their strength.
“1049. Last Stone Weight II”  This problem involves handling subsets and their sums in a slightly different context, but still provides a good practice.
These cover dynamic programming, handling products in arrays, understanding subarrays, and forming subsets with certain properties, which are helpful for solving the “Maximum Strength of a Group” problem.


Problem Classification
Problem Statement:You are given a 0indexed integer array nums representing the score of students in an exam. The teacher would like to form one nonempty group of students with maximal strength, where the strength of a group of students of indices i0, i1, i2, … , ik is defined as nums[i0] * nums[i1] * nums[i2] * … * nums[ik].
Return the maximum strength of a group the teacher can create.
Example 1:
Input: nums = [3,1,5,2,5,9] Output: 1350 Explanation: One way to form a group of maximal strength is to group the students at indices [0,2,3,4,5]. Their strength is 3 * (5) * 2 * 5 * (9) = 1350, which we can show is optimal.
Example 2:
Input: nums = [4,5,4] Output: 20 Explanation: Group the students at indices [0, 1] . Then, we’ll have a resulting strength of 20. We cannot achieve greater strength.
Constraints:
1 <= nums.length <= 13 9 <= nums[i] <= 9
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.
Clarification Questions
What are the clarification questions we can ask about this problem?
Problem Analysis and Key Insights
What are the key insights from analyzing the problem statement?
Problem Boundary
What is the scope of this problem?
How to establish the boundary of this problem?
Distilling the Problem to Its Core Elements
Can you identify the fundamental concept or principle this problem is based upon? Please explain. What is the simplest way you would describe this problem to someone unfamiliar with the subject? What is the core problem we are trying to solve? Can we simplify the problem statement? Can you break down the problem into its key components? What is the minimal set of operations we need to perform to solve this problem?
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?
Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific realworld 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.
Provide names by categorizing these cases
What are the edge cases?
How to visualize these cases?
What are the key insights from analyzing the different cases?
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?
Simple Explanation
Can you explain this problem in simple terms or like you would explain to a nontechnical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?
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 ProblemSolving Approach from the Problem Statement
Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method. How can I recognize these properties by drawing tables or diagrams?
How did you infer from the problem statement that this problem can be solved using ?
Simple Explanation of the Proof
I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?
Stepwise Refinement
Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the highlevel 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
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.
Identify Invariant
What is the invariant in this problem?
Identify Loop Invariant
What is the loop invariant in this problem?
Is invariant and loop invariant the same for this problem?
Thought Process
Can you explain the basic thought process and steps involved in solving this type of problem?
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.
Establishing Preconditions and Postconditions
Parameters:
 What are the inputs to the method?
 What types are these parameters?
 What do these parameters represent in the context of the problem?
Preconditions:
 Before this method is called, what must be true about the state of the program or the values of the parameters?
 Are there any constraints on the input parameters?
 Is there a specific state that the program or some part of it must be in?
Method Functionality:
 What is this method expected to do?
 How does it interact with the inputs and the current state of the program?
Postconditions:
 After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
 What does the return value represent or indicate?
 What side effects, if any, does the method have?
Error Handling:
 How does the method respond if the preconditions are not met?
 Does it throw an exception, return a special value, or do something else?
Problem Decomposition
Problem Understanding:
 Can you explain the problem in your own words? What are the key components and requirements?
Initial Breakdown:
 Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?
Subproblem Refinement:
 For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?
Task Identification:
 Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?
Task Abstraction:
 For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?
Method Naming:
 Can you give each task a simple, descriptive name that makes its purpose clear?
Subproblem Interactions:
 How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?
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?
Code Explanation and Design Decisions
Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?
If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.
If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?
Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.
Coding Constructs
Consider the code for the solution of this problem.
What are the highlevel problemsolving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a nonprogrammer, 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 languageagnostic 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 problemsolving 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 stepbystep 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 Pythonbased 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 problemspecific 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.
Q&A
Similar Problems
Can you suggest 10 problems from LeetCode that require similar problemsolving strategies or use similar underlying concepts as the problem we’ve just solved? These problems can be from any domain or topic, but they should involve similar steps or techniques in the solution process. Also, please briefly explain why you consider each of these problems to be related to our original problem. The response text is of the following format:
Here are 10 problems that use similar underlying concepts:
You are given a 0indexed integer array nums representing the score of students in an exam. The teacher would like to form one nonempty group of students with maximal strength, where the strength of a group of students of indices i0, i1, i2, … , ik is defined as nums[i0] * nums[i1] * nums[i2] * … * nums[ik].
Return the maximum strength of a group the teacher can create.
Example 1:
Input: nums = [3,1,5,2,5,9] Output: 1350 Explanation: One way to form a group of maximal strength is to group the students at indices [0,2,3,4,5]. Their strength is 3 * (5) * 2 * 5 * (9) = 1350, which we can show is optimal. Example 2:
Input: nums = [4,5,4] Output: 20 Explanation: Group the students at indices [0, 1] . Then, we’ll have a resulting strength of 20. We cannot achieve greater strength.
Constraints:
1 <= nums.length <= 13 9 <= nums[i] <= 9
Language Agnostic Coding Drills
Variable Definitions: Understand how to define variables and assign them values.
List Manipulation: Familiarize yourself with lists, which are ordered, mutable collections of items. Also, learn how to append items to a list.
Loops: Grasp how to use a
for
loop to iterate over a sequence of elements.Conditional Statements: Learn to use
if
elif
else
statements to perform different actions based on certain conditions.Functions: Learn how to use builtin functions. The code uses the
max()
andlen()
functions.Boolean Expressions: Understand how to use boolean expressions in the context of conditionals.
Classes and Methods: Learn how to define classes and methods within those classes.
Bitwise Operations: The code uses a bitwise AND operation (
&
) to check whether the length of the listnegative
is odd.
Targeted Drills in Python
 Variable Definitions:


 List Manipulation:


 Loops:


 Conditional Statements:


 Functions:


 Boolean Expressions:


 Classes and Methods:


 Bitwise Operations:


The given Python code utilizes the concepts described in the drills to solve a specific problem. Let’s see how these concepts can be used to build up to the final solution:
Variable Definitions: Define variables to hold boolean flags and lists for storing different categories of numbers.
1 2 3
contains_zero = False positive = [] negative = []
Loops & Conditional Statements: Iterate over the given list of numbers, categorize them into positive, negative, and zero, and store them in the respective lists.
1 2 3 4 5 6 7
for num in nums: if num > 0: positive.append(num) elif num < 0: negative.append(num) else: contains_zero = True
Functions & Conditional Statements: Perform different actions based on certain conditions. In this case, checking if the length of the negative list is odd. If yes, and there are other elements in the list, remove the maximum negative number. If no other negative elements are present, check if positive numbers exist or if a zero is present, otherwise return the maximum negative number.
1 2 3 4 5 6 7 8 9
if len(negative) & 1: if len(negative) > 1: negative.remove(max(negative)) elif positive: return prod(positive) elif contains_zero: return 0 else: return max(negative)
Functions & Conditional Statements: Based on the existence of positive and negative numbers, return the appropriate product.
1 2 3 4
if not (negative or positive): return 0 return prod(negative) * prod(positive)
In summary, the program breaks down the input into positive, negative and zero categories, and performs actions based on certain conditions to return the maximum possible product. This solution uses the fundamental concepts of variable definitions, list manipulations, loops, conditional statements, and functions, combined with bitwise operations for determining the length parity.
The prod()
function is a mathematical function which is used to compute the product of a sequence of numbers. The sequence can be any kind of iterable.
In Python, the prod()
function is a part of the math
module.
Here is how you can use it:


In the above example, prod()
computes the product of the numbers in the list, which is 1 * 2 * 3 * 4 * 5 = 120
.
If the prod()
function is not available (for example, if you’re using a Python version older than 3.8), you can use functools.reduce()
with operator.mul
instead:


This does the same thing: it multiplies all the numbers in the list together. The reduce()
function applies the mul()
function (which multiplies its arguments) to the first two items in the list, then to the result and the next item, and so on, effectively computing the product of all the items in the list.


Problem Classification
This problem is in the domain of Computer Science, particularly focusing on the subdomain of Algorithms, Data Structures, and Optimization.
Here are the ‘What’ components of the problem:
You have an integer array “nums” which represents the score of students in an exam.
The task is to form a nonempty group of students with the maximum strength.
The strength of a group of students is the product of their scores.
Based on these components, this problem can be further classified as an Array Processing and Optimization problem. The goal is to identify a subset of the array such that the product of its elements is maximized.
Here’s why the problem is classified as such:
Array Processing: The problem involves handling an array, accessing its elements, and determining a subset of these elements.
Optimization: The main goal of the problem is to maximize the product of the scores, which falls into the category of optimization problems. You are tasked with finding the best possible solution from a set of possible solutions.
This problem may also involve sorting or dynamic programming depending upon the method used to solve it. The aim is to optimize the grouping of students based on their scores to achieve the maximum product, indicating the possible usage of these algorithms.
Language Agnostic Coding Drills
Distinct Concepts:
a) Variable Assignment and Initialization: The code begins with the initialization of multiple variables.
b) List Comprehension/Array Manipulation: The code divides the given array into positive, negative numbers, and zeros.
c) Conditional Statements: The code includes various ifelse conditions to determine how to handle the negative numbers list.
d) Builtin Functions: The code uses builtin functions like
prod()
andmax()
to perform calculations on the list.e) Bitwise Operation: The code uses a bitwise operation to determine if the length of an array is odd.
f) Return Statement: The code uses a return statement to provide the final result.
Increasing Difficulty Order:
a) Variable Assignment and Initialization: This is a basic coding concept, necessary for almost all programs. Difficulty Level: Easy.
b) List Comprehension/Array Manipulation: Requires understanding of how to access and manipulate array data. Difficulty Level: Easy to Medium.
c) Conditional Statements: Necessary for controlling the flow of the program, might require deeper understanding of the problem. Difficulty Level: Medium.
d) Return Statement: Basic concept to return the output from a function. Difficulty Level: Easy.
e) Builtin Functions: Need understanding of various builtin functions and their usage. Difficulty Level: Medium.
f) Bitwise Operation: Requires understanding of binary numbers and how bitwise operations work. Difficulty Level: Hard.
Problemsolving Approach:
a) The first step is to initialize variables and categorize the input array into three separate lists based on their sign.
b) If there’s an odd number of negative numbers and it’s more than one, remove the maximum negative number from the list to maximize the product. This is because the maximum negative number is closest to zero and removing it will have the least impact on the product.
c) If there’s an odd number of negative numbers and no positive numbers or zeros exist, return the maximum negative number. It’s the only group that can be created, so its strength is the maximum.
d) If there are no positive or negative numbers, return 0.
e) If none of the above conditions are met, return the product of all the numbers in the positive and negative lists.
Each of these coding drills contributes to the final solution by dividing the problem into smaller, manageable tasks. This allows for clearer, easier debugging, and more understandable code. Each drill handles a specific task that works towards achieving the final solution.
Targeted Drills in Python
Distinct Concepts:
a) Variable Assignment and Initialization: The code begins with the initialization of multiple variables.
b) List Comprehension/Array Manipulation: The code divides the given array into positive, negative numbers, and zeros.
c) Conditional Statements: The code includes various ifelse conditions to determine how to handle the negative numbers list.
d) Builtin Functions: The code uses builtin functions like
prod()
andmax()
to perform calculations on the list.e) Bitwise Operation: The code uses a bitwise operation to determine if the length of an array is odd.
f) Return Statement: The code uses a return statement to provide the final result.
Increasing Difficulty Order:
a) Variable Assignment and Initialization: This is a basic coding concept, necessary for almost all programs. Difficulty Level: Easy.
b) List Comprehension/Array Manipulation: Requires understanding of how to access and manipulate array data. Difficulty Level: Easy to Medium.
c) Conditional Statements: Necessary for controlling the flow of the program, might require deeper understanding of the problem. Difficulty Level: Medium.
d) Return Statement: Basic concept to return the output from a function. Difficulty Level: Easy.
e) Builtin Functions: Need understanding of various builtin functions and their usage. Difficulty Level: Medium.
f) Bitwise Operation: Requires understanding of binary numbers and how bitwise operations work. Difficulty Level: Hard.
Problemsolving Approach:
a) The first step is to initialize variables and categorize the input array into three separate lists based on their sign.
b) If there’s an odd number of negative numbers and it’s more than one, remove the maximum negative number from the list to maximize the product. This is because the maximum negative number is closest to zero and removing it will have the least impact on the product.
c) If there’s an odd number of negative numbers and no positive numbers or zeros exist, return the maximum negative number. It’s the only group that can be created, so its strength is the maximum.
d) If there are no positive or negative numbers, return 0.
e) If none of the above conditions are met, return the product of all the numbers in the positive and negative lists.
Each of these coding drills contributes to the final solution by dividing the problem into smaller, manageable tasks. This allows for clearer, easier debugging, and more understandable code. Each drill handles a specific task that works towards achieving the final solution.