Find the Value of the Partition
The given problem asks to partition the input array nums
into two nonempty arrays, nums1
and nums2
, in such a way that the value max(nums1)  min(nums2)
is minimized. Let’s understand how to do this by breaking it down into smaller steps.
Approach
Sort the Array: First, we need to sort the input array. This will help us in identifying the elements that would lead to the minimum value of the partition.
Partition the Array: Once sorted, we can partition the array into two parts at each index
i
and calculate the value for that partition. The idea is to minimize the absolute difference between the last element of the left partition (maximum ofnums1
) and the first element of the right partition (minimum ofnums2
).Calculate the Value: For each partition, calculate the value
max(nums1)  min(nums2)
. Keep track of the minimum value as you iterate through all the possible partitions.
Code Solution
Here’s the Python code:


Insights
 Sorting the array ensures that we are comparing adjacent elements, which would give us the smallest possible absolute difference.
 By partitioning at every index and comparing adjacent elements, we are guaranteed to find the minimum possible value for the partition.
 The overall time complexity of this solution is O(n log n) due to the sorting step, where n is the length of the
nums
array.
Identifying Problem Isomorphism
This problem can be mapped approximately to “Partition to K Equal Sum Subsets” (#698). The base concept is similar in both problems  we’re looking to partition an array based on certain conditions.
In “Partition to K Equal Sum Subsets”, we have to divide the array into K nonempty subsets such that the sum of all subsets is equal. This problem requires us to explore all possible partitions, similarly to your problem. However, while the conditions for partitioning are different (one is based on sums and the other is based on the max and min values), the methods to solve them could be similar.
In both problems, we need to consider all possible partitions and pick the best one based on the problem’s requirements. Therefore, while the specifics are not the same, the fundamental techniques of exploring partitions apply to both.
Here, we are partitioning the array such that max(nums1)  min(nums2) is minimized. We could first sort the array and then start partitioning the sorted array from the smallest value to the largest. Initially, we keep adding the smallest elements to nums1 and the largest elements to nums2. We stop as soon as max(nums1) > min(nums2) and then return the difference. It’s because, after this point, adding any more elements will only increase the difference.
This is an approximate mapping and the exact methodologies to solve these two problems might differ.
 Find the Value of the Partition
You are given a positive integer array nums.
Partition nums into two arrays, nums1 and nums2, such that:
Each element of the array nums belongs to either the array nums1 or the array nums2. Both arrays are nonempty. The value of the partition is minimized. The value of the partition is max(nums1)  min(nums2).
Here, max(nums1) denotes the maximum element of the array nums1, and min(nums2) denotes the minimum element of the array nums2. Return the integer denoting the value of such partition.
Example 1:
Input: nums = [1,3,2,4] Output: 1 Explanation: We can partition the array nums into nums1 = [1,2] and nums2 = [3,4].
 The maximum element of the array nums1 is equal to 2.
 The minimum element of the array nums2 is equal to 3. The value of the partition is 2  3 = 1. It can be proven that 1 is the minimum value out of all partitions. Example 2:
Input: nums = [100,1,10] Output: 9 Explanation: We can partition the array nums into nums1 = [10] and nums2 = [100,1].
 The maximum element of the array nums1 is equal to 10.
 The minimum element of the array nums2 is equal to 1. The value of the partition is 10  1 = 9. It can be proven that 9 is the minimum value out of all partitions.
Constraints:
2 <= nums.length <= 105 1 <= nums[i] <= 109


Problem Classification
The problem falls under the category of “Combinatorial Optimization” problems. It requires you to partition a set (an array in this case) into two nonempty subsets in such a way that a specific cost function (the difference between the maximum element of one subset and the minimum element of the other subset) is minimized.
“What” Components:
 Input: A positive integer array nums.
 Output: The minimum possible value of the partition defined by max(nums1)  min(nums2).
 Constraints: Each element of the array nums belongs to either the array nums1 or the array nums2, and both arrays must be nonempty.
This is an optimization problem since the objective is to find the partition that minimizes the given cost function. It also has elements of “partition problems,” a subset of combinatorial optimization problems, since it involves partitioning an array into two subsets. It may also be seen as a type of “search problem” since it requires searching over all possible partitions of the array to find the one that minimizes the cost function.
Language Agnostic Coding Drills
Dissection of the Code: The provided code includes a range of distinct concepts:
 Class and method definition in Python: This is the general structure of the code where a class and method are defined.
 List sorting: The code sorts the input list in ascending order.
 Array slicing: The code uses slicing to access elements from the sorted array.
 Iterating over an array with a loop: The code iterates over the array with a loop to find differences between consecutive elements.
 Use of list comprehension: The code uses list comprehension to create a list of differences between consecutive elements.
 Min function: The min function is used to find the smallest difference from the generated list.
Listing of Concepts (Ordered by Increasing Difficulty):
 Class and method definition in Python: This is a basic concept that provides structure to Python code. All Python programs should be comfortable with this.
 Array slicing: This is another basic concept, though it might require some understanding of array indexing.
 List sorting: This is a common operation in many algorithms. Python’s builtin sort function makes it straightforward.
 Iterating over an array with a loop: This is a fundamental concept that’s used in almost every piece of code.
 Use of list comprehension: This concept is a bit more advanced as it involves understanding Python’s syntax for concise, inline loops.
 Min function: The use of the min function is straightforward but understanding that it can be applied to a list comprehension directly can be considered slightly more advanced.
Problemsolving Approach:
The problem is solved using a greedy strategy. The insight is that to minimize the difference max(nums1)  min(nums2), one needs to find the smallest possible difference between any two elements in the array. This is because once the array is sorted, the smallest difference between any two elements will be between some element and the next one. Thus, one partition will have the element on one side of the difference, and the other partition will have the element on the other side. This guarantees that no other pair can produce a smaller difference.
In terms of the coding concepts, first, the method is defined within a class according to Python’s syntax. Then, the input list is sorted to enable efficient searching for the smallest difference. The code then iterates over the sorted list, using array slicing and a loop, to find the differences between consecutive elements. This is done using list comprehension to create a new list. Finally, the min function is applied to this list to find and return the smallest difference.
Targeted Drills in Python
Python Drills for each Concept:
 Class and method definition in Python:
class HelloWorld: def hello(self): print("Hello, World!") HelloWorld().hello()
 Array slicing:
array = [1, 2, 3, 4, 5] sliced = array[1:3] print(sliced) # Output: [2, 3]
 List sorting:
list_to_sort = [5, 1, 9, 3, 7] sorted_list = sorted(list_to_sort) print(sorted_list) # Output: [1, 3, 5, 7, 9]
 Iterating over an array with a loop:
array = [1, 2, 3, 4, 5] for i in array: print(i)
 Use of list comprehension:
array = [1, 2, 3, 4, 5] squared = [i ** 2 for i in array] print(squared) # Output: [1, 4, 9, 16, 25]
 Min function:
array = [5, 1, 9, 3, 7] min_val = min(array) print(min_val) # Output: 1
 Class and method definition in Python:
Drills for Specific Needs of the Problem:
 Finding difference between consecutive elements in a list:
array = [5, 1, 9, 3, 7] diffs = [array[i+1]  array[i] for i in range(len(array)  1)] print(diffs) # Output: [4, 8, 6, 4]
 Finding minimum difference between consecutive elements in a sorted list:
array = [1, 3, 5, 7, 9] min_diff = min(array[i+1]  array[i] for i in range(len(array)  1)) print(min_diff) # Output: 2
 Finding difference between consecutive elements in a list:
Merging these Drills for the Final Solution:
Start by defining a class and a method to house the solution as per Python’s syntax. Then, receive an input list. Use Python’s builtin sort function to sort the list. Now, with the sorted list, iterate over it to find the difference between every pair of consecutive elements. This is done using a combination of list comprehension and array slicing. Finally, apply the min function to the list of differences to find the smallest difference, which would be the result. Return this result as the final output of the method.
10 Prerequisite LeetCode Problems
For “2740. Find the Value of the Partition”, the following are a good preparation:
“215. Kth Largest Element in an Array”  This problem provides practice on finding kth largest element which can be utilized to find max and min values in array partitions.
“414. Third Maximum Number”  This problem is useful for understanding how to find maximum values in an array, which is important for determining max(nums1) in the main problem.
“88. Merge Sorted Array”  Understanding this problem will help you to understand how to merge and manage sorted arrays, which is a potential strategy for solving the main problem.
“209. Minimum Size Subarray Sum”  This problem can help you understand how to handle subarray problems, which is crucial for partitioning the main array.
“1480. Running Sum of 1d Array”  Solving this problem provides experience with the processing of array elements, which is a fundamental operation in the main problem.
“905. Sort Array By Parity”  The partition logic in this problem can be helpful for understanding how to divide the main array.
“75. Sort Colors”  This problem can help you understand sorting and partitioning on an array which is a key operation in the main problem.
“283. Move Zeroes”  This problem helps to understand how to arrange elements in an array, which could be useful when thinking about partitioning strategies.
“704. Binary Search”  Understanding binary search can help in searching elements in the sorted array after partition.
“242. Valid Anagram”  Although not directly related, this problem’s focus on managing array elements can provide useful background understanding.
These cover array operations, including sorting, partitioning, and finding maximum and minimum values, which will be helpful to solve problem “2740. Find the Value of the Partition”.
Problem Classification
Problem Statement:You are given a positive integer array nums. Partition nums into two arrays, nums1 and nums2, such that: Each element of the array nums belongs to either the array nums1 or the array nums2. Both arrays are nonempty. The value of the partition is minimized. The value of the partition is max(nums1)  min(nums2). Here, max(nums1) denotes the maximum element of the array nums1, and min(nums2) denotes the minimum element of the array nums2. Return the integer denoting the value of such partition.
Example 1:
Input: nums = [1,3,2,4] Output: 1 Explanation: We can partition the array nums into nums1 = [1,2] and nums2 = [3,4].
 The maximum element of the array nums1 is equal to 2.
 The minimum element of the array nums2 is equal to 3. The value of the partition is 2  3 = 1. It can be proven that 1 is the minimum value out of all partitions.
Example 2:
Input: nums = [100,1,10] Output: 9 Explanation: We can partition the array nums into nums1 = [10] and nums2 = [100,1].
 The maximum element of the array nums1 is equal to 10.
 The minimum element of the array nums2 is equal to 1. The value of the partition is 10  1 = 9. It can be proven that 9 is the minimum value out of all partitions.
Constraints:
2 <= nums.length <= 105 1 <= nums[i] <= 109
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?
Identifying Problem Isomorphism
Can you help me with finding the isomorphism for this problem?
Which problem does it map to on Leetcode for 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?
Identify Recursion Invariant
Is there an invariant during recursion in this problem?
Is invariant and invariant during recursion 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. Do not include the original problem. The response text is of the following format. First provide this as the first sentence: Here are 10 problems that use similar underlying concepts: