4Sum II
You can solve this problem by using a hashmap to store the sum of the elements from the first two arrays and then using that to count the number of tuples from the other two arrays that sum up to zero.
Here’s the code:


The time complexity of this solution is (O(n^2)), and the space complexity is (O(n^2)) because we store all the possible sums of the first two arrays.
Identifying Problem Isomorphism
“4Sum II” asks for the count of i, j, k, l from four lists A, B, C, D, where A[i] + B[j] + C[k] + D[l] is zero.
An approximate isomorphic problem to this is “Two Sum”. This problem requires you to return the indices of two numbers such that they add up to a target.
The isomorphism between these two problems lies in their shared goal: finding elements in arrays that satisfy a specific sum condition. In “4Sum II”, you are looking for four elements from different arrays that add up to zero, while in “Two Sum”, you are looking for two elements that add up to a target.
“Two Sum” is simpler due to it involving only a pair of numbers and a single array. “4Sum II” is more complex because it deals with four different arrays and four elements, thereby increasing the potential combinations to check.
10 Prerequisite LeetCode Problems
The problem “4Sum II” involves concepts of hash maps, array manipulation, and counting pairs in arrays. Here are 10 problems to prepare for this problem:
Basic Array Manipulation
 1. Two Sum: Given an array of integers nums and an integer target, return indices of the two numbers such that they add up to target.
 15. 3Sum: Given an integer array nums, return all the triplets [nums[i], nums[j], nums[k]] such that i != j, i != k, and j != k, and nums[i] + nums[j] + nums[k] == 0.
Hash Map Use
 136. Single Number: Given a nonempty array of integers nums, every element appears twice except for one. Find that single one.
 349. Intersection of Two Arrays: Given two arrays, write a function to compute their intersection.
Counting Pairs
 167. Two Sum II  Input array is sorted: Similar to the “Two Sum” problem, but the input array is sorted.
 532. Kdiff Pairs in an Array: Given an array of integers and an integer k, you need to find the number of unique kdiff pairs in the array.
Advanced Array/HashMap Manipulation
 18. 4Sum: Given an array nums of n integers, are there elements a, b, c, and d in nums such that a + b + c + d = target?
 454. 4Sum II: Similar to the problem you’re preparing for, but the condition is different.
Subarray Sum
 560. Subarray Sum Equals K: Given an array of integers and an integer k, you need to find the total number of continuous subarrays whose sum equals to k.
These cover how to manipulate arrays and use hash maps to track counts, which are the main techniques you’ll need to solve the “4Sum II” problem.
Clarification Questions
Clarification questions about the problem might include:
Duplicates: Are duplicate values allowed within or across the arrays? If yes, how should they be handled in the tuples?
Output Format: Is there a specific format required for the output, or is it only the count of tuples that is required?
Unique Tuples: Should the tuples be unique, or is it acceptable to count the same tuple more than once if it can be formed through different combinations?
Order of Elements: Does the order of elements within the tuples matter? If so, what is the specified order?
Handling of Zero: What if all the arrays consist of only zero? Should this be treated as a special case, or does the standard logic apply?
Constraint Clarification: Are the constraints on the array length and element values hard constraints, or could they potentially vary?
Negative Numbers: How should negative numbers be handled? The problem mentions them but doesn’t give specific instructions.
Performance Expectations: What are the performance expectations? Is there a maximum acceptable runtime or memory usage for the solution?
These questions can help to clarify ambiguities or uncertainties in the problem statement, ensuring that the solution is aligned with the expected requirements and constraints.
Identifying Problem Isomorphism
Isomorphism in the context of problemsolving refers to a structural similarity or onetoone correspondence between the components of two different problems. By identifying an isomorphism, you can apply knowledge and insights from one problem to understand or solve another.
Given the problem of finding the number of tuples (i, j, k, l) such that nums1[i] + nums2[j] + nums3[k] + nums4[l] == 0
, we can identify an isomorphism with the wellknown problem of 2Sum, 3Sum, or 4Sum, where the objective is to find elements from arrays that sum to a target number.
Isomorphism:
Elements and Constraints: The given problem deals with four arrays and seeks to find combinations that sum to zero. This resembles the 4Sum problem, where you are asked to find quadruples in an array that add up to a target number.
Problem Structure: Both problems involve iterating through elements, working with indexes, and forming combinations that meet specific criteria.
Solution Approach: A common approach to the 4Sum problem is to break it down into 2Sum problems, and this can be adapted to the given problem as well. By creating a mapping of the sums of pairs from
nums1
andnums2
, and then checking for corresponding pairs fromnums3
andnums4
that sum to the negative of that value, you can find the tuples that meet the criteria.Goal: In both problems, the goal is not merely to find individual elements but combinations of elements that meet specific criteria.
By recognizing this isomorphism, you can leverage existing knowledge and algorithms related to the 2Sum and 4Sum problems to design an efficient solution to the given problem. This approach of identifying similarities with known problems is a powerful tool in problemsolving and can lead to more efficient and insightful solutions.
Problem Analysis and Key Insights
The key insights from analyzing the problem statement are:
Multiple Arrays: There are four separate arrays to consider, and the solution must involve elements from each array.
Equality Condition: The problem is defined by an equality condition that must be met (sum of selected elements must equal zero), which can guide the search or matching process.
Tuple Formation: Solutions must be found in the form of tuples, indicating that multiple elements must be selected and their relationships considered.
Combinatorial Search: The problem requires examining a large number of possible combinations, leading to potential computational complexity.
Counting Objective: The ultimate goal is not to list or use the tuples but merely to count how many meet the given conditions, which might allow for optimizations.
Bounded Values: The constraints on the length of arrays and the values within them set clear bounds on the problem space, possibly aiding in the formulation of an efficient solution.
Equal Length of Arrays: The fact that all arrays are of equal length might simplify the iteration or search process across arrays.
These insights help in understanding the core aspects of the problem and provide a foundation for selecting appropriate problemsolving strategies and techniques.
Problem Boundary
What is the scope of this problem?
How to establish the boundary of this problem?
Problem Classification
The given problem falls into the domain of Combinatorial Mathematics and Search Algorithms. It requires finding specific tuples from given sets that satisfy a particular condition.
‘What’ Components
 Input Arrays: Four integer arrays
nums1
,nums2
,nums3
, andnums4
of equal lengthn
.  Constraints: Constraints on the values of the integers within the arrays and the length of the arrays.
 Tuples: A need to identify tuples
(i, j, k, l)
that adhere to specific conditions.  Condition: The condition is that the sum of elements at the corresponding indices in the four arrays must equal zero.
 Output: The number of such tuples that satisfy the given condition.
Classification of the Problem
 Type: This is a Counting Problem, as the main objective is to count the number of tuples that meet a certain condition.
 Search Strategy: It involves a Search Problem, where you have to search through all possible combinations to find those that satisfy the given equation.
 Constraints Handling: Constraints are given on the values of the integers and the length of the arrays, leading to a bounded search space.
The problem requires the identification and counting of specific combinations (tuples) from four given sets (arrays) that satisfy a mathematical condition (sum equals zero). Therefore, it’s a counting problem that involves searching through a combinatorial space. The constraints on the values and lengths of the arrays limit the search space, making it a finite problem that likely involves both combinatorial mathematics and search algorithms in its solution.
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?
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 following piece of complex software code.
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.