Number of Pairs Satisfying Inequality

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from sortedcontainers import SortedList

class Solution:
    def numberOfPairs(self, nums1: List[int], nums2: List[int], diff: int) -> int:
        sorted_list = SortedList()
        count = 0

        for n1, n2 in zip(nums1, nums2):
            count += sorted_list.bisect_right(n1 - n2 + diff)
            sorted_list.add(n1 - n2)

        return count

10 Prerequisite LeetCode Problems

“2426. Number of Pairs Satisfying Inequality” involves elements of two-pointer technique, binary search, and sorting or array manipulation. Here are some problems to understand these concepts better:

LeetCode 167. Two Sum II - Input array is sorted
- This problem helps to understand the basic idea of using two-pointer technique to find pairs in a sorted array.
LeetCode 345. Reverse Vowels of a String
- This problem introduces the concept of the two-pointer technique, which is important for the main problem.
LeetCode 15. 3Sum
- This problem involves finding triplets that sum to zero and uses the two-pointer approach. This problem is slightly more complex and will further improve your understanding of the two-pointer technique.
LeetCode 704. Binary Search
- This problem will help you master the basics of binary search.
LeetCode 349. Intersection of Two Arrays
- This problem requires understanding of set operations and binary search which could be beneficial for the main problem.
LeetCode 977. Squares of a Sorted Array
- This problem is about sorting and square of numbers which can help you practice array manipulations.
LeetCode 167. Two Sum II - Input array is sorted
- This problem gives you practice with the two-pointer technique in a simple context.
LeetCode 75. Sort Colors
- This problem involves understanding of sorting and manipulating arrays to arrange elements.
LeetCode 33. Search in Rotated Sorted Array
- This problem is a bit more advanced application of binary search, which will give you more confidence with this method.
LeetCode 34. Find First and Last Position of Element in Sorted Array

This problem involves binary search and deals with finding specific elements in a sorted array, which will be helpful in the main problem.

By working through these problems, you should get a solid understanding of two-pointer technique, binary search, and array manipulation which are likely important concepts for the problem “2426. Number of Pairs Satisfying Inequality”.

“2426. Number of Pairs Satisfying Inequality” involves working with arrays and finding pairs of elements that satisfy a certain inequality condition. This typically involves using a two-pointer approach, sorting, or sometimes hashing.

Here are 10 problems of lesser complexity that could help you prepare for this problem:

18. 4Sum: This problem is another extension of the two sum problem and requires finding all unique quadruplets that add up to a target.
454. 4Sum II: This problem involves finding the count of quadruplets across four arrays that add up to zero.
611. Valid Triangle Number: This problem requires counting the number of triplets that can form a valid triangle, which involves an inequality condition.
217. Contains Duplicate: This problem requires finding if an array contains any duplicates, which can be solved by sorting or hashing.
283. Move Zeroes: This problem involves rearranging elements in an array, which can be done using a two-pointer approach.
209. Minimum Size Subarray Sum: This problem requires finding a contiguous subarray with a sum at least as large as a target, which can be solved using a two-pointer approach.

These cover sorting, two-pointer approach, and hashing, which are useful for “2426. Number of Pairs Satisfying Inequality”.

Problem Classification

Problem Statement:You are given two 0-indexed integer arrays nums1 and nums2, each of size n, and an integer diff. Find the number of pairs (i, j) such that:

0 <= i < j <= n - 1 and nums1[i] - nums1[j] <= nums2[i] - nums2[j] + diff. Return the number of pairs that satisfy the conditions.

Example 1:

Input: nums1 = [3,2,5], nums2 = [2,2,1], diff = 1 Output: 3 Explanation: There are 3 pairs that satisfy the conditions:

i = 0, j = 1: 3 - 2 <= 2 - 2 + 1. Since i < j and 1 <= 1, this pair satisfies the conditions.
i = 0, j = 2: 3 - 5 <= 2 - 1 + 1. Since i < j and -2 <= 2, this pair satisfies the conditions.
i = 1, j = 2: 2 - 5 <= 2 - 1 + 1. Since i < j and -3 <= 2, this pair satisfies the conditions. Therefore, we return 3. Example 2:

Input: nums1 = [3,-1], nums2 = [-2,2], diff = -1 Output: 0 Explanation: Since there does not exist any pair that satisfies the conditions, we return 0.

Constraints:

n == nums1.length == nums2.length 2 <= n <= 105 -104 <= nums1[i], nums2[i] <= 104 -104 <= diff <= 104

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?

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.