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Count the Number of Fair Pairs

This README provides a step-by-step breakdown of the solution to the problem of counting fair pairs in an array. We'll cover the approach for each language: C++, Java, JavaScript, Python, and Go.

Problem Summary

Given a 0-indexed integer array nums and two integers lower and upper, we need to count the number of pairs (i, j) such that:

  • ( 0 \leq i < j < n ) (i.e., ( i ) should be less than ( j ))
  • ( \text{lower} \leq \text{nums}[i] + \text{nums}[j] \leq \text{upper} ) (i.e., the sum of the elements should lie between lower and upper)

Approach Overview

To solve this problem efficiently:

  1. Sort the Array: We start by sorting the array nums to simplify the process of finding pairs.
  2. Iterate and Search: For each element nums[i], calculate the range of values that nums[j] should lie within to meet the conditions. Use binary search to quickly find the count of valid j indices for each i.
  3. Binary Search Utility Functions: Implement functions for binary search that will find the lower and upper bounds of the range that nums[j] can fall into for a valid pair.

Complexity Analysis

  • Time Complexity: (O(n \log n)) due to sorting and binary search for each element.
  • Space Complexity: (O(1)) (or (O(n)) if the sorted array copy is counted as extra space).

C++ Solution Explanation

Step 1: Sort the Array

  • Start by sorting nums. This enables us to efficiently find the range of valid pairs for each element.

Step 2: Iterate Over nums

  • For each element at index i, calculate the minimum and maximum values (minVal and maxVal) that nums[j] (where ( j > i )) must satisfy.

Step 3: Find Lower and Upper Bounds Using Binary Search

  • Use lower_bound to find the smallest index j where nums[j] >= minVal.
  • Use upper_bound to find the smallest index j where nums[j] > maxVal.

Step 4: Count Valid Pairs

  • For each i, add the number of valid pairs (i, j) by subtracting the lower bound index from the upper bound index.

Java Solution Explanation

Step 1: Sort the Array

  • Start by sorting nums to make the search for valid pairs faster.

Step 2: Define Binary Search Functions

  • Implement custom lowerBound and upperBound methods.
    • lowerBound finds the first index j where nums[j] >= minVal.
    • upperBound finds the first index j where nums[j] > maxVal.

Step 3: Iterate Over nums

  • For each index i, calculate minVal and maxVal based on the range [lower, upper].

Step 4: Use Binary Search to Count Valid Pairs

  • For each i, find the range of valid indices j using the custom binary search functions.
  • Add the difference between upper bound and lower bound indices to the total count.

JavaScript Solution Explanation

Step 1: Sort the Array

  • Sort nums to streamline finding valid pairs later.

Step 2: Define Binary Search Helper Functions

  • Create lowerBound and upperBound helper functions.
    • lowerBound finds the first index j where nums[j] >= minVal.
    • upperBound finds the first index j where nums[j] > maxVal.

Step 3: Iterate Over the Array

  • For each index i, determine minVal and maxVal for the current element as the required bounds for nums[j].

Step 4: Count Valid Pairs with Binary Search

  • For each i, use the helper functions to find the valid range of indices for j.
  • Increment the count based on the difference between the indices found by the helper functions.

Python Solution Explanation

Step 1: Sort the Array

  • Begin by sorting nums to enable efficient range searching.

Step 2: Define Binary Search Functions

  • Use custom functions lower_bound and upper_bound.
    • lower_bound finds the first position where nums[j] >= minVal.
    • upper_bound finds the first position where nums[j] > maxVal.

Step 3: Iterate and Calculate the Range

  • For each element nums[i], calculate minVal and maxVal for valid pairs.

Step 4: Count Pairs in the Range

  • For each i, determine the valid range for j indices and count the pairs by subtracting the lower bound index from the upper bound index.

Go Solution Explanation

Step 1: Sort the Array

  • Start by sorting nums to make it easier to locate valid pairs.

Step 2: Define Binary Search Functions

  • Implement lowerBound and upperBound functions.
    • lowerBound finds the index where nums[j] >= minVal.
    • upperBound finds the index where nums[j] > maxVal.

Step 3: Iterate and Set Bounds

  • For each element in nums, calculate minVal and maxVal based on the lower and upper constraints.

Step 4: Count Valid Pairs Using Range

  • For each i, use lowerBound and upperBound to get the range of valid j indices.
  • Add the count of valid pairs to the total based on the difference in indices.

Summary

The main strategy is to:

  1. Sort the array.
  2. For each index i, find the valid range for j using binary search functions.
  3. Count the number of pairs in this range.

Each language implements the same approach, with minor variations due to syntax and standard library differences.

This structured approach ensures that the solution is both efficient and easy to understand across different programming languages. Each step in the implementation is designed to make the most of binary search and sorting for an optimal solution.