Given an array of integers, write a function to optimize the calculation of the number of inversions in the array, where an inversion is defined as a pair of indices (i, j) such that i arr[j]. Additionally, provide a follow-up solution that improves the time complexity of your initial implementation. The function should return the total count of inversions.

The problem is to count the number of inversions in an array, where an inversion is defined as a pair of indices (i, j) such that i arr[j]. The underlying pattern here is a sorting problem because inversions are inherently related to the order of elements. A naive approach would involve a nested loop, resulting in O(n^2) time complexity, which would be inefficient for large inputs. Instead, we can utilize a modified merge sort algorithm that counts inversions while sorting, achieving O(n log n) time complexity. This approach benefits from the divide-and-conquer nature of merge sort, where inversions can be counted during the merge step. 1. **Merge Sort Algorithm**: We'll implement a merge sort that counts inversions during the merging step. 2. **Counting Inversions**: While merging two sorted halves of the array, if an element from the right half is less than an element from the left half, it means all remaining elements in the left half form an inversion with this element. 3. **Steps**: - Divide the array into two halves. - Recursively sort each half while counting inversions. - Merge the sorted halves and count additional inversions. 4. **Example**: For the array [2, 3, 8, 6, 1], we would split it into [2, 3] and [8, 6, 1], sort each half, and during the merge step, we would identify that 8 is greater than both 6 and 1, thus counting those inversions. Here is the complete Python solution using the modified merge sort: ```python def merge_and_count(arr, temp_arr, left, mid, right): i = left # Starting index for left subarray j = mid + 1 # Starting index for right subarray k = left # Starting index to be sorted inv_count = 0 while i <= mid and j <= right: if arr[i] <= arr[j]: temp_arr[k] = arr[i] i += 1 else: temp_arr[k] = arr[j] inv_count += (mid - i + 1) # Count inversions j += 1 k += 1 while i <= mid: temp_arr[k] = arr[i] i += 1 k += 1 while j <= right: temp_arr[k] = arr[j] j += 1 k += 1 for i in range(left, right + 1): arr[i] = temp_arr[i] return inv_count def merge_sort_and_count(arr, temp_arr, left, right): inv_count = 0 if left < right: mid = (left + right) // 2 inv_count += merge_sort_and_count(arr, temp_arr, left, mid) inv_count += merge_sort_and_count(arr, temp_arr, mid + 1, right) inv_count += merge_and_count(arr, temp_arr, left, mid, right) return inv_count def count_inversions(arr): temp_arr = [0] * len(arr) return merge_sort_and_count(arr, temp_arr, 0, len(arr) - 1) ``` You can call `count_inversions([2, 3, 8, 6, 1])` to get the total count of inversions. The time complexity of the above solution is O(n log n) due to the merge sort's divide-and-conquer approach, where each merge operation takes O(n) time and there are log n levels of recursion. The space complexity is O(n) because of the temporary array used for merging. For inputs with a maximum array length of 10^5, this optimized solution is efficient and feasible within time limits. 1. **Empty Array**: If the input array is empty, the function should return 0, as there are no elements to form inversions. 2. **Single Element**: An array with a single element also has 0 inversions. 3. **All Elements Same**: For an array where all elements are the same (e.g., [1, 1, 1]), the function should return 0, as no inversions exist. 4. **Sorted Array**: For a sorted array (e.g., [1, 2, 3, 4]), the function should again return 0. 5. **Reverse Sorted Array**: For an array sorted in reverse (e.g., [4, 3, 2, 1]), the function should return the maximum number of inversions, which is n(n-1)/2 for n elements.

Optimize inversion for with follow-up | Lyft Machine Learning Engineer Interview

Optimize inversion for with follow-up

Last updated: September 11, 2025

Quick Overview

Given an array of integers, write a function to optimize the calculation of the number of inversions in the array, where an inversion is defined as a pair of indices (i, j) such that i < j and arr[i] > arr[j]. Additionally, provide a follow-up solution that improves the time complexity of your initial implementation. The function should return the total count of inversions.

Lyft

Coding & Algorithms

Machine Learning Engineer

Lyft

September 11, 2025

Machine Learning Engineer

Onsite

Coding & Algorithms

Medium

510

2,511 solved

Coding interviews at Lyft focus on problem-solving approach as much as the final solution. The interviewer wants to see you break down the problem, consider edge cases, and optimize iteratively. Communication throughout the process is key.

What the Interviewer Expects

Recognize the underlying problem pattern (sliding window, two pointers, BFS/DFS, etc.)
Discuss multiple approaches and trade-offs before coding
Implement an optimal solution with clean, production-quality code
Handle all edge cases including boundary conditions and invalid input
Optimize both time and space complexity with clear justification
Test your solution systematically with well-chosen examples

Key Topics to Cover

Tree structures and recursion

Data structure selection and trade-offs

Sorting and searching

Dynamic programming and memoization

Edge cases and input validation

How to Approach This

Clarify input constraints and edge cases before writing code.
Walk through your approach verbally and confirm with the interviewer before coding.
Start with a brute force solution, then optimize. Mention time and space complexity.
Test your solution with examples, including edge cases like empty input or duplicates.
Consider common patterns: sliding window, two pointers, hash map, BFS/DFS, dynamic programming.

Possible Follow-up Questions

What if the input doesn't fit in memory?
Can you solve this in a single pass?
How would you modify your solution to handle streaming input?

Sharpen Your Skills on Codemia

Practice similar problems with our interactive workspace, get AI feedback, and track your progress.

Practice DSA Problems

Sample Answer

Problem Analysis

The problem is to count the number of inversions in an array, where an inversion is defined as a pair of indices (i, j) such that i < j and arr[i] > arr[j]. The underlying pattern here is a sorting pr...

Approach

Merge Sort Algorithm: We'll implement a merge sort that counts inversions during the merging step.
Counting Inversions: While merging two sorted halves of the array, if an element from ...

Submit Your Answer

Markdown supported

Lyft Machine Learning Engineer Interview Guide

Interview process, tips, and preparation timeline

Optimize inversion for with follow-up

Lyft

What the Interviewer Expects

Key Topics to Cover

How to Approach This

Possible Follow-up Questions

Sharpen Your Skills on Codemia

Sample Answer

Problem Analysis

Approach

Submit Your Answer

Lyft Machine Learning Engineer Interview Guide

Related Questions

Binary Search on graph

Count kth largest element in array

Find connected components in string

Merge Three Sorted strings

Binary Search on string