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What difficulty level is this interview question?

This is a hard difficulty Coding & Algorithms question, commonly asked during Onsite rounds at Google.

What role is this question designed for?

This question is commonly asked for Software Engineer candidates at Google during technical interviews.

Solve order-statistics and XOR-triplet problems

Q: Solve order-statistics and XOR-triplet problems

This multi-part question evaluates skills in order-statistics and efficient counting for array inversion-like problems, alongside bitwise manipulation and prefix-XOR reasoning for combinatorial triplet counting.

You are given two separate algorithmic problems.

Problem 1: Count smaller elements to the right

Given an integer array nums of length n, for each index i compute the number of indices j > i such that nums[j] < nums[i].

Input: nums (may contain duplicates and negative values)
Output: an array ans of length n , where ans[i] is the count of smaller elements strictly to the right of i
Constraints (typical interview scale): 1 ≤ n ≤ 10^5 , |nums[i]| ≤ 10^9
Expectation: design an appropriate data structure / approach to meet an efficient time bound (e.g., ~ O(n log n) ), and be ready to discuss complexity and edge cases.

Problem 2: Count triplets satisfying an XOR inequality on subarray XORs

You are given n integers a1, a2, ..., an (1 ≤ n ≤ 10^5, 0 ≤ ai ≤ 10^9). Define the function

F(i, j) = ai XOR a(i+1) XOR ... XOR aj for 1 ≤ i ≤ j ≤ n .

Count the number of index triplets (x, y, z) with 1 ≤ x ≤ y ≤ z ≤ n such that:

F(x, y) XOR F(y, z) > F(x, z) .

Notes:

XOR is the bitwise exclusive-or.
The output should be the count of valid triplets (an O(n^3) enumeration is not acceptable for the given constraints).
You may assume the answer fits in 64-bit signed integer unless otherwise specified.
Interview expectation: simplify using prefix XOR and reason about the inequality bit-by-bit to derive an efficient algorithm.

Problem 1: Count smaller elements to the right

Given an integer array nums of length n, for each index i compute the number of indices j > i such that nums[j] < nums[i].

Input: nums (may contain duplicates and negative values)

Output: an array ans of length n , where ans[i] is the count of smaller elements strictly to the right of i

Constraints (typical interview scale): 1 ≤ n ≤ 10^5 , |nums[i]| ≤ 10^9

Expectation: design an appropriate data structure / approach to meet an efficient time bound (e.g., ~ O(n log n) ), and be ready to discuss complexity and edge cases.

Problem 2: Count triplets satisfying an XOR inequality on subarray XORs

You are given n integers a1, a2, ..., an (1 ≤ n ≤ 10^5, 0 ≤ ai ≤ 10^9). Define the function

F(i, j) = ai XOR a(i+1) XOR ... XOR aj for 1 ≤ i ≤ j ≤ n .

Count the number of index triplets (x, y, z) with 1 ≤ x ≤ y ≤ z ≤ n such that:

F(x, y) XOR F(y, z) > F(x, z) .

Notes:

XOR is the bitwise exclusive-or.

The output should be the count of valid triplets (an O(n^3) enumeration is not acceptable for the given constraints).

You may assume the answer fits in 64-bit signed integer unless otherwise specified.

Interview expectation: simplify using prefix XOR and reason about the inequality bit-by-bit to derive an efficient algorithm.

Solve order-statistics and XOR-triplet problems

Quick Overview

Solve order-statistics and XOR-triplet problems

Problem 1: Count smaller elements to the right

Problem 2: Count triplets satisfying an XOR inequality on subarray XORs

Submit Your Answer to Earn 20XP

Solve order-statistics and XOR-triplet problems

Quick Overview

Solve order-statistics and XOR-triplet problems

Problem 1: Count smaller elements to the right

Problem 2: Count triplets satisfying an XOR inequality on subarray XORs

Submit Your Answer to Earn 20XP