Number of Unique XOR Triplets II

To solve Number of Unique XOR Triplets II, iterate over all valid triplets with i <= j <= k and compute their XOR. Store results in a set to automatically track unique values. This ensures the output counts only distinct XOR outcomes while handling array length and value constraints efficiently.

Problem Statement

Given an integer array nums, a XOR triplet consists of three elements nums[i], nums[j], nums[k] such that i <= j <= k. Your task is to compute the XOR of each such triplet and determine how many unique results exist.

Return the total count of distinct XOR values generated from all possible triplets in the array. The input array can be of moderate length, so consider an approach that balances correctness with feasible computation time.

Examples

Example 1

Input: nums = [1,3]

Output: 2

The possible XOR triplet values are: The unique XOR values are {1, 3} . Thus, the output is 2.

Example 2

Input: nums = [6,7,8,9]

Output: 4

The possible XOR triplet values are {6, 7, 8, 9} . Thus, the output is 4.

Constraints

• 1 <= nums.length <= 1500
• 1 <= nums[i] <= 1500

Solution Approach

Brute Force Enumeration

Loop through all indices i, j, k with i <= j <= k and calculate nums[i] XOR nums[j] XOR nums[k]. Store each result in a set to filter unique values and return the set size.

Prefix XOR Optimization

Use a prefix XOR array to quickly compute XOR for ranges, reducing repeated computation. For each triplet, derive the XOR using prefix values and maintain a set of unique results.

Bit Manipulation Insight

Recognize patterns where certain XOR results cannot exceed the maximum element value multiplied in combinations. Use bit-level observations to prune unnecessary computations when possible.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity is O(n^3) for brute force, but using prefix XOR can reduce redundant calculations; space complexity is O(n^3) in the worst case for storing unique XOR values, but practically limited by the range of XOR results.

What Interviewers Usually Probe

• Focus on handling the i <= j <= k condition correctly without missing triplets.
• Expect recognition of how XOR properties affect uniqueness and maximum value constraints.
• Watch for optimized approaches that reduce redundant XOR computations using prefix arrays or bit tricks.

Common Pitfalls or Variants

Common pitfalls

• Incorrectly counting duplicate XOR results due to not using a set.
• Misordering indices violating i <= j <= k which leads to wrong triplet computation.
• Assuming XOR results grow linearly with array values, leading to unnecessary computation or pruning errors.

Follow-up variants

• Count unique XOR quadruplets instead of triplets for larger combinations.
• Return the list of unique XOR values instead of just the count.
• Compute XOR triplets under modular constraints to test arithmetic variations.

FAQ

What is the best approach to count unique XOR triplets efficiently?

Use a set to store XOR results while iterating over all triplets, optionally applying prefix XOR to avoid repeated computation.

Does the problem Number of Unique XOR Triplets II require i < j < k or i <= j <= k?

The problem specifies i <= j <= k, so triplets with repeated indices are valid and must be included in XOR calculations.

Can we prune triplet combinations using maximum element value?

Yes, observing the maximum possible XOR value can help skip combinations that cannot yield new unique results.

Is using bit manipulation necessary for correctness?

Bit manipulation helps optimize performance but correctness can be achieved using brute force with a set.

What pattern does this problem follow?

It follows an array plus math pattern with XOR operations, emphasizing enumeration and uniqueness in combinatorial results.