Number of Unique XOR Triplets I

To solve Number of Unique XOR Triplets I, iterate through all triplets in the permutation array using nested loops or prefix XORs to track unique values. Apply bit manipulation to compute XOR efficiently, ensuring duplicates are avoided. This approach balances array traversal with mathematical XOR properties to achieve the correct count.

Problem Statement

Given an integer array nums of length n representing a permutation of numbers from 1 to n, compute the number of unique XOR values formed by all triplets (i, j, k) where 0 <= i <= j <= k < n. Each triplet value is calculated as nums[i] XOR nums[j] XOR nums[k].

Return the total count of distinct XOR results across all possible triplets. Consider how the combination of array ordering and XOR properties affects duplicate and unique results.

Examples

Example 1

Input: nums = [1,2]

Output: 2

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

Example 2

Input: nums = [3,1,2]

Output: 4

The possible XOR triplet values include: The unique XOR values are {0, 1, 2, 3} , so the output is 4.

Constraints

• 1 <= n == nums.length <= 105
• 1 <= nums[i] <= n
• nums is a permutation of integers from 1 to n.

Solution Approach

Brute Force Triplet Enumeration

Iterate over all possible triplets (i, j, k) in nums using three nested loops. Compute the XOR for each triplet and store results in a set to track unique values. This is simple but may be too slow for large n.

Prefix XOR Optimization

Compute a prefix XOR array to efficiently calculate XOR for any subarray segment. Use the property that XOR(i, j, k) = prefix[k] XOR prefix[i-1] and reduce redundant calculations, improving performance over full brute force.

Mathematical Counting Insight

Analyze patterns of numbers in the permutation to predict XOR outcomes. Leverage the fact that XOR is associative and that certain ranges produce repeated values. Count unique results using frequency or range observations instead of full enumeration.

Complexity Analysis

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

Time complexity can range from O(n^3) for naive triple iteration to O(n^2) or O(n) using prefix XOR and mathematical insights. Space complexity ranges from O(n) for prefix arrays to O(n^2) for storing unique XOR values depending on the approach.

What Interviewers Usually Probe

• Notice if candidate identifies the permutation property and its impact on XOR results.
• Check if they optimize nested loops using prefix XOR or other bitwise properties.
• Observe whether they consider duplicate elimination and efficient unique counting.

Common Pitfalls or Variants

Common pitfalls

• Forgetting that triplet indices must satisfy i <= j <= k, causing incorrect counts.
• Overlooking repeated XOR values and failing to track unique results.
• Assuming a simple formula for XOR count without considering array permutation constraints.

Follow-up variants

• Count unique XOR pairs instead of triplets.
• Allow arrays with repeated elements rather than strict permutations.
• Compute maximum XOR value among all triplets instead of counting unique results.

FAQ

What is a XOR triplet in Number of Unique XOR Triplets I?

A XOR triplet consists of three elements nums[i], nums[j], nums[k] with i <= j <= k, and the triplet value is nums[i] XOR nums[j] XOR nums[k].

Can I use brute force for large arrays?

Brute force works but is inefficient for n up to 10^5. Using prefix XOR or mathematical patterns is recommended.

Do I need to worry about duplicate XOR results?

Yes, counting only unique XOR values is required, so use a set or similar structure to track distinct results.

How does array being a permutation affect the solution?

Permutation ensures all numbers 1 to n are present, affecting maximum and minimum XOR values and helping with pattern recognition.

Which data structures help compute unique XOR triplets efficiently?

Sets for tracking unique results and prefix XOR arrays to reduce repeated XOR calculations are most effective.