#2527

Medium

LeetCode Problem Workspace

Find Xor-Beauty of Array

Find the xor-beauty of an array by XORing the effective values of all possible triplets of indices.

Problem Statement

You are given a 0-indexed integer array nums. For each triplet of indices (i, j, k) where 0 <= i, j, k < n, compute the effective value as ((nums[i] | nums[j]) & nums[k]).

The xor-beauty of the array is the result of XORing the effective values for all triplets. Your task is to find the xor-beauty of the array efficiently.

Examples

Example 1

Input: nums = [1,4]

Output: 5

The triplets and their corresponding effective values are listed below:

(0,0,0) with effective value ((1 | 1) & 1) = 1
(0,0,1) with effective value ((1 | 1) & 4) = 0
(0,1,0) with effective value ((1 | 4) & 1) = 1
(0,1,1) with effective value ((1 | 4) & 4) = 4
(1,0,0) with effective value ((4 | 1) & 1) = 1
(1,0,1) with effective value ((4 | 1) & 4) = 4
(1,1,0) with effective value ((4 | 4) & 1) = 0
(1,1,1) with effective value ((4 | 4) & 4) = 4 Xor-beauty of array will be bitwise XOR of all beauties = 1 ^ 0 ^ 1 ^ 4 ^ 1 ^ 4 ^ 0 ^ 4 = 5.

Example 2

Input: nums = [15,45,20,2,34,35,5,44,32,30]

Output: 34

The xor-beauty of the given array is 34.

Constraints

1 <= nums.length <= 105
1 <= nums[i] <= 109

Solution Approach

Simplifying the XOR Expression

The key to solving this problem efficiently lies in simplifying the given expression. Recognizing patterns in bitwise operations can help reduce unnecessary calculations. Focus on bit manipulation tricks to compute the effective value of triplets more quickly.

Brute Force Approach

You can iterate over all possible triplets (i, j, k) and calculate the effective value for each. Although this brute force approach guarantees correctness, it may not scale well for larger arrays due to the O(n^3) complexity.

Optimizing with Bitwise Operations

To optimize the solution, consider reducing the number of operations by leveraging bitwise properties. Look for opportunities to avoid recalculating common values or bits across multiple triplet evaluations.

Complexity Analysis

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

The time complexity depends on the approach chosen. A brute force solution has O(n^3) time complexity, while optimized bitwise operations can help reduce it to O(n^2). The space complexity depends on the storage of intermediate values, potentially O(n) in optimized solutions.

What Interviewers Usually Probe

Check for the candidate's understanding of bitwise operations and their ability to identify optimization opportunities.
Evaluate whether the candidate can recognize the trade-off between brute force and optimized solutions.
Assess the candidate's ability to explain their approach clearly and in detail.

Common Pitfalls or Variants

Common pitfalls

Not simplifying the XOR expression leads to inefficient solutions.
Overcomplicating the solution by missing opportunities for bitwise optimization.
Neglecting edge cases such as when the array contains only one element.

Follow-up variants

Consider variants where the XOR expression is changed to use other bitwise operators.
Implement a solution where the array is sorted first to see how it affects performance.
Change the problem to compute a different bitwise combination or condition between indices.

FAQ

What is the xor-beauty of an array?

The xor-beauty of an array is the result of XORing the effective values of all possible triplets of indices in the array, where each triplet's effective value is calculated as ((nums[i] | nums[j]) & nums[k]).

How do I simplify the XOR expression in this problem?

You can simplify the XOR expression by leveraging properties of bitwise operations, focusing on reducing redundant calculations and optimizing the evaluation of each triplet.

What is the time complexity of the brute force solution?

The brute force solution has a time complexity of O(n^3) since it evaluates all possible triplets of indices in the array.

How does GhostInterview help with the bitwise optimization for this problem?

GhostInterview guides you in applying bitwise operations efficiently, helping you identify redundant steps and optimize the solution to avoid unnecessary recalculations.

What is the optimal solution for the Find Xor-Beauty of Array problem?

The optimal solution involves reducing the time complexity by recognizing patterns in bitwise operations, enabling you to calculate the effective values more efficiently than brute force.

terminal

Solution

Solution 1: Bit Manipulation

We first consider the case where $i$ and $j$ are not equal. In this case, `((nums[i] | nums[j]) & nums[k])` and `((nums[j] | nums[i]) & nums[k])` produce the same result, and their XOR result is $0$.

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class Solution:
    def xorBeauty(self, nums: List[int]) -> int:
        return reduce(xor, nums)

Continue Practicing

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