#2317

Medium

LeetCode Problem Workspace

Maximum XOR After Operations

Maximize the bitwise XOR of an array after applying a special operation with non-negative integers multiple times.

Problem Statement

You are given a 0-indexed integer array nums. In one operation, select any non-negative integer x and an index i, then update nums[i] to be equal to nums[i] AND (nums[i] XOR x).

Return the maximum possible bitwise XOR of all elements of nums after applying the operation any number of times.

Examples

Example 1

Input: nums = [3,2,4,6]

Output: 7

Apply the operation with x = 4 and i = 3, num[3] = 6 AND (6 XOR 4) = 6 AND 2 = 2. Now, nums = [3, 2, 4, 2] and the bitwise XOR of all the elements = 3 XOR 2 XOR 4 XOR 2 = 7. It can be shown that 7 is the maximum possible bitwise XOR. Note that other operations may be used to achieve a bitwise XOR of 7.

Example 2

Input: nums = [1,2,3,9,2]

Output: 11

Apply the operation zero times. The bitwise XOR of all the elements = 1 XOR 2 XOR 3 XOR 9 XOR 2 = 11. It can be shown that 11 is the maximum possible bitwise XOR.

Constraints

1 <= nums.length <= 105
0 <= nums[i] <= 108

Solution Approach

Bitwise XOR Operation Insights

Start by observing how the AND and XOR operations interact. The XOR operation changes bits, while AND can only retain or reset them. The goal is to find the maximum bitwise XOR of the array by manipulating the array elements iteratively through these operations.

Strategic Application of XOR Operation

Each number in the array can be manipulated with different x values. The key to maximizing the XOR is to evaluate how the operation affects the bits across different elements. This allows for the creation of higher XOR values by considering the impact of applying XOR at various indices.

Efficient Optimization of XOR

By strategically selecting indices and XOR values, you can optimize the XOR result across the entire array. The challenge is to apply these operations efficiently without redundant calculations, ensuring the maximum XOR is obtained.

Complexity Analysis

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

The time and space complexity depend on the specific approach used to apply the operations. Efficiently handling bitwise operations and ensuring no redundant computations are necessary for the desired output will impact both time and space complexity.

What Interviewers Usually Probe

Tests understanding of bitwise operations.
Evaluates ability to optimize algorithms for bit manipulation.
Assesses knowledge of iterative optimization techniques.

Common Pitfalls or Variants

Common pitfalls

Failing to consider all possible values for x, limiting potential XOR results.
Not optimizing the algorithm for large input sizes, leading to inefficiency.
Misunderstanding the impact of applying multiple XOR operations on a single element.

Follow-up variants

Limit the number of allowed operations and evaluate the effect on XOR.
Introduce constraints on the values of x and explore its effect on the result.
Test with large input arrays and optimize for both time and space complexity.

FAQ

What is the key concept for solving the Maximum XOR After Operations problem?

The key concept is understanding how to strategically apply XOR and AND operations to maximize the final XOR result of all array elements.

How do I optimize my solution for the Maximum XOR After Operations problem?

Optimizing requires efficient use of bitwise operations, reducing redundant computations, and applying operations only when necessary to achieve the maximum result.

What is the difficulty level of the Maximum XOR After Operations problem?

The problem is categorized as Medium difficulty due to the need to understand bitwise operations and optimize them over multiple iterations.

What are the main techniques used in the Maximum XOR After Operations problem?

The main techniques are bitwise manipulation using AND and XOR operations, followed by strategic evaluation to maximize the XOR result.

Can the Maximum XOR After Operations problem be solved without considering all possible x values?

No, considering all possible x values is essential, as different values influence the XOR outcome, and missing some can result in suboptimal solutions.

terminal

Solution

Solution 1: Bit Manipulation

In one operation, we can update $\textit{nums}[i]$ to $\textit{nums}[i] \text{ AND } (\textit{nums}[i] \text{ XOR } x)$. Since $x$ is any non-negative integer, the result of $\textit{nums}[i] \oplus x$ can be any value. By performing a bitwise AND operation with $\textit{nums}[i]$, we can change some of the $1$ bits in the binary representation of $\textit{nums}[i]$ to $0$.

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

Continue Practicing

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