What is the time complexity of Maximum XOR After Operations ?

The optimal solution has a time complexity of O(n).

What data structures are used to solve Maximum XOR After Operations ?

This problem uses Array, Math, Bit Manipulation. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Maximum XOR After Operations ?

Insight 1: The operation `nums[i] = nums[i] AND (nums[i] XOR x)` can only clear bits (set them to 0) in `nums[i]`. It cannot set a bit that was originally 0 to 1. Insight 2: If a bit is set (1) in any of the numbers in the input array, it's possible to propagate that bit to all other numbers by carefully choosing 'x' in the given operation.

Which companies ask LeetCode 2317. Maximum XOR After Operations in interviews?

This problem is frequently asked at American Express.

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Maximum XOR After Operations - LeetCode 2317 Solution

Maximum XOR After Operations - Complete Solution Guide

Maximum XOR After Operations is LeetCode problem 2317, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

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) . Note that AND is the bitwise AND operation and XOR is the bitwise XOR operation. Return the maximum possible bitwise XOR of all elements of nums after applying the operation any number of times . Example 1: Input: nums = [3,2,4,6] Output: 7 Explanation: Apply the operation with x = 4 and i = 3, num[3] = 6 AND (6 XOR 4

Detailed Explanation

The problem asks us to find the maximum possible bitwise XOR of all elements in a given array `nums` after applying a specific operation any number of times. The operation involves choosing an element `nums[i]` and a non-negative integer `x`, and then updating `nums[i]` to `nums[i] AND (nums[i] XOR x)`. The goal is to maximize the XOR sum of all elements after these operations.

Solution Approach

The solution approach leverages the insight that the operation can only clear bits. By taking the bitwise OR of all numbers in the array, we effectively capture all bits that are set in at least one of the numbers. The problem asks for the maximum XOR after applying the operation, which can be achieved by setting every bit position which is 1 in any of the numbers to 1 in the result.

Step-by-Step Algorithm

Step 1: Initialize a variable `result` to 0. This variable will store the bitwise OR of all the numbers.
Step 2: Iterate through the `nums` array.
Step 3: For each number `num` in the array, perform a bitwise OR operation between `result` and `num` (`result |= num`). This operation sets any bit in `result` to 1 if it is 1 in either `result` or `num`.
Step 4: After iterating through all the numbers, `result` will contain the bitwise OR of all numbers in the array. Return the value of `result`.

Key Insights

Insight 1: The operation `nums[i] = nums[i] AND (nums[i] XOR x)` can only clear bits (set them to 0) in `nums[i]`. It cannot set a bit that was originally 0 to 1.
Insight 2: If a bit is set (1) in any of the numbers in the input array, it's possible to propagate that bit to all other numbers by carefully choosing 'x' in the given operation.
Insight 3: Therefore, the maximum XOR sum we can achieve is when all bits that are set in at least one of the input numbers are also set in the final XOR result. This is equivalent to performing a bitwise OR of all the numbers in the input array.

Complexity Analysis

Time Complexity: O(n)

Space Complexity: O(1)

Topics

This problem involves: Array, Math, Bit Manipulation.

Companies

Asked at: American Express.

Maximum XOR After Operations | LeetCode 2317

Medium

Array Math Bit Manipulation

Time: O(n)Space: O(1)

Solution Code

Available in:

class Solution {
    public int maximumXOR(int[] nums) {
        int result = 0;
        for (int num : nums) {
            result |= num;
        }
        return result;
    }
}

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Problem Info

Description

Problem 2317: Maximum XOR After Operations - Detailed explanation and solution approaches.

Topics

Array Math Bit Manipulation

Companies

American Express

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:2317

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

4 steps

1
Step 1: Initialize a variable `result` to 0. This variable will store the bitwise OR of all the numbers.
2
Step 2: Iterate through the `nums` array.
3
Step 3: For each number `num` in the array, perform a bitwise OR operation between `result` and `num` (`result |= num`). This operation sets any bit in `result` to 1 if it is 1 in either `result` or `num`.
4
Step 4: After iterating through all the numbers, `result` will contain the bitwise OR of all numbers in the array. Return the value of `result`.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n)

👍 Linear - Good! Visits each element once.

💾

Space Complexity

O(1)

💪 Constant space - Uses same memory regardless of input!

📝

Why?

The time complexity is linear because the solution involves a single loop that iterates through each element in the input array `nums`. The space complexity is constant as we only use a fixed number of variables regardless of the input size.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Using Reduce Function (Python)

In Python, the `reduce` function from the `functools` module can be used to perform the bitwise OR operation cumulatively across all elements of the list.

Time: O(n)Space: O(1)

Trade-offs: This approach can be more concise but might be slightly less readable for those unfamiliar with the `reduce` function. The performance is comparable to the iterative approach.

✨

Key Insights

1Insight 1: The operation `nums[i] = nums[i] AND (nums[i] XOR x)` can only clear bits (set them to 0) in `nums[i]`. It cannot set a bit that was originally 0 to 1.
2Insight 2: If a bit is set (1) in any of the numbers in the input array, it's possible to propagate that bit to all other numbers by carefully choosing 'x' in the given operation.
3Insight 3: Therefore, the maximum XOR sum we can achieve is when all bits that are set in at least one of the input numbers are also set in the final XOR result. This is equivalent to performing a bitwise OR of all the numbers in the input array.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Edge Cases

Edge case 1: `nums` is empty. The code will still work because the loop will not execute, and the initial value of `result` (0) will be returned, which is the correct answer for an empty array (although the problem states that the array has at least length 1).
Edge case 2: All numbers in `nums` are 0. The result will correctly be 0, as the bitwise OR of all zeros is zero.
Edge case 3: A single number in the array is the maximum integer value (2^31 - 1 if integers are 32 bit). The bitwise OR operation will still work correctly and set all bits to 1.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Trying to actually perform the given operation and trying all possible values of `x` and `i`. This would lead to an exponential time complexity and is not feasible given the constraints.

✗Common mistake 2: Incorrectly assuming that the maximum XOR is achieved by simply XORing all the numbers directly. This is not true because the given operation allows us to potentially set more bits to 1 than the original XOR sum would allow.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: In database indexing, bitwise OR operations are used to efficiently search for records that match any of several criteria. By representing the criteria as bit patterns, a single OR operation can combine multiple criteria before searching.

🏦Real-world scenario 2: In networking, bitwise OR operations are used in network masks to identify network addresses and subnets. Combining multiple masks can be done efficiently with OR.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

If stuck, try working through a small example by hand. The pattern often reveals itself.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: November 5, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJul 13

Try solving this without using extra space first. It's a great exercise for understanding the underlying pattern.

@CodeNinjaJul 6

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJun 29

❓

Frequently Asked Questions

When should I use in-place modification vs creating a new array?

Use in-place modification when space complexity matters (O(1) space requirement) and the original array can be modified. Create a new array when you need to preserve the original data or when the problem involves significant restructuring that would complicate in-place logic.

How much time should I spend on a medium problem in interviews?

In a 45-minute interview, aim for: 5-10 minutes understanding and clarifying, 5-10 minutes discussing approach, 20-25 minutes coding, and 5-10 minutes testing. If stuck after 15 minutes of thinking, verbalize your thought process and ask for hints.

What is the importance of time and space complexity analysis?

Complexity analysis is crucial because: 1) Interviewers always ask about it, 2) It helps you choose between approaches, 3) It demonstrates CS fundamentals. Always state both time and space complexity, and be prepared to explain how you derived them.

What is the optimal approach for "Maximum XOR After Operations "?

For this medium-level Array problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether math techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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