What is the time complexity of Maximum OR?

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

What data structures are used to solve Maximum OR?

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

What are the key insights for solving Maximum OR?

Insight 1: The key is to realize that to maximize the overall OR value, we should apply all `k` operations to a single element in the array. Since we want to maximize the OR, we should shift the bits of one element as much as possible. Insight 2: It's beneficial to precompute the bitwise OR of the prefix and suffix of the array. This way, for each element, we can efficiently calculate the bitwise OR of all the other elements without including the current element.

Which companies ask LeetCode 2680. Maximum OR in interviews?

This problem is frequently asked at DE Shaw.

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Maximum OR - LeetCode 2680 Solution

Maximum OR - Complete Solution Guide

Maximum OR is LeetCode problem 2680, 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 of length n and an integer k . In an operation, you can choose an element and multiply it by 2 . Return the maximum possible value of nums[0] | nums[1] | ... | nums[n - 1] that can be obtained after applying the operation on nums at most k times . Note that a | b denotes the bitwise or between two integers a and b . Example 1: Input: nums = [12,9], k = 1 Output: 30 Explanation: If we apply the operation to index 1, our new array nums will be equal to

Detailed Explanation

The problem asks us to maximize the bitwise OR of all elements in an array `nums` after applying an operation at most `k` times. The operation involves choosing an element in the array and multiplying it by 2. We need to find the maximum possible result of the bitwise OR of the array elements after applying the operation optimally.

Solution Approach

The solution calculates the prefix OR and suffix OR of the input array. Then, it iterates through the array, considering each element as the one to be multiplied by 2^k. For each element, it calculates the OR of all other elements using the prefix and suffix OR arrays. It then calculates the OR of the boosted element (multiplied by 2^k) with the OR of all other elements, and maintains the maximum value found so far.

Step-by-Step Algorithm

Step 1: Initialize two arrays, `prefix_or` and `suffix_or`, of size `n+1` with all elements set to 0. These will store the prefix and suffix bitwise ORs, respectively.
Step 2: Calculate the prefix OR array. For each index `i` from 0 to `n-1`, `prefix_or[i+1]` is the bitwise OR of all elements from `nums[0]` to `nums[i]`. So, `prefix_or[i+1] = prefix_or[i] | nums[i]`.
Step 3: Calculate the suffix OR array. For each index `i` from `n-1` down to 0, `suffix_or[i]` is the bitwise OR of all elements from `nums[i]` to `nums[n-1]`. So, `suffix_or[i] = suffix_or[i+1] | nums[i]`.
Step 4: Initialize a variable `max_val` to 0 to store the maximum OR value found so far.
Step 5: Iterate through the `nums` array from `i = 0` to `n-1`. For each index `i`:
Step 6: Calculate the OR of all other elements excluding `nums[i]`. This is done by taking the bitwise OR of `prefix_or[i]` and `suffix_or[i+1]`. `others_or = prefix_or[i] | suffix_or[i+1]`.
Step 7: Calculate the value of `nums[i]` after applying the multiplication by `2^k`. `boosted_num = nums[i] << k`.
Step 8: Calculate the current OR value by taking the bitwise OR of the `boosted_num` and `others_or`. `current_or = boosted_num | others_or`.
Step 9: Update the `max_val` with the maximum of the current `max_val` and `current_or`. `max_val = max(max_val, current_or)`.
Step 10: Return the `max_val`.

Key Insights

Insight 1: The key is to realize that to maximize the overall OR value, we should apply all `k` operations to a single element in the array. Since we want to maximize the OR, we should shift the bits of one element as much as possible.
Insight 2: It's beneficial to precompute the bitwise OR of the prefix and suffix of the array. This way, for each element, we can efficiently calculate the bitwise OR of all the other elements without including the current element.
Insight 3: We iterate through each element in the array, assuming it's the element that gets multiplied by 2^k, then calculate and compare the resultant OR sum.

Complexity Analysis

Time Complexity: O(n)

Space Complexity: O(n)

Topics

This problem involves: Array, Greedy, Bit Manipulation, Prefix Sum.

Companies

Asked at: DE Shaw.

Maximum OR | LeetCode 2680

Medium

Array Greedy Bit Manipulation Prefix Sum

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

Solution Code

Available in:

class Solution {
    public long maximumOr(int[] nums, int k) {
        int n = nums.length;
        
        long[] prefix_or = new long[n + 1];
        for (int i = 0; i < n; i++) {
            prefix_or[i+1] = prefix_or[i] | nums[i];
        }
        
        long[] suffix_or = new long[n + 1];
        for (int i = n - 1; i >= 0; i--) {
            suffix_or[i] = suffix_or[i+1] | nums[i];
        }
        
        long max_val = 0;
        for (int i = 0; i < n; i++) {
            // Calculate the OR of all other numbers
            long others_or = prefix_or[i] | suffix_or[i+1];
            
            // Boost the current number
            long boosted_num = (long)nums[i] << k;
            
            // Combine and update the maximum
            long current_or = boosted_num | others_or;
            max_val = Math.max(max_val, current_or);
        }
        
        return max_val;
    }
}

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Problem Info

Description

Problem 2680: Maximum OR - Detailed explanation and solution approaches.

Topics

Array Greedy Bit Manipulation Prefix Sum

Companies

DE Shaw

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:2680

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

10 steps

1
Step 1: Initialize two arrays, `prefix_or` and `suffix_or`, of size `n+1` with all elements set to 0. These will store the prefix and suffix bitwise ORs, respectively.
2
Step 2: Calculate the prefix OR array. For each index `i` from 0 to `n-1`, `prefix_or[i+1]` is the bitwise OR of all elements from `nums[0]` to `nums[i]`. So, `prefix_or[i+1] = prefix_or[i] | nums[i]`.
3
Step 3: Calculate the suffix OR array. For each index `i` from `n-1` down to 0, `suffix_or[i]` is the bitwise OR of all elements from `nums[i]` to `nums[n-1]`. So, `suffix_or[i] = suffix_or[i+1] | nums[i]`.
4
Step 4: Initialize a variable `max_val` to 0 to store the maximum OR value found so far.
5
Step 5: Iterate through the `nums` array from `i = 0` to `n-1`. For each index `i`:
6
Step 6: Calculate the OR of all other elements excluding `nums[i]`. This is done by taking the bitwise OR of `prefix_or[i]` and `suffix_or[i+1]`. `others_or = prefix_or[i] | suffix_or[i+1]`.
7
Step 7: Calculate the value of `nums[i]` after applying the multiplication by `2^k`. `boosted_num = nums[i] << k`.
8
Step 8: Calculate the current OR value by taking the bitwise OR of the `boosted_num` and `others_or`. `current_or = boosted_num | others_or`.
9
Step 9: Update the `max_val` with the maximum of the current `max_val` and `current_or`. `max_val = max(max_val, current_or)`.
10
Step 10: Return the `max_val`.

💡

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(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The loops for prefix/suffix OR take O(n) time each. The main loop for finding the maximum OR also takes O(n) time. The space complexity is derived from the auxiliary space used by `prefix_or` and `suffix_or`, both of which are O(n).

📖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

Brute Force

Try every possible combination of applying the `k` operations to the elements of the array. This would involve iterating through each possible selection of the index to which the multiplication occurs.

Time: O(n * n^k) or O(n * 2^(k * log(n))), since we potentially need to examine 2^k possible subsets of size nSpace: O(1), constant space is only used for storing variables.

Trade-offs: The brute force approach is very slow for larger values of k. The presented solution is more efficient for the given constraints, and is much faster.

✨

Key Insights

1Insight 1: The key is to realize that to maximize the overall OR value, we should apply all `k` operations to a single element in the array. Since we want to maximize the OR, we should shift the bits of one element as much as possible.
2Insight 2: It's beneficial to precompute the bitwise OR of the prefix and suffix of the array. This way, for each element, we can efficiently calculate the bitwise OR of all the other elements without including the current element.
3Insight 3: We iterate through each element in the array, assuming it's the element that gets multiplied by 2^k, then calculate and compare the resultant OR sum.

📚

Key Terms Explained

For Students

Array

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

Greedy Algorithm

Making the best choice at each step without looking ahead.

Edge Cases

Edge case 1: When the array is empty. The provided code handles this case implicitly as the loops will not execute, and the initial max_val of 0 will be returned. Although the constraint mentions that the array length is at least 1.
Edge case 2: When k is very large, the left shift operation might cause overflow if the data type used isn't large enough. The code uses `long` or `long long` to mitigate this risk by providing a wider range.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Not using a data type large enough (e.g., `int`) to store the results of the left shift operation, potentially leading to overflow errors. Using `long` or `long long` helps to avoid this.

✗Common mistake 2: Incorrectly calculating the bitwise OR of the other elements. Failing to exclude the current element when calculating the OR of others can result in an incorrect maximum value.

✗Common mistake 3: Not understanding bitwise OR operations and how they can maximize the final result.

✅

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: Data compression algorithms often use bitwise operations to efficiently represent data. Optimizing bitwise OR operations can lead to better compression ratios.

🏦Real-world scenario 2: Cryptographic algorithms heavily rely on bitwise operations for encryption and decryption. Finding efficient ways to maximize or manipulate bits can have implications for algorithm security and performance.

💡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.

👨‍💻 From Experience

This problem pattern appears frequently in real codebases. I've used similar logic when building production systems.

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 8, 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

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 13

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJul 6

❓

Frequently Asked Questions

What is the most efficient way to solve array problems?

Most array problems can be solved efficiently using techniques like two-pointer approach, sliding window, or hash maps. The key is identifying whether the array is sorted (enabling binary search) or if tracking seen elements (using hash maps) can help achieve O(n) time complexity.

How do I verify if a greedy approach is correct?

Prove the greedy choice property: the locally optimal choice leads to globally optimal solution. Use exchange argument: show that swapping any element with the greedy choice doesn't improve the solution. If you can't prove it, test extensively or consider DP.

What techniques are commonly tested in medium-difficulty problems?

Medium problems often require combining multiple techniques: two-pointer with hash maps, BFS/DFS with state tracking, dynamic programming with optimization, or divide-and-conquer approaches. They test your ability to recognize patterns and choose the right combination of techniques.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Maximum OR"?

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 greedy techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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