The task requires maximizing the bitwise OR of an array with up to k operations, where each operation multiplies an element by 2. The optimal approach involves greedy choices, applying all k operations to one element to maximize the final OR value. This problem tests your understanding of greedy algorithms and bit manipulation.
Problem Statement
You are given a 0-indexed integer array nums of length n and an integer k. You can apply at most k operations on the elements of nums, where each operation involves multiplying one element of nums by 2.
The goal is to maximize the bitwise OR of all the elements of nums after performing the operations. You must return the maximum OR value achievable after at most k operations.
Examples
Example 1
Input: nums = [12,9], k = 1
Output: 30
If we apply the operation to index 1, our new array nums will be equal to [12,18]. Thus, we return the bitwise or of 12 and 18, which is 30.
Example 2
Input: nums = [8,1,2], k = 2
Output: 35
If we apply the operation twice on index 0, we yield a new array of [32,1,2]. Thus, we return 32|1|2 = 35.
Constraints
- 1 <= nums.length <= 105
- 1 <= nums[i] <= 109
- 1 <= k <= 15
Solution Approach
Greedy Approach with Focus on One Element
The optimal solution applies all k operations to a single element of the array. By focusing on one element, we maximize the power of that element's value, which contributes most significantly to the OR result.
Bitwise OR Understanding
Bitwise OR results in a value that contains all the bits set in either of the operands. A greedy approach will ensure that the element receiving the most operations contributes the highest possible value, affecting the final OR outcome.
Maximizing with Prefix Sum
By calculating a prefix sum of the OR values, we can efficiently track and update the OR result as we modify elements. This helps in reducing recalculations and ensures we are optimizing the operation usage.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time complexity depends on the approach chosen for calculating the OR value and applying operations. A naive approach could be O(n * k), but optimized solutions may reduce this to O(n log(max(nums))) with careful manipulation of the OR and operations.
What Interviewers Usually Probe
- Candidate should demonstrate understanding of bitwise operations and greedy algorithms.
- Look for efficient handling of the k operations on a single element to maximize OR.
- Candidate may struggle if they fail to prioritize which element to operate on.
Common Pitfalls or Variants
Common pitfalls
- Failing to apply all k operations on a single element, leading to suboptimal OR values.
- Not recognizing the importance of bitwise OR and how each operation doubles the value of an element.
- Inefficient brute-force approaches that involve recalculating the OR too often.
Follow-up variants
- Allowing operations on multiple elements, not just one.
- Changing the type of operation (e.g., multiplying by a factor other than 2).
- Considering a scenario where the OR operation also involves other bit manipulations (e.g., XOR).
How GhostInterview Helps
- GhostInterview can guide you through bitwise operations and their impact on the OR result.
- With step-by-step hints, GhostInterview helps you break down the greedy approach and understand its application in the problem.
- GhostInterview assists in refining your time complexity analysis by walking you through possible optimizations.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
How does the greedy approach work in the Maximum OR problem?
The greedy approach applies all k operations to a single element to maximize its value, which contributes most to the final OR result.
What is the optimal solution for the Maximum OR problem?
The optimal solution focuses all k operations on a single element, maximizing its value and subsequently the bitwise OR result.
Can the Maximum OR problem be solved without bitwise operations?
No, bitwise OR is central to the problem, as it determines the final value after applying operations on the elements.
What is the time complexity of the Maximum OR problem?
The time complexity can range from O(n * k) to O(n log(max(nums))) depending on the approach and optimizations applied.
How does GhostInterview help with the Maximum OR problem?
GhostInterview provides insights into the greedy approach and helps refine time complexity analysis, offering structured hints and solutions.
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