Maximum OR
You are given an integer array nums
and an integer k
. You can multiply any element in the array by 2 for at most k
times. The goal is to find the maximum possible bitwise OR value that can be obtained from the array after applying the operation at most k
times.
Approach
To solve this problem, we will use a greedy approach:
 Identify Bits: Identify the most significant bit that is set to 0 in the bitwise OR of all elements. This is the bit that can contribute the most to the final result if changed to 1.
 Find Element: Find the element in the array that has the most number of trailing zeros in the bit representation after the identified bit. This is the element that will contribute the most to changing the identified bit to 1 when multiplied by 2.
 Apply Operation: Multiply the selected element by 2 and decrease the counter
k
by 1.  Repeat: Repeat the above steps until
k
becomes 0.  Calculate Result: Calculate the bitwise OR of the modified array.
Implementation


Examples
 For
nums = [12,9], k = 1
, the output will be30
, as explained in the problem statement.  For
nums = [8,1,2], k = 2
, the output will be35
.
Time Complexity
The time complexity of this solution is (O(k \cdot n)), where (n) is the length of the given array.
Summary
 Identify the most significant bit that is 0 in the bitwise OR of all elements.
 Find the element that can most efficiently change the identified bit to 1.
 Apply the multiplication operation on the selected element at most
k
times.  Calculate the final bitwise OR of the modified array.
10 Prerequisite LeetCode Problems
For the problem “Maximum OR”, the following problems are a good preparation:
“476. Number Complement”  This problem introduces the concept of bitwise operations, specifically bitwise not, which is crucial in understanding bitwise or operation.
“191. Number of 1 Bits”  This problem introduces bit manipulation and helps you understand how to count bits, an important step towards bitwise operations.
“371. Sum of Two Integers”  It helps in understanding how to perform addition using bitwise operations, which could lead to better understanding of such operations.
“338. Counting Bits”  This problem helps you get used to bit manipulation, specifically counting bits, which is a crucial part of this problem.
“136. Single Number”  To practice the concept of XOR and understanding of bit manipulation.
“231. Power of Two”  It reinforces the concept of bitwise operations and dealing with binary representations.
“260. Single Number III”  It’s a more challenging problem involving bitwise operations and is a good practice for the given problem.
“137. Single Number II”  This problem further explores bitwise manipulation, building up the complexity a bit more.
“190. Reverse Bits”  It will help to understand the binary number system and the effect of operations like shift and or on it.
“342. Power of Four”  This problem solidifies understanding of bitwise operations and specifically focuses on the effect of ‘and’ and ‘or’ operations.
These cover bitwise operations and bit manipulation, which are crucial for solving the “Maximum OR”.


Problem Classification
Problem Statement:You are given a 0indexed 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 [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 Explanation: If we apply the operation twice on index 0, we yield a new array of [32,1,2]. Thus, we return 3212 = 35.
Constraints:
1 <= nums.length <= 105 1 <= nums[i] <= 109 1 <= k <= 15 Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.
Clarification Questions
What are the clarification questions we can ask about this problem?
Problem Analysis and Key Insights
What are the key insights from analyzing the problem statement?
Problem Boundary
What is the scope of this problem?
How to establish the boundary of this problem?
Distilling the Problem to Its Core Elements
Can you identify the fundamental concept or principle this problem is based upon? Please explain. What is the simplest way you would describe this problem to someone unfamiliar with the subject? What is the core problem we are trying to solve? Can we simplify the problem statement? Can you break down the problem into its key components? What is the minimal set of operations we need to perform to solve this problem?
Visual Model of the Problem
How to visualize the problem statement for this problem?
Problem Restatement
Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.
Abstract Representation of the Problem
Could you help me formulate an abstract representation of this problem?
Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific realworld details?
Terminology
Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?
Problem Simplification and Explanation
Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?
Constraints
Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.
What are the key insights from analyzing the constraints?
Case Analysis
Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.
Provide names by categorizing these cases
What are the edge cases?
How to visualize these cases?
What are the key insights from analyzing the different cases?
Identification of Applicable Theoretical Concepts
Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?
Simple Explanation
Can you explain this problem in simple terms or like you would explain to a nontechnical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?
Problem Breakdown and Solution Methodology
Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.
Inference of ProblemSolving Approach from the Problem Statement
Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method. How can I recognize these properties by drawing tables or diagrams?
How did you infer from the problem statement that this problem can be solved using ?
Simple Explanation of the Proof
I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?
Stepwise Refinement
Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the highlevel solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?
Solution Approach and Analysis
Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.
Identify Invariant
What is the invariant in this problem?
Identify Loop Invariant
What is the loop invariant in this problem?
Is invariant and loop invariant the same for this problem?
Thought Process
Can you explain the basic thought process and steps involved in solving this type of problem?
Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.
Establishing Preconditions and Postconditions
Parameters:
 What are the inputs to the method?
 What types are these parameters?
 What do these parameters represent in the context of the problem?
Preconditions:
 Before this method is called, what must be true about the state of the program or the values of the parameters?
 Are there any constraints on the input parameters?
 Is there a specific state that the program or some part of it must be in?
Method Functionality:
 What is this method expected to do?
 How does it interact with the inputs and the current state of the program?
Postconditions:
 After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
 What does the return value represent or indicate?
 What side effects, if any, does the method have?
Error Handling:
 How does the method respond if the preconditions are not met?
 Does it throw an exception, return a special value, or do something else?
Problem Decomposition
Problem Understanding:
 Can you explain the problem in your own words? What are the key components and requirements?
Initial Breakdown:
 Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?
Subproblem Refinement:
 For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?
Task Identification:
 Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?
Task Abstraction:
 For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?
Method Naming:
 Can you give each task a simple, descriptive name that makes its purpose clear?
Subproblem Interactions:
 How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?
From Brute Force to Optimal Solution
Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?
Code Explanation and Design Decisions
Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?
If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.
If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?
Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.
Coding Constructs
Consider the code for the solution of this problem.
What are the highlevel problemsolving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a nonprogrammer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?
Language Agnostic Coding Drills
Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.
Dissect the code and identify each distinct concept it contains. Remember, this process should be languageagnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problemsolving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the stepbystep process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.
Targeted Drills in Python
Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Pythonbased coding drills for each of those concepts.
Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problemspecific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.
Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.
Q&A
Similar Problems
Can you suggest 10 problems from LeetCode that require similar problemsolving strategies or use similar underlying concepts as the problem we’ve just solved? These problems can be from any domain or topic, but they should involve similar steps or techniques in the solution process. Also, please briefly explain why you consider each of these problems to be related to our original problem. The response text is of the following format:
Here are 10 problems that use similar underlying concepts:
Analysis:
Domain Categorization: This problem falls within the domain of Computer Science and specifically within the subdomain of Bit Manipulation and Array manipulation. The problem requires a thorough understanding of bitwise operations, in particular, the bitwise OR operation.
‘What’ Components:
 An integer array ’nums’ of length ’n’.
 An integer ‘k’, the maximum number of times an operation (multiplying an element by 2) can be performed on the elements of ’nums’.
 The task is to return the maximum possible value of the bitwise OR of all elements in ’nums’ after applying the operation at most ‘k’ times.
Problem Classification: This problem is an optimization problem as it seeks to find the maximum possible value. It requires the ability to manipulate array elements and employ bitwise operations. The problem involves aspects of Dynamic Programming, as the decision to multiply a number could affect the overall result. Additionally, due to the repetition of operations ‘k’ times, elements of Greedy algorithms may also be present, as at each step we will attempt to make a decision that seems the best at that moment.
Language Agnostic Coding Drills
Dissection of the Code into Distinct Concepts:
a. Python List Manipulation: Creation and manipulation of lists (arrays) is fundamental to this problem.
b. Looping Constructs: Iterating over elements of an array is a key concept in this code. In this case, forloops are used.
c. Bitwise Operations: This code utilizes two types of bitwise operations: the bitwise OR (’’) and the left shift operation (’«’).
d. Array Prefix and Suffix Calculations: This involves creating two arrays (prefix and suffix) that store the bitwise OR of elements up to a certain index from the beginning or from the end.
e. Use of Max Function: The code keeps track of the maximum OR value observed so far.
Coding Concepts/Drills in Increasing Difficulty:
a. Python List Manipulation: This is one of the most basic concepts in Python. Difficulty Level: Easy.
b. Looping Constructs: Understanding how to iterate over an array using loops is essential for almost all algorithmic problems. Difficulty Level: Easy.
c. Bitwise Operations: Bitwise operations, particularly bitwise OR and shift operations, require a bit more conceptual understanding. Difficulty Level: Medium.
d. Array Prefix and Suffix Calculations: It requires understanding of the problem and good control over array manipulation. Difficulty Level: Medium.
e. Use of Max Function: Using the max function to maintain the maximum result is a standard strategy in optimization problems. Difficulty Level: Medium.
ProblemSolving Approach:
The problemsolving approach involves iterating over the given array, applying the operation (i.e., multiplying by 2, which is equivalent to shifting left by 1 in binary representation) to each element ‘k’ times, and then finding the maximum OR of all the elements. The bitwise OR of the elements to the left and right of the current element is precomputed and stored in the prefix and suffix arrays respectively. The maximum OR value is updated for each element after shifting ‘k’ times and OR’ing with the corresponding prefix and suffix values.
Each of the identified coding drills contributes to the overall solution. Array manipulation and looping constructs provide the basic structure of the solution. Bitwise operations are used to perform the operation on elements and compute the OR values. Prefix and Suffix calculations help in optimizing the OR calculation for each element. The use of the max function helps in keeping track of the maximum OR value obtained so far.
Targeted Drills in Python
Python Code for Each Identified Concept:
a. Python List Manipulation:
1 2 3 4 5 6 7 8 9
# Creating a list nums = [1, 2, 3, 4, 5] # Accessing an element in the list print(nums[0]) # prints 1 # Modifying an element in the list nums[0] = 6 print(nums) # prints [6, 2, 3, 4, 5]
b. Looping Constructs:
1 2 3 4
# Looping over a list nums = [1, 2, 3, 4, 5] for num in nums: print(num) # prints 1, 2, 3, 4, 5 in separate lines
c. Bitwise Operations:
1 2 3 4 5
# Bitwise OR operation print(6  3) # prints 7 # Left shift operation print(2 << 1) # prints 4
d. Array Prefix and Suffix Calculations:
1 2 3 4 5 6 7 8
nums = [1, 2, 3, 4, 5] prefix = [0] * len(nums) suffix = [0] * len(nums) for i in range(1, len(nums)): prefix[i] = prefix[i  1]  nums[i  1] suffix[len(nums)  i  1] = suffix[len(nums)  i]  nums[len(nums)  i] print(prefix) # prints [0, 1, 3, 3, 7] print(suffix) # prints [7, 7, 5, 4, 0]
e. Use of Max Function:
1 2
# Finding the maximum of two numbers print(max(5, 3)) # prints 5
ProblemSpecific Coding Drills:
In addition to the general concepts, the problemspecific concept here involves iterating over each element of the array, applying the operation (shifting left by k positions), and updating the maximum result so far with the OR of the current element (after operation), prefix[i], and suffix[i]. This requires knowledge of loops, bitwise operations, the max function, and prefixsuffix calculations.
Integration of Coding Drills:
Start by initializing the result, prefix, and suffix variables. Then, calculate the prefix and suffix OR values for all elements. Now, iterate over each element of the array. For each element, apply the operation (i.e., shift left by k positions), calculate the OR with the corresponding prefix and suffix values, and update the result if it is greater than the current maximum. After the iteration is over, the result will hold the maximum OR value that can be obtained after applying the operation at most k times. This involves integrating all the identified coding drills in a sequential manner.