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Longest Subarray With Maximum Bitwise AND

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Longest Subarray With Maximum Bitwise AND

This article delves into the problem "Longest Subarray With Maximum Bitwise AND", a challenging coding problem that often arises in competitive programming and software development. We'll explore the intricacies of bitwise AND operations, analyze efficient solutions, and gain a deeper understanding of how to approach similar problems.

Problem Statement

Given an array of integers nums, find the length of the longest subarray where the maximum bitwise AND of all elements in the subarray is equal to the maximum bitwise AND of all elements in the original array.

For example:

Input: nums = [1,2,3,4]
Output: 4
Explanation: 
The maximum bitwise AND of all elements in nums is 0. 
The whole array [1,2,3,4] is the longest subarray with maximum bitwise AND equal to 0.

Understanding the Problem

The core of this problem lies in understanding the behavior of the bitwise AND operator (&) and how it interacts with finding the maximum value within a subarray. Let's break down these concepts:

Bitwise AND Operator

The bitwise AND operator performs a logical AND operation on corresponding bits of two operands. For each bit position, if both bits are 1, the resulting bit is 1; otherwise, it's 0.

Example:

  1011 (Decimal: 11)
&amp; 0101 (Decimal: 5)
-------
  0001 (Decimal: 1)

Maximum Bitwise AND

The maximum bitwise AND of an array is the largest value you can obtain by performing bitwise AND on any combination of elements within the array. This value is crucial because it determines the target AND value for our subarray.

Solution Approach

The key to solving this problem is to analyze the properties of the bitwise AND operation and how it relates to finding the longest subarray.

Preprocessing: Finding the Maximum Bitwise AND

First, we need to determine the maximum bitwise AND (max_and) of the entire input array. This value will be our target for identifying the longest subarray.

The simplest way to find max_and is to iterate through the array and perform bitwise AND with a running max_and variable. Here's a Python implementation:

def find_max_and(nums):
    max_and = nums[0]
    for num in nums:
        max_and &amp;= num
    return max_and

Iterating Through the Array

After finding max_and, we iterate through the array nums to identify subarrays that meet our condition. For each element, we maintain:

current_and: The bitwise AND of elements encountered so far within the current subarray.
max_length: The length of the longest subarray found so far.
current_length: The length of the current subarray being analyzed.

Updating the Subarray Length

For each element num in the array:

Calculate the current bitwise AND: current_and &= num
Check if current_and equals the max_and:
If it does, we've found a subarray that satisfies the condition, and we update max_length if current_length is greater.
If current_and is not equal to max_and:
We reset current_length to 1, as we need to start a new subarray.

Return the Maximum Length

After iterating through the entire array, we return max_length as the length of the longest subarray with the maximum bitwise AND.

Code Implementation (Python)

def longestSubarray(nums):
    max_and = find_max_and(nums)  # Find the maximum bitwise AND
    max_length = 0
    current_length = 0
    current_and = 1  # Initialize current_and to 1 (all bits set)

    for num in nums:
        current_and &amp;= num
        current_length += 1
        if current_and == max_and:
            max_length = max(max_length, current_length)
        else:
            current_length = 1
            current_and = num  # Reset current_and for the next subarray

    return max_length

def find_max_and(nums):
    max_and = nums[0]
    for num in nums:
        max_and &amp;= num
    return max_and

Example Walkthrough

Let's walk through an example with nums = [1,2,3,4].

Find max_and: max_and = 0 (calculated using find_max_and(nums)).
1. Iteration:
2. num = 1, current_and = 1, current_length = 1. Since current_and != max_and, current_length is reset to 1.
3. num = 2, current_and = 0, current_length = 1. Since current_and == max_and, max_length is updated to 1.
4. num = 3, current_and = 0, current_length = 2. Since current_and == max_and, max_length is updated to 2.
5. num = 4, current_and = 0, current_length = 3. Since current_and == max_and, max_length is updated to 3.
6. Return max_length: The function returns 3.
  Key Optimizations
Preprocessing: Calculating max_and beforehand avoids redundant computations within the main loop.
- Efficient Bitwise Operations: Using the bitwise AND operator directly on integers is efficient for calculating the maximum AND value.
  Time and Space Complexity
Time Complexity: O(N), where N is the length of the input array nums. We perform a single pass through the array.
- Space Complexity: O(1). We use a constant amount of additional space to store variables.
  Conclusion
  
  This article provided a comprehensive guide to solving the "Longest Subarray With Maximum Bitwise AND" problem. We explored the core concepts of bitwise AND, analyzed the solution approach, and implemented a Python code solution. By understanding the properties of bitwise AND and using efficient iteration, we can effectively identify the longest subarray that meets our criteria. This problem highlights the importance of applying bitwise operations to solve complex array manipulation challenges.

2419. Longest Subarray With Maximum Bitwise AND