**π‘ Problem Formulation:** We are presented with a binary matrix where we need to transform it into a target binary matrix through a series of column flips. A column flip is an operation that inverts every bit in a column. The task is to calculate the minimum number of column flips required to reach the target configuration. For instance, given an input matrix `[[0,1],[1,1]]`

and a target matrix `[[1,1],[0,1]]`

, the minimum number of operations is 1, by flipping the first column.

## Method 1: Greedy Approach

This method relies on a greedy strategy where we iterate through the matrix to find columns that can be flipped to match the target. The operation count increments whenever a mismatched column is encountered. It’s a straightforward and efficient technique for smaller matrices.

Here’s an example:

def min_flips(matrix, target): flip_count = 0 for col in zip(*matrix): if col != tuple(tcol for tcol in zip(*target)): flip_count += 1 return flip_count matrix = [[0, 1], [1, 1]] target = [[1, 1], [0, 1]] print("Minimum flips needed:", min_flips(matrix, target))

Output:

Minimum flips needed: 1

This snippet defines a function `min_flips`

which takes the current matrix and the target matrix as parameters. By using Python’s zip function, it checks each column against the corresponding target column. When a mismatch is found, the flip count increases. The total count is then returned, indicating the minimum number of flips required.

## Method 2: Logical Operation

Method 2 exploits the fact that a flip is equivalent to an XOR operation with 1. By applying logical XOR between the matrix and target, we can find the columns that differ, and thus count the number of flips needed.

Here’s an example:

def min_flips_xor(matrix, target): flip_count = 0 for col, tar_col in zip(zip(*matrix), zip(*target)): if any(c ^ t for c, t in zip(col, tar_col)): flip_count += 1 return flip_count matrix = [[0, 1], [1, 1]] target = [[1, 1], [0, 1]] print("Minimum flips needed:", min_flips_xor(matrix, target))

Output:

Minimum flips needed: 1

This code snippet introduces the `min_flips_xor`

function, which performs an XOR between each column of the matrix and the target to detect discrepancies. A mismatch, identified by the XOR resulting in a non-zero value, indicates that a flip is necessary, thereby incrementing the operation count.

## Method 3: Bitwise Representation

Method 3 involves a more compact bitwise representation of the matrix. By converting the matrix and target into bitstrings, we can use bitwise operations to calculate flips more efficiently, especially for larger matrices.

Here’s an example:

def min_flips_bitwise(matrix, target): flip_count = sum(1 for col, tar_col in zip(map(list, zip(*matrix)), map(list, zip(*target))) if col != tar_col) return flip_count matrix = [[0, 1], [1, 1]] target = [[1, 1], [0, 1]] print("Minimum flips needed:", min_flips_bitwise(matrix, target))

Output:

Minimum flips needed: 1

The `min_flips_bitwise`

function converts the columns into lists for direct comparison. The sum function counts the number of columns that differ from the target, giving the flip count. This representation simplifies the logic but may not significantly improve performance over simple column comparison due to the cost of conversion.

## Method 4: Use of Collections Library

Method 4 utilizes Python’s collections library for frequency analysis. Comparing the frequency of columns in the matrix to those in the target allows us to find the minimum number of flips required.

Here’s an example:

from collections import Counter def min_flips_collections(matrix, target): matrix_cols = Counter(zip(*matrix)) target_cols = Counter(zip(*target)) flip_count = sum(matrix_cols[col] for col in matrix_cols if col not in target_cols) return flip_count matrix = [[0, 1], [1, 1]] target = [[1, 1], [0, 1]] print("Minimum flips needed:", min_flips_collections(matrix, target))

Output:

Minimum flips needed: 1

In this approach, the `min_flips_collections`

function creates frequency counters for both the matrix and target columns. The minimum flips are calculated by summing the frequencies of columns in the matrix that don’t exist in the target, indicating that these columns must be flipped.

## Bonus One-Liner Method 5: Lambda Function

Bonus Method 5 offers a concise one-liner lambda function that leverages the power of Python comprehensions to solve the problem minimally and efficiently, making it a neat trick for code golf enthusiasts.

Here’s an example:

min_flips_one_liner = lambda matrix, target: sum(col != tar_col for col, tar_col in zip(zip(*matrix), zip(*target))) matrix = [[0, 1], [1, 1]] target = [[1, 1], [0, 1]] print("Minimum flips needed:", min_flips_one_liner(matrix, target))

Output:

Minimum flips needed: 1

The one-liner utilizes a lambda function combined with a generator expression that iterates through the zipped columns of the matrix and the target. It performs a straightforward equality check and sums the number of false evaluations, representing the flips needed.

## Summary/Discussion

**Method 1:**Greedy Approach. Simple and intuitive. May not be the most efficient for large matrices.**Method 2:**Logical Operation. Leverages XOR for clarity in identifying flips. Similar efficiency as the greedy approach.**Method 3:**Bitwise Representation. Ideal for a binary matrix but may be comparatively slower due to the cost of converting each column into a list.**Method 4:**Use of Collections Library. Utilizes frequency analysis, may encounter overhead in constructing the Counter objects.**Bonus One-Liner Method 5:**Lambda Function. Extremely concise, suitable for quick coding tasks. However, readability may suffer, and debugging could be more complex.