**π‘ Problem Formulation:** Python developers often face the challenge of determining whether all rows in one matrix share at least one common element with another matrix. This might arise in data analysis, where compatibility or correlation between datasets is necessary. For example, given two matrices A and B, we aim to identify whether each row in A has at least one element that is present in any row of B.

## Method 1: Using Nested Loops and Sets

This approach involves iterating through each row of the first matrix and converting it into a set. Then, for each row, we check if this set has any intersection with sets created from each row of the second matrix. While straightforward, this method’s complexity can increase with the size of the matrices.

Here’s an example:

def check_common_elements(matrix1, matrix2): for row1 in matrix1: row1_set = set(row1) for row2 in matrix2: if row1_set.intersection(row2): break else: return False return True matrix_A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]] matrix_B = [[3, 10, 11], [12, 13, 5], [14, 15, 16]] print(check_common_elements(matrix_A, matrix_B))

Output: `True`

This code defines a function `check_common_elements()`

that takes two matrices as arguments. It iterates over every row of the first matrix to check if there exists any common element with any row of the second matrix. With this method, the presence of any common element across all rows from the first matrix means the function returns `True`

.

## Method 2: Using Set Unions and Intersections

A more Pythonic approach is to union sets of elements from each row of the second matrix beforehand and then check for intersections with each row of the first matrix. This method might be faster since it calculates the union only once.

Here’s an example:

def check_common_with_union(matrix1, matrix2): union_matrix2 = set().union(*matrix2) for row in matrix1: if not set(row).intersection(union_matrix2): return False return True print(check_common_with_union(matrix_A, matrix_B))

Output: `True`

The function `check_common_with_union()`

utilizes the union across all rows of the second matrix, making the intersection check for each row of the first matrix more efficient. If any row in the first matrix does not contain a common element with the second matrix’s unified set, it returns `False`

.

## Method 3: Using List Comprehensions and Set Operations

For fans of list comprehensions, this method encapsulates the logic into a succinct and readable one-liner that is idiomatic to Python. However, it may not be as efficient for large matrices due to its compact but potentially complex expression.

Here’s an example:

all(any(set(row1).intersection(row2) for row2 in matrix_B) for row1 in matrix_A)

The one-liner checks for intersections between all pairs of rows from both matrices and wraps the entire logic within the `all()`

function to ensure every row in the first matrix has a common element with at least one row in the second matrix.

## Method 4: Using itertools and Set Intersections

Leveraging Python’s itertools module, you can create product pairs of rows from both matrices and check intersections. While itertools optimizes iteration and is useful for larger datasets, the logic may be less transparent to those unfamiliar with itertools.

Here’s an example:

import itertools def check_common_itertools(matrix1, matrix2): for row1 in matrix1: if not any(set(row1).intersection(row2) for row2 in itertools.product(*matrix2)): return False return True print(check_common_itertools(matrix_A, matrix_B))

Output: `True`

The `check_common_itertools()`

function iterates through the first matrix and uses `itertools.product()`

to form all possible combinations with the second matrix, then checks these combinations for intersecting elements. If a row from the first matrix has no common elements with any combination from the second matrix, it returns `False`

.

## Bonus One-Liner Method 5: Using numpy

For those using numpy arrays, here’s a bonus one-liner that utilizes numpy’s broadcasting and vectorization features to efficiently find common elements. Note that this method requires numpy and performs well on large arrays.

Here’s an example:

import numpy as np matrix_A_np = np.array(matrix_A) matrix_B_np = np.array(matrix_B) common_elements = np.any(np.isin(matrix_A_np[:, None, :], matrix_B_np), axis=(1, 2)) print(all(common_elements))

Output: `True`

By converting the matrices into numpy arrays and using broadcasting with `np.isin()`

, this code efficiently computes common elements between the two matrices. If all rows in the first matrix have common elements with any row in the second matrix, `all(common_elements)`

will be `True`

.

## Summary/Discussion

**Method 1:**Nested Loops and Sets. Straightforward and simple. Potentially inefficient for large matrices.**Method 2:**Using Set Unions and Intersections. Efficient due to a single computation of the union. Still requires iteration over the first matrix.**Method 3:**List Comprehensions and Set Operations. Compact and pythonic. May be less efficient and readable for complex expressions or large datasets.**Method 4:**Using itertools and Set Intersections. Good for large datasets. Less readable for those unfamiliar with itertools.**Bonus Method 5:**Using numpy. Highly efficient with numpy’s broadcasting and vectorization. Requires numpy and more suitable for large numerical arrays.