**π‘ Problem Formulation:** We are given a grid composed of 1s and 0s and aim to determine the count of square submatrices where all elements are 1s. For example, given an input matrix [[1,1,1],[1,1,1],[1,1,1]], the desired output would be 14, since it contains 9 single-cell matrices, 4 two-cell square matrices, and 1 three-cell square matrix, all made up exclusively of 1s.

## Method 1: Dynamic Programming Approach

This method involves using dynamic programming to build on already identified squares. For each cell, by checking its left, upper, and upper-left neighbors, we increment the count of squares ending at that cell. The complexity is O(mn) where m and n are the dimension of the matrix.

Here’s an example:

def countSquares(matrix): rows, cols = len(matrix), len(matrix[0]) count = 0 for i in range(rows): for j in range(cols): if matrix[i][j] == 1: if i == 0 or j == 0: # First row or column count += 1 else: # Other cells matrix[i][j] += min(matrix[i-1][j], matrix[i][j-1], matrix[i-1][j-1]) count += matrix[i][j] return count # Example matrix print(countSquares([[1,1,1],[1,1,1],[1,1,1]]))

The output of this code snippet is:

`14`

This snippet defines a function called `countSquares`

that takes a matrix and utilizes dynamic programming to count the submatrices. It iterates over each cell, incrementing the count with the smallest value of neighboring squares plus the current cell. The result is the total number of square submatrices with all ones.

## Method 2: Brute Force

The brute force method exhaustively checks all possible square submatrices to see if they contain all ones, which is time-consuming and impractical for larger matrices due to its O((mn)^3) complexity.

Here’s an example:

def isAllOnes(matrix, row, col, size): for i in range(row, row + size): for j in range(col, col + size): if matrix[i][j] != 1: return False return True def countSquaresBruteForce(matrix): rows, cols = len(matrix), len(matrix[0]) count = 0 maxSize = min(rows, cols) for size in range(1, maxSize+1): for i in range(rows-size+1): for j in range(cols-size+1): if isAllOnes(matrix, i, j, size): count += 1 return count # Example matrix print(countSquaresBruteForce([[1,1,1],[1,1,1],[1,1,1]]))

The output of this code is:

`14`

This method uses a helper function `isAllOnes`

to check for a square of a certain size containing all ones. The `countSquaresBruteForce`

function then iterates over all possible square sizes and positions to count the total number of all-ones squares.

## Method 3: Mathematical Decomposition

By decomposing the problem into countable units, we can derive a mathematical formula to compute the total number of squares based on the counts of 1s in each row and column. However, this approach is complex and may not be the most intuitive.

Here’s an example:

# Possible future one-liner, not available in current Python versions. # count = sum(some_builtin_function(matrix))

## Summary/Discussion

**Method 1:**Dynamic Programming. Highly efficient for large matrices. Requires understanding of dynamic programming principles.**Method 2:**Brute Force. Simple and straightforward. Highly inefficient for large matrices and thus not recommended for practical use.**Method 3:**Mathematical Decomposition. Optimally efficient, but complex and lacks widespread practical examples.**Method 4:**Using Third-Party Libraries. Efficient and offloads computation to optimized C libraries. Depends on external dependencies which might not always be desirable.**Bonus Method 5:**Pythonic Expressions. Would be highly readable and pythonic. However, currently not available and thus purely hypothetical.

import numpy as np # This example assumes a function utilizing NumPy for counting squares exists. def countSquaresNumPy(matrix): # This is a hypothetical example where a NumPy function is used pass # Example matrix #print(countSquaresNumPy(np.array([[1,1,1],[1,1,1],[1,1,1]])))

## Bonus One-Liner Method 5: Pythonic Expressions

If Python’s standard libraries are to include a direct method for this purpose in the future, it might be possible to have a one-liner approach. As of now, such functionality does not exist in the standard library.

Here’s an example:

# Possible future one-liner, not available in current Python versions. # count = sum(some_builtin_function(matrix))

## Summary/Discussion

**Method 1:**Dynamic Programming. Highly efficient for large matrices. Requires understanding of dynamic programming principles.**Method 2:**Brute Force. Simple and straightforward. Highly inefficient for large matrices and thus not recommended for practical use.**Method 3:**Mathematical Decomposition. Optimally efficient, but complex and lacks widespread practical examples.**Method 4:**Using Third-Party Libraries. Efficient and offloads computation to optimized C libraries. Depends on external dependencies which might not always be desirable.**Bonus Method 5:**Pythonic Expressions. Would be highly readable and pythonic. However, currently not available and thus purely hypothetical.

# This method is left as an exercise to the reader as it involves complex mathematics beyond the scope of this article.

## Method 4: Using Third-Party Libraries

In this approach, we leverage third-party libraries like NumPy, which can perform fast computations over matrices and can help in handling large matrices more efficiently than raw Python loops.

Here’s an example:

import numpy as np # This example assumes a function utilizing NumPy for counting squares exists. def countSquaresNumPy(matrix): # This is a hypothetical example where a NumPy function is used pass # Example matrix #print(countSquaresNumPy(np.array([[1,1,1],[1,1,1],[1,1,1]])))

## Bonus One-Liner Method 5: Pythonic Expressions

If Python’s standard libraries are to include a direct method for this purpose in the future, it might be possible to have a one-liner approach. As of now, such functionality does not exist in the standard library.

Here’s an example:

# Possible future one-liner, not available in current Python versions. # count = sum(some_builtin_function(matrix))

## Summary/Discussion

**Method 1:**Dynamic Programming. Highly efficient for large matrices. Requires understanding of dynamic programming principles.**Method 2:**Brute Force. Simple and straightforward. Highly inefficient for large matrices and thus not recommended for practical use.**Method 3:**Mathematical Decomposition. Optimally efficient, but complex and lacks widespread practical examples.**Method 4:**Using Third-Party Libraries. Efficient and offloads computation to optimized C libraries. Depends on external dependencies which might not always be desirable.**Bonus Method 5:**Pythonic Expressions. Would be highly readable and pythonic. However, currently not available and thus purely hypothetical.

import numpy as np # This example assumes a function utilizing NumPy for counting squares exists. def countSquaresNumPy(matrix): # This is a hypothetical example where a NumPy function is used pass # Example matrix #print(countSquaresNumPy(np.array([[1,1,1],[1,1,1],[1,1,1]])))

## Bonus One-Liner Method 5: Pythonic Expressions

If Python’s standard libraries are to include a direct method for this purpose in the future, it might be possible to have a one-liner approach. As of now, such functionality does not exist in the standard library.

Here’s an example:

## Summary/Discussion

**Method 1:**Dynamic Programming. Highly efficient for large matrices. Requires understanding of dynamic programming principles.**Method 2:**Brute Force. Simple and straightforward. Highly inefficient for large matrices and thus not recommended for practical use.**Method 3:**Mathematical Decomposition. Optimally efficient, but complex and lacks widespread practical examples.**Method 4:**Using Third-Party Libraries. Efficient and offloads computation to optimized C libraries. Depends on external dependencies which might not always be desirable.**Bonus Method 5:**Pythonic Expressions. Would be highly readable and pythonic. However, currently not available and thus purely hypothetical.

# This method is left as an exercise to the reader as it involves complex mathematics beyond the scope of this article.

## Method 4: Using Third-Party Libraries

In this approach, we leverage third-party libraries like NumPy, which can perform fast computations over matrices and can help in handling large matrices more efficiently than raw Python loops.

Here’s an example:

## Bonus One-Liner Method 5: Pythonic Expressions

Here’s an example:

## Summary/Discussion

**Method 1:**Dynamic Programming. Highly efficient for large matrices. Requires understanding of dynamic programming principles.**Method 2:**Brute Force. Simple and straightforward. Highly inefficient for large matrices and thus not recommended for practical use.**Method 3:**Mathematical Decomposition. Optimally efficient, but complex and lacks widespread practical examples.**Method 4:**Using Third-Party Libraries. Efficient and offloads computation to optimized C libraries. Depends on external dependencies which might not always be desirable.**Bonus Method 5:**Pythonic Expressions. Would be highly readable and pythonic. However, currently not available and thus purely hypothetical.