Finding the Column with Maximum Zeros in a Matrix

Finding the Column with Maximum Zeros in a Matrix

When working with a 2D matrix filled with only 0s and 1s, it often becomes essential to extract meaningful information from it. In this article, we'll explore a problem that requires finding the column in such a matrix with the maximum number of 0s. If more than one column has the maximum number of 0s, we will print the one that comes first. If there are no columns with 0s, we will return -1. We will provide an intuitive understanding of the problem, a step-by-step approach to solve it, and a Python code implementation. Additionally, we'll include a brief dry run of the provided optimized code, analyze its time and space complexity, and conclude our discussion.

Introduction

Consider a square matrix of size N*N, where each element can be either 0 or 1. We want to determine which column in this matrix contains the most 0s. If multiple columns have the same maximum number of 0s, we will select the leftmost column among them. If no column contains any 0s, we will return -1.

Intuition

To tackle this problem efficiently, we need to iterate through each column in the matrix, counting the number of 0s in each column. We'll keep track of the column with the maximum count of 0s and return its index.

Approach

We'll use a Python class with a method called `columnWithMaxZeros` to implement our solution. This method will take the matrix and its size `N` as input. Here's a step-by-step approach to solving the problem:

1. Initialize two variables: `min_non_zero_count` to keep track of the minimum non-zero count in any column (initialize to positive infinity), and `max_zero_column` to store the index of the column with the maximum number of 0s (initialize to -1).
2. Iterate through each column in the matrix (from 0 to N-1).
3.  For each column, count the number of 0s by iterating through the rows and summing up the 0s.
4. Check if the current column has fewer non-zeros than the column with the current minimum non-zero count. If this is the case, update `min_non_zero_count` to the new minimum and `max_zero_column` to the current column index.
5. After looping through all columns, return the index stored in "max_zero_column".

Here's the Python code implementing the above approach:

class Solution:
    def columnWithMaxZeros(self, matrix, N):
        min_non_zero_count, max_zero_column = float("inf"), -1

        for column_index in range(N):
            zero_count = 0
            for row_index in range(N):
                zero_count += matrix[row_index][column_index]

            if zero_count < N and min_non_zero_count > zero_count:
                min_non_zero_count, max_zero_column = zero_count, column_index

        return max_zero_column

Time and Space Complexity

Time Complexity: The code iterates through each element of the matrix exactly once, resulting in a time complexity of O(N^2).

Space Complexity: The code uses a constant amount of space for variables, so the space complexity is O(1).

Conclusion

In this article, we explored a common problem involving 2D matrices. We learned how to efficiently find the column with the maximum number of 0s in such a matrix. The provided Python code, along with a detailed explanation and a dry run example, should help you grasp the concept and solution. By following the intuitive approach and optimized code, you can easily solve similar problems that involve matrix manipulation and searching for specific patterns.