This problem takes us into the world of binary matrices and challenges us to flip columns to make rows identical. Using the power of Python and some creative pattern recognition, we can solve this efficiently. Let’s dive into the solution step by step! 🌟

📝 Problem Statement

We are given a binary matrix of dimensions \(m \times n\), where each cell contains either 0 or 1. You can flip any column, i.e., change all 0s in the column to 1s and all 1s to 0s.

Your goal is to determine the maximum number of rows that can have all values equal (all 0s or all 1s) after flipping some columns.

💡 Key Insights

1. Focus on Patterns: Rows that can be converted into identical rows by flipping columns share a relationship. Specifically, rows that are complements of each other can be unified by flipping specific columns.
2. Standardize Rows: To simplify, we transform each row into a standardized form, either as-is or flipped entirely, depending on its first value.
3. Count and Maximize: Using a counter, we track the frequency of each standardized row and return the maximum frequency.

🔍 Observations and Examples

Example 1

matrix = [[0, 1], 
          [1, 1]]

• Explanation: Without flipping any column, the second row ([1, 1]) is already identical. Maximum identical rows = 1.
• Output: 1

Example 2

matrix = [[0, 1], 
          [1, 0]]

• Explanation: Flip the first column to make both rows [1, 1]. Maximum identical rows = 2.
• Output: 2

Example 3

matrix = [[0, 0, 0], 
          [0, 0, 1], 
          [1, 1, 0]]

• Explanation: Flip the first two columns to make the last two rows [0, 0, 1]. Maximum identical rows = 2.
• Output: 2

🧪 Edge Cases

1. Single Row Matrix

   matrix = [[0, 1, 0]]
   

   Output: 1.

2. All Rows Identical

   matrix = [[1, 1], [1, 1], [1, 1]]
   

   Output: 3.

3. No Flipping Needed

   matrix = [[0, 0], [1, 1]]
   

   Output: 1.

🛠️ Solution

Core Idea

The key is to standardize rows based on their first value:

• If the first element of a row is 0, keep the row as is.
• If the first element of a row is 1, flip all the bits in the row.
  This ensures that complementary rows map to the same pattern.

We then count occurrences of each standardized row using a Counter and return the maximum count.

Python Code Implementation

from collections import Counter
from typing import List

class Solution:
    def maxEqualRowsAfterFlips(self, matrix: List[List[int]]) -> int:
        # Initializing a counter to keep track of the frequency of each standardized row
        row_counter = Counter()
      
        # Iterating through each row in the matrix
        for row in matrix:
            # Standardizing the row:
            # If the first element is 0, keep the row unchanged; otherwise flip all bits
            standardized_row = tuple(row) if row[0] == 0 else tuple(1 - x for x in row)
          
            # Updating the counter for the standardized row
            row_counter[standardized_row] += 1
      
        # Returning the maximum frequency found in the counter as the result
        return max(row_counter.values())

Step-by-Step Walkthrough

Let’s work through an example step by step!

Example Input:

matrix = [[0, 1, 0], 
          [1, 0, 1], 
          [1, 1, 0], 
          [0, 0, 0]]

1. Initialization:
   Start with an empty counter:

   row_counter = {}
   

2. Process Rows:

   • Row 1: [0, 1, 0]
     • First element is 0, so keep as is: (0, 1, 0).
     • Counter: {(0, 1, 0): 1}.
   • Row 2: [1, 0, 1]
     • First element is 1, so flip all bits: (0, 1, 0).
     • Counter: {(0, 1, 0): 2}.
   • Row 3: [1, 1, 0]
     • First element is 1, so flip all bits: (0, 0, 1).
     • Counter: {(0, 1, 0): 2, (0, 0, 1): 1}.
   • Row 4: [0, 0, 0]
     • First element is 0, so keep as is: (0, 0, 0).
     • Counter: {(0, 1, 0): 2, (0, 0, 1): 1, (0, 0, 0): 1}.
3. Find Maximum Frequency:
   The maximum value in the counter is 2 for (0, 1, 0).

Output: 2

⏱ Time Complexity

• Row Standardization: \(O(m \times n)\), where \(m\) is the number of rows and \(n\) is the number of columns.
• Counting Rows: \(O(m)\).
• Overall: \(O(m \times n)\).

🔑 Conclusion

This problem demonstrates the power of pattern recognition and leveraging hashing techniques to optimize complex problems. By transforming rows into a standardized format and counting occurrences, we avoid brute-force computations and achieve an efficient solution.

🎉 Key Takeaways

1. Always look for ways to reduce the problem space by identifying patterns or relationships.
2. Hashable structures like tuples and counters are invaluable for optimizing solutions.
3. Simplify complex transformations into a few logical steps.

With these insights, you’re ready to tackle similar matrix transformation problems. 🚀 Happy coding! 🌟