#1072

Medium

auto_awesomeArray scanning plus hash lookup

LeetCode Problem Workspace

Flip Columns For Maximum Number of Equal Rows

Maximize the number of equal rows in a binary matrix by flipping any number of columns.

Problem Statement

You are given an m x n binary matrix, where each element is either 0 or 1. You can flip any number of columns in the matrix, meaning you can toggle each cell's value in a column (0 becomes 1 and 1 becomes 0). The goal is to return the maximum number of rows that can be made equal after performing some number of column flips.

For example, if you are given the matrix [[0,1],[1,1]], you cannot flip any columns, and only one row has equal values. On the other hand, if the matrix is [[0,1],[1,0]], flipping the first column will make both rows equal, so the result would be 2.

Examples

Example 1

Input: matrix = [[0,1],[1,1]]

Output: 1

After flipping no values, 1 row has all values equal.

Example 2

Input: matrix = [[0,1],[1,0]]

Output: 2

After flipping values in the first column, both rows have equal values.

Example 3

Input: matrix = [[0,0,0],[0,0,1],[1,1,0]]

Output: 2

After flipping values in the first two columns, the last two rows have equal values.

Constraints

m == matrix.length
n == matrix[i].length
1 <= m, n <= 300
matrix[i][j] is either 0 or 1.

Solution Approach

Array Scanning

The problem requires scanning each row to identify potential patterns of equal rows after column flips. The idea is to consider flipping certain columns to create uniformity across rows. By treating each row as a binary string, flipping columns becomes equivalent to flipping specific bits in the string representation of each row.

Hash Table Lookup

After scanning the rows, use a hash table to track the number of occurrences of each unique row pattern. This helps in determining which row patterns can be maximized by flipping columns. The key insight is to flip columns such that the most common row pattern can be achieved across the matrix.

Bitwise XOR Optimization

Flipping a subset of columns can be seen as performing a bitwise XOR operation. For a given row X, flipping columns results in XORing the row with a number K. To achieve uniform rows, we look for rows that can be transformed into all zeros or all ones using XOR. This helps us optimize the column flipping process to maximize the number of equal rows.

Complexity Analysis

Metric	Value
Time	O(n \cdot m)
Space	O(n \cdot m)

The time complexity of the solution is O(n * m), where n is the number of rows and m is the number of columns. This is due to scanning each row and processing its columns for pattern matching. The space complexity is O(n * m) because we store the transformed rows in a hash table for lookup and comparison.

What Interviewers Usually Probe

Look for the candidate's understanding of binary operations and pattern matching within arrays.
Gauge the candidate's ability to utilize hash tables effectively for optimizing solutions.
Evaluate the candidate's approach to reducing unnecessary computations when flipping columns.

Common Pitfalls or Variants

Common pitfalls

Forgetting to account for the XOR operation's effect on the row patterns when flipping columns.
Overcomplicating the solution by trying to brute-force all possible column flips instead of using a hash table for optimization.
Missing edge cases such as matrices where no flips are needed or where flipping results in fewer equal rows.

Follow-up variants

How would the solution change if we could only flip a single column?
What if the matrix had more than two distinct values (e.g., 0, 1, and 2)?
Can the solution be optimized further using advanced techniques like bitwise manipulation or dynamic programming?

FAQ

How can I maximize the number of equal rows in a binary matrix?

To maximize equal rows, scan each row and use a hash table to track patterns. Flip columns to match the most frequent pattern.

What is the time complexity of the solution?

The time complexity is O(n * m), where n is the number of rows and m is the number of columns.

What is the key insight behind solving this problem?

The key insight is to use a hash table to track row patterns and flip columns to maximize the occurrence of the most frequent row pattern.

How does the XOR operation help in flipping columns?

The XOR operation helps by transforming rows in a way that mimics the effect of flipping specific columns. It ensures that rows are either all zeros or all ones after flipping columns.

What are the common pitfalls in this problem?

Common pitfalls include not properly considering the XOR operation's effects, overcomplicating the solution, or missing edge cases like matrices with no flips required.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def maxEqualRowsAfterFlips(self, matrix: List[List[int]]) -> int:
        cnt = Counter()
        for row in matrix:
            t = tuple(row) if row[0] == 0 else tuple(x ^ 1 for x in row)
            cnt[t] += 1
        return max(cnt.values())

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