Flip Columns For Maximum Number of Equal Rows

1072. Flip Columns For Maximum Number of Equal Rows

Difficulty: Medium

Topics: Array, Hash Table, Matrix

You are given an m x n binary matrix matrix.

You can choose any number of columns in the matrix and flip every cell in that column (i.e., Change the value of the cell from 0 to 1 or vice versa).

Return the maximum number of rows that have all values equal after some number of flips.

Example 1:

• Input: matrix = [[0,1],[1,1]]
• Output: 1
• Explanation: After flipping no values, 1 row has all values equal.

Example 2:

• Input: matrix = [[0,1],[1,0]]
• Output: 2
• Explanation: 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
• Explanation: After flipping values in the first two columns, the last two rows have equal values.

Constraints:

• m == matrix.length
• n == matrix[i].length
• 1
• matrix[i][j] is either 0 or 1.

Hint:

1. Flipping a subset of columns is like doing a bitwise XOR of some number K onto each row. We want rows X with X ^ K = all 0s or all 1s. This is the same as X = X^K ^K = (all 0s or all 1s) ^ K, so we want to count rows that have opposite bits set. For example, if K = 1, then we count rows X = (00000...001, or 1111....110).

Solution:

We can utilize a hash map to group rows that can be made identical by flipping certain columns. Rows that can be made identical have either the same pattern or a complementary pattern (bitwise negation).

Here’s the step-by-step solution:

Algorithm:

1. For each row, calculate its pattern and complementary pattern:
   • The pattern is the row as it is.
   • The complementary pattern is the result of flipping all bits in the row.
2. Use a hash map to count occurrences of patterns and their complements.
3. The maximum count for any single pattern or its complement gives the result.

Let's implement this solution in PHP: 1072. Flip Columns For Maximum Number of Equal Rows

<?php /**
 * @param Integer[][] $matrix
 * @return Integer
 */
function maxEqualRowsAfterFlips($matrix) {
    ...
    ...
    ...
    /**
     * go to ./solution.php
     */
}

// Example usage
$matrix1 = [[0, 1], [1, 1]];
$matrix2 = [[0, 1], [1, 0]];
$matrix3 = [[0, 0, 0], [0, 0, 1], [1, 1, 0]];

echo maxEqualRowsAfterFlips($matrix1) . "\n"; // Output: 1
echo maxEqualRowsAfterFlips($matrix2) . "\n"; // Output: 2
echo maxEqualRowsAfterFlips($matrix3) . "\n"; // Output: 2
?>

Explanation:

1. Pattern and Complement:
   • For each row, the pattern is the concatenated row (e.g., 010).
   • The complement flips all bits of the row (e.g., 101).
2. Hash Map: Count the occurrences of each pattern and its complement. This helps group rows that can be made identical.
3. Max Count: Find the maximum count of a single pattern or its complement to determine how many rows can be made identical.

Complexity:

• Time Complexity: O(m x n), where m is the number of rows and n is the number of columns.
• Space Complexity: O(m x n), for storing patterns in the hash map.

This solution adheres to the constraints and is efficient for the problem size.

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