Flip Columns For Maximum Number of Equal Rows

Flip Columns For Maximum Number of Equal Rows

1. What is this problem about?

The Flip Columns For Maximum Number of Equal Rows interview question is a matrix optimization problem. You are given a binary matrix. You can choose any number of columns and flip all bits in those columns (0 to 1, and 1 to 0). Your goal is to maximize the number of rows that have all identical values (either all 0s or all 1s).

2. Why is this asked in interviews?

Companies like Microsoft and Bloomberg ask the Flip Columns coding problem to see if a candidate can find a clever "canonical representation" for data. Instead of trying all $2^{cols}$ possible flips, the goal is to realize that two rows can be made equal if they are either identical or bitwise inverses of each other. It evaluation your Hash Table interview pattern skills and logical deduction.

3. Algorithmic pattern used

This problem follows the Canonical Grouping pattern.

1. The Insight: Two rows can become "all same values" simultaneously only if they already follow the same pattern or the exact opposite pattern. For example, [0, 1, 0] and [1, 0, 1] can both become [1, 1, 1] if we flip the first and third columns.
2. Normalization: For each row, if it starts with a '1', flip all its bits. This makes every row start with '0'.
3. Frequency Count: Use a Hash Map to count the frequency of these "normalized" row strings.
4. Result: The maximum frequency in the map is the answer.

4. Example explanation

Matrix:

[0, 1]
[1, 1]
[1, 0]

1. Row 0: [0, 1]. Starts with 0. Keep: "01".
2. Row 1: [1, 1]. Starts with 1. Flip: "00".
3. Row 2: [1, 0]. Starts with 1. Flip: "01".

• Pattern "01" appears twice.
• Pattern "00" appears once. Result: 2. (By flipping column 1, row 0 and row 2 both become [0, 0]).

5. Common mistakes candidates make

• Brute Force: Trying all combinations of column flips, which is exponential ($O(2^C)$).
• Ignoring Inverses: Only looking for identical rows and forgetting that a row of [1, 0, 1] is just as useful as [0, 1, 0].
• Matrix Mutation: Actually trying to flip columns in the matrix during the search process.

6. Interview preparation tip

Always look for a "Canonical Form." If multiple inputs can be transformed into the same target, find a way to map them to a single key in a Hash Table. This is a powerful technique for many grouping and optimization problems.