Problem Description

You are presented with a binary matrix grid, which is a 2D array containing 0s and 1s. Your task is to maximize the "score" of this matrix. The score is calculated by summing up all the rows considered as binary numbers. In order to raise the score, you can perform as many "moves" as you want. A move consists of picking any row or column and flipping all the bits in it; that is, all 0s are changed to 1s and vice versa.

For example, if you have a row [1, 0, 1] and you perform a move, this row will become [0, 1, 0].

The primary goal is to find the highest score possible after performing an optimal sequence of moves.

Intuition

The key observation to solve this problem is to realize that the most significant bit in a binary number has the highest value, so it should always be set to 1 to maximize the number. Therefore, the first step is to ensure all rows start with a 1. If a row starts with a 0, we flip the entire row.

The next step is to consider each column, starting from the second one since the first column should already have all 1s after the initial step. At each column, we can maximize the score by making sure that the number of 1s is greater than or equal to the number of 0s. If there are more 0s than 1s, we flip the entire column. This is because flipping a column doesn't change the leftmost 1s that we have set in the first step but increases the number of 1s in the current column, therefore increasing the overall score.

By applying these two steps, first row-wise, then column-wise, we ensure that each bit contributes the maximum possible value to the score of the matrix.

Here's the reasoning for the solution:

1. Flip the rows if the first bit is 0 (outer loop over rows).
2. For each column, count the number of 1s. If the number is less than half of the total rows, flip the column (inner loop over columns).
3. Calculate the score by adding the value of each column's bits to the total score.

In the provided code, grid[i][j] ^= 1 is used to flip the bits (using the XOR operator). max(cnt, m - cnt) * (1 << (n - j - 1)) is the calculation of the contribution of each column to the score, choosing to flip the column or not depending on which option gives more 1s, and then multiplying by the value of the bit position, which decreases as we move right.

Learn more about Greedy patterns.

Solution Approach

The solution takes a greedy approach to maximizing the score of the binary matrix by making sure that the most significant bit of every row is set to 1 and then ensuring that each column contains as many 1s as possible.

Here's the breakdown of how the code implements the solution:

1. The first loop iterates through each row in the binary matrix grid. The size of the matrix is determined by m rows and n columns.

2. Inside this loop, we check if the first element of the row (grid[i][0]) is 0. If so, we flip the entire row using a nested loop which toggles each element grid[i][j] ^= 1. This is done using the XOR operator ^, which will flip a 0 to 1 and vice versa.

3. The outer loop with the variable j goes through each column. The inner loop counts the number of 1s in the current column. Instead of flipping each bit when necessary, we're just counting the number of 1s because we can compute the column contribution to the score mathematically, without modifying the matrix.

4. For each column, we calculate the maximum possible value it can contribute to the score based on the current state of bits in that column. This is computed by max(cnt, m - cnt). We take the larger value between the number of 1s (cnt) and the number of 0s (m - cnt), considering that if the 0s were in the majority, a column flip would change them to 1s.

5. The column's contribution to the overall score is then computed by multiplying this maximum number with the binary value of the position. In a binary number, the leftmost bit is the most significant; hence, the value is computed by 1 << (n - j - 1), which is equivalent to raising 2 to the power of the column's index from the right (0-based). For example, if the column is the second from the right in a 4-column grid, its value is 2^(4-2-1) = 2^(1) = 2.

6. This contribution is added to ans for each column to build up the total score.

7. Once all columns have been processed, the variable ans holds the highest possible score, which is returned.

The algorithm efficiently calculates the maximum score possible without needing to actually perform the flips visually or record the state of the matrix after each flip, thanks to mathematical binary operations and the properties of binary numbers.

Example Walkthrough

Let's take a simple example and walk through the solution approach to understand it better.

Consider the binary matrix, grid, represented by the following 2D array:

0 1
1 1
0 1

The matrix has 3 rows (m = 3) and 2 columns (n = 2). We want to maximize the "score" of this matrix through flips.

Step by Step Solution:

1. Make sure that all rows start with 1:

   • Row 1 starts with 0, so we flip it: 0 1 becomes 1 0.
   • Row 3 also starts with 0, we flip it: 0 1 becomes 1 0.

   After this step, our grid looks like this:

    1 0
    1 1
    1 0

2. Now, we need to maximize 1s in each column. We note that the first column already has all 1s because of the initial row flips. So we consider the second column.

3. Count the number of 1s in the second column:

   • There is only 1 1 in the second column.

   We want the majority of the bits in each column to be 1. Since there is only 1 '1' and 2 '0's in the second column, we'll consider this column flipped for maximization purposes.

4. Calculate the contribution of each column to the total score:

   • The first column, having all 1s, contributes 2^(n-1-0) = 2^(2-1-0) = 2^1 = 2 for each row, totaling 2 * 3 = 6.
   • The second column, we consider flipped (for scoring purposes), so it effectively has 2 1s, and each contributes 2^(n-2-0) = 2^(2-2-0) = 2^0 = 1 to the score. The total from this column will be 1 * 2 = 2.

5. The total score is then the sum of the contributions from step 4:

   • Total score = Score from column 1 + Score from column 2
   • Total score = 6 + 2
   • Total score = 8

The highest possible score for the given matrix after the optimal sequence of moves is 8.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given solution involves iterating over each cell of the grid to potentially toggle its value, followed by counting the number of 1's in each column to maximize the row score.

• The first loop, which toggles the cells if the leftmost bit is 0, traverses all cells in the grid and performs a constant-time XOR operation. This double loop runs in O(m * n) time, where m is the number of rows and n is the number of columns.

• The second loop calculates the count of 1's in each column and determines the maximum score by considering the current and the inverse value count. This also runs in O(m * n) time since it involves iterating over each column for all rows.

The two operations above are consecutive, resulting in a total time complexity of O(m * n) + O(m * n), which simplifies to O(m * n) because constant factors are dropped.

Space Complexity

The space complexity of the algorithm is determined by any additional space that we use relative to the input size:

• No extra space is used for processing besides a few variables for iteration (i, j) and storing intermediate results (such as cnt and ans). These use a constant amount of space irrespective of the input size.

• Since the grid is modified in place, and no additional data structures are used to store intermediate results, the space complexity remains constant.

Given the fixed number of variables with space independent of the input size, the space complexity is O(1), which signifies constant space complexity.

Learn more about how to find time and space complexity quickly using problem constraints.