Largest Submatrix With Rearrangements

Problem Description

You are given a binary matrix of size m x n, and you are allowed to rearrange the columns of the matrix in any order. Return the area of the largest submatrix within the matrix where every element of the submatrix is 1 after reordering the columns optimally.

Key Insights

• The problem requires the identification of the largest contiguous submatrix of 1s after optimally rearranging the columns.
• The height of 1s in each column can be calculated, creating a histogram-like representation for each row.
• Sorting the heights can help determine the largest rectangle area that can be formed at each row level.
• The area can be computed as the product of the height of 1s and the number of columns up to that height.

Space and Time Complexity

Time Complexity: O(m * n log n)
Space Complexity: O(n)

Solution

To solve this problem, we can utilize a greedy approach combined with sorting. The algorithm follows these steps:

1. Calculate the height of 1s for each column for every row, effectively creating a histogram for each row.
2. For each row, sort the heights in descending order.
3. The area of the largest rectangle that can be formed at that row is determined by multiplying the height with its corresponding index (1-based).
4. Track the maximum area found across all rows.

Code Solutions