#1727

Medium

auto_awesomeGreedy choice plus invariant validation

LeetCode Problem Workspace

Largest Submatrix With Rearrangements

Rearrange columns of a binary matrix to find the largest submatrix of 1s.

Problem Statement

You are given a binary matrix of size m x n, and you are allowed to rearrange its columns. Your task is to return the area of the largest submatrix that consists entirely of 1s after optimally rearranging the columns.

To achieve this, you must rearrange the columns of the matrix and find the largest contiguous area made up of 1s. The matrix contains only 0s and 1s, and the maximum number of elements is 10^5, meaning your approach must be efficient.

Examples

Example 1

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

Output: 4

You can rearrange the columns as shown above. The largest submatrix of 1s, in bold, has an area of 4.

Example 2

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

Output: 3

You can rearrange the columns as shown above. The largest submatrix of 1s, in bold, has an area of 3.

Example 3

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

Output: 2

Notice that you must rearrange entire columns, and there is no way to make a submatrix of 1s larger than an area of 2.

Constraints

m == matrix.length
n == matrix[i].length
1 <= m * n <= 105
matrix[i][j] is either 0 or 1.

Solution Approach

Greedy Column Rearrangement

For each row, count the number of consecutive 1s ending at each column. Once this is done, sort the columns based on these counts, allowing you to group 1s together in the most optimal way. This step ensures that you can maximize the area of the 1s submatrix.

Iterative Row Expansion

After sorting, iterate through the rows and calculate the possible submatrix area using the current row's counts of consecutive 1s. For each row, consider it as the base of a potential submatrix and compute the area by multiplying the minimum number of consecutive 1s in each column by the number of rows considered.

Dynamic Validation of Submatrix

To validate the submatrix area, check the consistency of the consecutive 1s for each column. Ensure that for every row, the sequence of consecutive 1s doesn't decrease as you expand the submatrix.

Complexity Analysis

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

The time complexity of the algorithm is O(m * n), where m is the number of rows and n is the number of columns in the matrix. This is because we need to process each element in the matrix to compute the consecutive 1s for each column. The space complexity is O(n) because we store the consecutive 1s counts for each column in a temporary array for each row.

What Interviewers Usually Probe

Candidate demonstrates a clear understanding of greedy algorithms and how to apply them to matrix problems.
The candidate is able to articulate the importance of sorting and invariant validation in optimization problems.
The candidate shows awareness of how to manage large inputs efficiently within the problem's constraints.

Common Pitfalls or Variants

Common pitfalls

Failing to account for the fact that columns must be rearranged as a whole, not individually.
Overcomplicating the solution by attempting brute force or nested loops instead of leveraging sorting and greedy methods.
Neglecting to validate the arrangement of columns after sorting and failing to track the consecutive 1s correctly.

Follow-up variants

Consider cases where multiple rows are completely filled with 1s, testing the scalability of the solution.
Extend the problem to non-binary matrices where other values are allowed.
Explore optimizations that allow solving even larger matrices while maintaining the time complexity.

FAQ

What is the greedy approach in the Largest Submatrix With Rearrangements problem?

The greedy approach involves sorting columns based on the number of consecutive 1s and then calculating the largest possible submatrix area using these sorted values.

How do I handle larger matrices efficiently in this problem?

Focus on optimizing your solution by processing each element in the matrix only once, and using sorting and greedy methods to ensure optimal performance.

Can the Largest Submatrix With Rearrangements problem be solved with dynamic programming?

While dynamic programming may be useful in some matrix problems, the greedy choice combined with invariant validation is more effective for this problem due to the rearrangement constraint.

How do I validate the largest submatrix area after rearranging columns?

After sorting the columns based on the consecutive 1s, you iterate through the rows and compute the area of potential submatrices, validating by ensuring consistent consecutive 1s.

What is the time complexity of the Largest Submatrix With Rearrangements problem?

The time complexity is O(m * n), where m is the number of rows and n is the number of columns. This is efficient given the problem constraints.

terminal

Solution

Solution 1: Preprocessing + Sorting

Since the matrix can be rearranged by columns, we can preprocess each column of the matrix first.

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class Solution:
    def largestSubmatrix(self, matrix: List[List[int]]) -> int:
        for i in range(1, len(matrix)):
            for j in range(len(matrix[0])):
                if matrix[i][j]:
                    matrix[i][j] = matrix[i - 1][j] + 1
        ans = 0
        for row in matrix:
            row.sort(reverse=True)
            for j, v in enumerate(row, 1):
                ans = max(ans, j * v)
        return ans

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