This problem requires counting the number of submatrices that consist entirely of ones in a given binary matrix. It can be approached with dynamic programming by constructing a nums array for each row, where each entry stores the consecutive number of ones up to that position. This allows for efficient counting of submatrices in O(m * n) time.
Problem Statement
Given an m x n binary matrix mat, your task is to return the number of submatrices that consist entirely of ones. Each submatrix is defined by its top-left and bottom-right corners. Submatrices can vary in size, and you must count all the possible submatrices that contain only 1s.
For example, consider the matrix mat = [[1,0,1],[1,1,0],[1,1,0]]. The task is to identify all the submatrices that contain only ones and return the total count. These submatrices include both smaller (1x1) and larger (2x2, 3x1) formations, requiring an efficient approach to avoid brute-force counting.
Examples
Example 1
Input: mat = [[1,0,1],[1,1,0],[1,1,0]]
Output: 13
There are 6 rectangles of side 1x1. There are 2 rectangles of side 1x2. There are 3 rectangles of side 2x1. There is 1 rectangle of side 2x2. There is 1 rectangle of side 3x1. Total number of rectangles = 6 + 2 + 3 + 1 + 1 = 13.
Example 2
Input: mat = [[0,1,1,0],[0,1,1,1],[1,1,1,0]]
Output: 24
There are 8 rectangles of side 1x1. There are 5 rectangles of side 1x2. There are 2 rectangles of side 1x3. There are 4 rectangles of side 2x1. There are 2 rectangles of side 2x2. There are 2 rectangles of side 3x1. There is 1 rectangle of side 3x2. Total number of rectangles = 8 + 5 + 2 + 4 + 2 + 2 + 1 = 24.
Constraints
- 1 <= m, n <= 150
- mat[i][j] is either 0 or 1.
Solution Approach
State Transition Dynamic Programming
To solve this problem, we use a dynamic programming approach where each row's nums array tracks the length of consecutive ones at each position. This helps in efficiently calculating the number of submatrices by iterating over each row and leveraging the nums array to identify possible submatrices.
Building the nums Array
For each row i, the nums array is constructed such that nums[j] represents the number of consecutive ones ending at mat[i][j]. If mat[i][j] is 0, nums[j] is set to 0; otherwise, it is incremented by 1 from nums[j-1]. This preprocessing step simplifies the submatrix counting process.
Counting Submatrices
After constructing the nums array for each row, count the submatrices by iterating through the matrix and using the values in nums to determine the number of submatrices that can be formed from each position. This approach ensures that the solution is both time and space efficient.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time complexity of this approach is O(m * n) due to the iteration over each element of the matrix, where m is the number of rows and n is the number of columns. The space complexity is O(n) as the nums array only requires storage for one row at a time.
What Interviewers Usually Probe
- The candidate should demonstrate familiarity with dynamic programming and matrix manipulation techniques.
- Look for a clear understanding of how to preprocess rows with the
numsarray to optimize the submatrix counting process. - Expect the candidate to discuss time and space complexities in the context of dynamic programming and matrix traversal.
Common Pitfalls or Variants
Common pitfalls
- Failing to optimize the solution by using the
numsarray can lead to inefficient brute-force counting. - Overlooking the need for row-wise preprocessing of the matrix, which can hinder performance in larger test cases.
- Misunderstanding the role of dynamic programming and treating the problem as a straightforward iteration over all submatrices.
Follow-up variants
- The problem can be generalized to non-binary matrices, where elements may have different values.
- A variant could involve considering rectangles instead of squares in submatrices.
- In a more challenging variant, you might be asked to find submatrices with a sum equal to a given target rather than all ones.
How GhostInterview Helps
- GhostInterview's dynamic programming hints can guide candidates through building the
numsarray efficiently, ensuring optimal space and time complexity. - The platform offers a focused approach to practicing matrix-related dynamic programming problems, honing skills in preprocessing and state transitions.
- With step-by-step problem-solving tools, GhostInterview helps candidates avoid common pitfalls by emphasizing best practices in dynamic programming for matrix-based problems.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
What is the main pattern to solve the Count Submatrices With All Ones problem?
The main pattern is state transition dynamic programming, where you preprocess each row into a nums array representing the count of consecutive ones.
How do you count submatrices in this problem?
Submatrices are counted by iterating through each row and using the nums array to determine the number of submatrices ending at each position.
What is the time complexity of solving Count Submatrices With All Ones?
The time complexity is O(m * n), where m is the number of rows and n is the number of columns in the matrix.
Can this problem be optimized further?
Yes, the solution can be optimized using dynamic programming by avoiding brute-force submatrix counting and using row-wise preprocessing with the nums array.
What are some common mistakes when solving Count Submatrices With All Ones?
Common mistakes include not using dynamic programming for efficient row preprocessing and brute-forcing the submatrix counting process without optimizing with the nums array.
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The problem asks for the minimum number of operations to transform an initial array of zeros into a target array using subarray increments.
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