Count Square Submatrices with All Ones

1277. Count Square Submatrices with All Ones

Difficulty: Medium

Topics: Array, Dynamic Programming, Matrix

Given a m * n matrix of ones and zeros, return how many square submatrices have all ones.

Example 1:

• Input: matrix = [[0,1,1,1], [1,1,1,1], [0,1,1,1]]
• Output: 15
• Explanation:
  • There are 10 squares of side 1.
  • There are 4 squares of side 2.
  • There is 1 square of side 3.
  • Total number of squares = 10 4 1 = 15.

Example 2:

• Input: matrix = [[1,0,1], [1,1,0], [1,1,0]]
• Output: 7
• Explanation:
  • There are 6 squares of side 1.
  • There is 1 square of side 2.
  • Total number of squares = 6 1 = 7.

Constraints:

• 1 <= arr.length <= 300
• 1 <= arr[0].length <= 300
• 0 <= arr[i][j] <= 1

Hint:

1. Create an additive table that counts the sum of elements of submatrix with the superior corner at (0,0).
2. Loop over all subsquares in O(n³) and check if the sum make the whole array to be ones, if it checks then add 1 to the answer.

Solution:

We can use Dynamic Programming (DP) to keep track of the number of square submatrices with all ones that can end at each cell in the matrix. Here's the approach to achieve this:

1. DP Matrix Definition:

   • Define a DP matrix dp where dp[i][j] represents the size of the largest square submatrix with all ones that has its bottom-right corner at cell (i, j).

2. Transition Formula:

   • For each cell (i, j) in the matrix:

     • If matrix[i][j] is 1, the value of dp[i][j] depends on the minimum of the squares that can be formed by extending from (i-1, j), (i, j-1), and (i-1, j-1). The transition formula is:

     dp[i][j] = min(dp[i-1][j], dp[i][j-1], dp[i-1][j-1]) + 1
     

  - If `matrix[i][j]` is 0, `dp[i][j]` will be 0 because a square of ones cannot end at a cell with a zero.

1. Count All Squares:

   • Accumulate the values of dp[i][j] for all (i, j) to get the total number of squares of all sizes.

2. Time Complexity:

   • The solution works in O(m X n), where m and n are the dimensions of the matrix.

Let's implement this solution in PHP: 1277. Count Square Submatrices with All Ones

dp[i][j] = min(dp[i-1][j], dp[i][j-1], dp[i-1][j-1]) + 1

Explanation:

1. We initialize a 2D array dp to keep track of the size of the largest square submatrix ending at each position (i, j).
2. For each cell in the matrix:
   • If the cell has a 1, we compute dp[i][j] based on neighboring cells and add its value to totalSquares.
3. Finally, totalSquares contains the count of all square submatrices with all ones.

This solution is efficient and meets the constraints provided in the problem.

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