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Submatrices That Sum to Target

Certification: Advanced Level Accuracy: 100% Submissions: 1 Points: 15

Write a Java program to count the number of submatrices in a given matrix that sum to a target value. A submatrix is a contiguous rectangular region within the matrix.

Example 1
    
  
  • Input: 
matrix = [
    [0, 1, 0],
    [1, 1, 1],
    [0, 1, 0]
]
target = 3
    • 
  
  • Output: 4
    • 
  
  • Explanation:
Step 1: The matrix has dimensions 3x3 and the target is 3.
Step 2: There are 4 submatrices that sum to 3:
   - The 1x3 submatrix at row 1, columns 0-2: [1, 1, 1]
   - The 3x1 submatrix at column 1, rows 0-2: [1, 1, 1]
   - The 2x2 submatrix at rows 0-1, columns 0-1: [[0, 1], [1, 1]]
   - The 2x2 submatrix at rows 0-1, columns 1-2: [[1, 0], [1, 1]]
Step 3: The total count is 4.
    •
Example 2
    
  
  • Input: 
matrix = [
    [1, -1],
    [-1, 1]
]
target = 0
    • 
  
  • Output: 5
    • 
  
  • Explanation:
Step 1: The matrix has dimensions 2x2 and the target is 0.
Step 2: There are 5 submatrices that sum to 0:
   - The 1x2 submatrix at row 0, columns 0-1: [1, -1]
   - The 1x2 submatrix at row 1, columns 0-1: [-1, 1]
   - The 2x1 submatrix at column 0, rows 0-1: [1, -1]
   - The 2x1 submatrix at column 1, rows 0-1: [-1, 1]
   - The 2x2 submatrix at rows 0-1, columns 0-1: [[1, -1], [-1, 1]]
Step 3: The total count is 5.
    •
Constraints
    
  
  • 1 <= matrix.length <= 100
    • 
  
  • 1 <= matrix[0].length <= 100
    • 
  
  • -1000 <= matrix[i][j] <= 1000
    • 
  
  • -10^8 <= target <= 10^8
    • 
  
  • Time Complexity: O(r^2 * c), where r is the number of rows and c is the number of columns
    • 
  
  • Space Complexity: O(c), where c is the number of columns
    •
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Hint
Solution

Solution Hints

    
  
  • Use prefix sums to efficiently calculate the sum of any submatrix
    • 
  
  • For each pair of rows (top and bottom), calculate the sum of each column between these rows
    • 
  
  • Convert the 2D problem to a 1D problem by computing the sum array for each fixed pair of rows
    • 
  
  • Use a hash map to count the number of subarrays that sum to the target
    • 
  
  • Iterate through the cumulative sums and check if (currentSum - target) exists in the hash map
    • 
  
  • Update the count based on the frequency found in the hash map
    • 
  
  • Return the final count of valid submatrices
    •

Steps to solve by this approach:

 Step 1: For each pair of rows (top and bottom), we reduce the 2D problem to a 1D problem by summing up the values in each column between these rows.

 Step 2: We create a colSums array that stores the sum of each column for the current range of rows.

 Step 3: For each row pair, we use a helper method subarraySum to count subarrays with the target sum.

 Step 4: The subarraySum method uses a hash map to track prefix sums and their frequencies.

 Step 5: For each element, we calculate the running sum and check if (currentSum - target) exists in our map.

 Step 6: If it exists, we add its frequency to our count, as each occurrence represents a subarray with the target sum.

 Step 7: We return the total count of all valid submatrices found.

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