# 1074. Number of Submatrices That Sum to Target

#### Hard

***

Given a `matrix` and a `target`, return the number of non-empty submatrices that sum to target.

A submatrix `x1, y1, x2, y2` is the set of all cells `matrix[x][y]` with `x1 <= x <= x2` and `y1 <= y <= y2`.

Two submatrices `(x1, y1, x2, y2)` and `(x1', y1', x2', y2')` are different if they have some coordinate that is different: for example, if `x1 != x1'`.

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**Example 1:**

![](https://assets.leetcode.com/uploads/2020/09/02/mate1.jpg)

```
Input: matrix = [[0,1,0],[1,1,1],[0,1,0]], target = 0
Output: 4
Explanation: The four 1x1 submatrices that only contain 0.
```

**Example 2:**

```
Input: matrix = [[1,-1],[-1,1]], target = 0
Output: 5
Explanation: The two 1x2 submatrices, plus the two 2x1 submatrices, plus the 2x2 submatrix.
```

**Example 3:**

```
Input: matrix = [[904]], target = 0
Output: 0
```

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**Constraints:**

* `1 <= matrix.length <= 100`
* `1 <= matrix[0].length <= 100`
* `-1000 <= matrix[i] <= 1000`
* `-10^8 <= target <= 10^8`

#### Same concept as <https://leetcode.com/problems/subarray-sum-equals-k/>

```python
class Solution:
    def numSubmatrixSumTarget(self, matrix: List[List[int]], target: int) -> int:
        #.
        prefix = matrix
        for i in range(len(matrix)):
            for j in range(1, len(matrix[0])):
                prefix[i][j] += prefix[i][j-1]
        counter = collections.Counter()
        count = 0
        # We calculate the prefix sum by row thus to get submatrix sum
        # we use startColumn and endColumn window to sub matrix sum.
        for i in range(len(matrix[0])):
            for j in range(i, len(matrix[0])):
                counter.clear()
                counter[0] = 1 # Simple denotion that sum == k
                s = 0
                for k in range(len(matrix)):
                    # Consider [j] as endColumn and [i-1] as startColumn
                    s += prefix[k][j] - (prefix[k][i-1] if i > 0 else 0)
                    count += counter[s - target]
                    counter[s] += 1
        return count
```