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# 1074. Number of Submatrices That Sum to Target

## LeetCode [1074. Number of Submatrices That Sum to Target](https://leetcode-cn.com/problems/number-of-submatrices-that-sum-to-target/)

### Description

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'`.

**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.
```

**Constraints:**

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

### Tags

Array, Dynamic Programming, Sliding Window

### Solution

From LeetCode [560. Subarray Sum Equals K](/leetcode-go-notes/solutions/560.-subarray-sum-equals-k.md) we know how to obtain the subarrays whose sum equal to a certain target value. This problem extends it to 2D scenario, but we can also take advantage of the solution of 1D version `subarraySum()`. We traverse all continuous rows permutations, and sum up these rows along each columns to obtain a 1D array. This array and target are the arguments of `subarraySum()`. The final result is the sum of all return values of this function.

### Complexity

* Time complexity: $$O(m^2\times n)$$
* Space complexity: $$O(n)$$

### Code

```go
func numSubmatrixSumTarget(matrix [][]int, target int) int {
	var ans int
	for i := range matrix { // upper bound
		colSum := make([]int, len(matrix[0])) // columns sum
		for _, row := range matrix[i:] {      // lower bound
			for j, v := range row {
				colSum[j] += v
			}
			ans += subarraySum(colSum, target)
		}
	}
	return ans
}

func subarraySum(nums []int, k int) int {
	ans, preSum, m := 0, 0, map[int]int{0: 1}
	for _, num := range nums {
		preSum += num
		ans += m[preSum-k]
		m[preSum]++
	}
	return ans
}
```

## Reference

1. [元素和为目标值的子矩阵数量](https://leetcode-cn.com/problems/number-of-submatrices-that-sum-to-target/solution/yuan-su-he-wei-mu-biao-zhi-de-zi-ju-zhen-8ym2/)