Skip to content

1074. 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:

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

 

Constraints:

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

Solutions

Solution 1

Thinking

Four nested loops over a \(100\times 100\) matrix are \(O(n^4)\). Fixing the top and bottom rows compresses each column to a sum, leaving “subarrays that add to \(\textit{target}\)” on a 1-D array.

That 1-D count uses a prefix-sum map: after adding \(x\), look up \(s-\textit{target}\). There are \(O(n^2)\) row pairs and a linear scan per pair.

Sum \(f(\textit{col})\) over every pair of bounds.

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
class Solution:
    def numSubmatrixSumTarget(self, matrix: List[List[int]], target: int) -> int:
        def f(nums: List[int]) -> int:
            d = defaultdict(int)
            d[0] = 1
            cnt = s = 0
            for x in nums:
                s += x
                cnt += d[s - target]
                d[s] += 1
            return cnt

        m, n = len(matrix), len(matrix[0])
        ans = 0
        for i in range(m):
            col = [0] * n
            for j in range(i, m):
                for k in range(n):
                    col[k] += matrix[j][k]
                ans += f(col)
        return ans

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
class Solution {
    public int numSubmatrixSumTarget(int[][] matrix, int target) {
        int m = matrix.length, n = matrix[0].length;
        int ans = 0;
        for (int i = 0; i < m; ++i) {
            int[] col = new int[n];
            for (int j = i; j < m; ++j) {
                for (int k = 0; k < n; ++k) {
                    col[k] += matrix[j][k];
                }
                ans += f(col, target);
            }
        }
        return ans;
    }

    private int f(int[] nums, int target) {
        Map<Integer, Integer> d = new HashMap<>();
        d.put(0, 1);
        int s = 0, cnt = 0;
        for (int x : nums) {
            s += x;
            cnt += d.getOrDefault(s - target, 0);
            d.merge(s, 1, Integer::sum);
        }
        return cnt;
    }
}

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
class Solution {
public:
    int numSubmatrixSumTarget(vector<vector<int>>& matrix, int target) {
        int m = matrix.size(), n = matrix[0].size();
        int ans = 0;
        for (int i = 0; i < m; ++i) {
            vector<int> col(n);
            for (int j = i; j < m; ++j) {
                for (int k = 0; k < n; ++k) {
                    col[k] += matrix[j][k];
                }
                ans += f(col, target);
            }
        }
        return ans;
    }

    int f(vector<int>& nums, int target) {
        unordered_map<int, int> d{{0, 1}};
        int cnt = 0, s = 0;
        for (int& x : nums) {
            s += x;
            if (d.count(s - target)) {
                cnt += d[s - target];
            }
            ++d[s];
        }
        return cnt;
    }
};

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
func numSubmatrixSumTarget(matrix [][]int, target int) (ans int) {
    m, n := len(matrix), len(matrix[0])
    for i := 0; i < m; i++ {
        col := make([]int, n)
        for j := i; j < m; j++ {
            for k := 0; k < n; k++ {
                col[k] += matrix[j][k]
            }
            ans += f(col, target)
        }
    }
    return
}

func f(nums []int, target int) (cnt int) {
    d := map[int]int{0: 1}
    s := 0
    for _, x := range nums {
        s += x
        if v, ok := d[s-target]; ok {
            cnt += v
        }
        d[s]++
    }
    return
}

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
function numSubmatrixSumTarget(matrix: number[][], target: number): number {
    const m = matrix.length;
    const n = matrix[0].length;
    let ans = 0;
    for (let i = 0; i < m; ++i) {
        const col: number[] = new Array(n).fill(0);
        for (let j = i; j < m; ++j) {
            for (let k = 0; k < n; ++k) {
                col[k] += matrix[j][k];
            }
            ans += f(col, target);
        }
    }
    return ans;
}

function f(nums: number[], target: number): number {
    const d: Map<number, number> = new Map();
    d.set(0, 1);
    let cnt = 0;
    let s = 0;
    for (const x of nums) {
        s += x;
        if (d.has(s - target)) {
            cnt += d.get(s - target)!;
        }
        d.set(s, (d.get(s) || 0) + 1);
    }
    return cnt;
}

Comments