How do you solve Number of Submatrices That Sum to Target on LeetCode?

Number of Submatrices That Sum to Target (LeetCode #1074) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(n^2 * m) time complexity and O(m) space complexity. Key insight: Using prefix sums allows for quick sum calculations of submatrices.

What is the brute force solution for Number of Submatrices That Sum to Target?

The brute force approach for Number of Submatrices That Sum to Target is Brute Force, with O(n^4) time complexity and O(1) space complexity. The brute force approach involves checking every possible submatrix in the given matrix and calculating its sum. This is straightforward but inefficient, especially for larger matrices. The optimal Optimal Solution approach improves this to O(n^2 * m).

What algorithm or data structure is used to solve Number of Submatrices That Sum to Target?

Number of Submatrices That Sum to Target uses the following concepts: Array, Hash Table, Matrix, Prefix Sum. The recommended patterns to study are: Hash Map, Prefix Sum.

What are the interview tips for Number of Submatrices That Sum to Target?

Always explain your thought process clearly when presenting your solution. Consider edge cases, such as empty matrices or matrices with all negative numbers. Practice optimizing brute force solutions to demonstrate problem-solving skills.

What are common mistakes when solving Number of Submatrices That Sum to Target?

Not considering all possible submatrices, leading to incorrect counts. Failing to reset cumulative sums correctly for different row pairs.

#1074

Number of Submatrices That Sum to Target

Hard

ArrayHash TableMatrixPrefix SumHash MapPrefix Sum

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n^4)	O(n^2 * m)
Space	O(1)	O(m)

💡

Intuition

Time O(n^2 * m)Space O(m)

The optimal approach uses a combination of prefix sums and a hash map to efficiently count the number of submatrices that sum to the target. This reduces the time complexity significantly by avoiding the need to calculate sums repeatedly.

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Algorithm

3 steps

1Step 1: Precompute the prefix sums for the matrix to allow O(1) sum queries for any submatrix.
2Step 2: For each pair of row indices (r1, r2), calculate the cumulative sums for each column between these rows.
3Step 3: Use a hash map to count how many times each cumulative sum has occurred, allowing us to quickly find how many submatrices sum to the target.

solution.py19 lines

1# Full working Python code
2
3def numSubmatrixSumTarget(matrix, target):
4    if not matrix: return 0
5    count = 0
6    rows, cols = len(matrix), len(matrix[0])
7    for r1 in range(rows):
8        cum_sum = [0] * cols
9        for r2 in range(r1, rows):
10            for c in range(cols):
11                cum_sum[c] += matrix[r2][c]
12            sum_count = {0: 1}
13            current_sum = 0
14            for s in cum_sum:
15                current_sum += s
16                if current_sum - target in sum_count:
17                    count += sum_count[current_sum - target]
18                sum_count[current_sum] = sum_count.get(current_sum, 0) + 1
19    return count

ℹ

Complexity note: The complexity is reduced because we only compute the sum of submatrices using cumulative sums and hash maps, which allows us to find the number of valid submatrices in linear time for each row pair.

1Using prefix sums allows for quick sum calculations of submatrices.
2Hash maps can efficiently track cumulative sums and their frequencies.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.