Sparse Matrix Multiplication

This problem asks for the multiplication of two sparse matrices. A sparse matrix is one with a relatively small number of non-zero elements. Standard matrix multiplication has a time complexity of O(mnk), where 'm' is the number of rows in the first matrix, 'n' is the number of columns in the second matrix, and 'k' is the number of columns in the first matrix (and rows in the second). For sparse matrices, we can optimize this.

Approach 1: Direct Multiplication

This is the straightforward approach, directly implementing the matrix multiplication formula. It iterates through each element of the resulting matrix, calculating its value by summing the products of corresponding elements in the input matrices.

Time Complexity: O(mnk) - This is because we iterate through all m rows, n columns, and k inner products.

Space Complexity: O(m*n) - We create a new matrix of size m x n to store the result.

Code: (Python example, similar implementations exist in other languages)

class Solution:
    def multiply(self, mat1: List[List[int]], mat2: List[List[int]]) -> List[List[int]]:
        m, n = len(mat1), len(mat2[0])
        ans = [[0] * n for _ in range(m)]
        for i in range(m):
            for j in range(n):
                for k in range(len(mat2)):
                    ans[i][j] += mat1[i][k] * mat2[k][j]
        return ans
 

Approach 2: Preprocessing (Optimizing for Sparsity)

This approach leverages the sparsity of the matrices to improve efficiency. Instead of iterating through every element, it focuses only on non-zero elements. It preprocesses each matrix to store only the non-zero values and their indices.

Time Complexity: While the worst-case remains O(mnk), in practice, for truly sparse matrices, this approach significantly reduces the number of calculations. The preprocessing step is O(mk + kn) where the number of non-zero elements is much smaller than mk or kn. The main multiplication loop will only involve non-zero entries.

Space Complexity: O(m*n + Z) where Z is the total number of non-zero elements in both matrices. The additional space is used to store the processed representations of the matrices.

Code: (Python example, similar implementations exist in other languages)

class Solution:
    def multiply(self, mat1: List[List[int]], mat2: List[List[int]]) -> List[List[int]]:
        def f(mat: List[List[int]]) -> List[List[int]]:
            g = [[] for _ in range(len(mat))]
            for i, row in enumerate(mat):
                for j, x in enumerate(row):
                    if x:
                        g[i].append((j, x))
            return g
 
        g1 = f(mat1)
        g2 = f(mat2)
        m, n = len(mat1), len(mat2[0])
        ans = [[0] * n for _ in range(m)]
        for i in range(m):
            for k, x in g1[i]:
                for j, y in g2[k]:
                    ans[i][j] += x * y
        return ans

In summary: The direct multiplication approach is simpler to understand and implement, but the preprocessing approach is considerably more efficient for sparse matrices because it avoids unnecessary computations with zero elements. The choice of which approach to use depends on the expected sparsity of the input matrices and the prioritization of code simplicity versus performance optimization.