The standard O(MKN) multiplication can be optimized for sparse matrices by only iterating over non-zero elements.

1. Initialize res matrix of size m x n with 0s.
2. Iterate i from 0 to m-1.
3. Iterate k from 0 to k-1.
4. If mat1[i][k] != 0:
   • Only then iterate j from 0 to n-1.
   • If mat2[k][j] != 0:
     • res[i][j] += mat1[i][k] * mat2[k][j]. This structure minimizes unnecessary multiplications.

• Time Complexity: O(M * K * N) worst case, but significantly faster for sparse matrices.
• Space Complexity: O(M * N) for the result.

def multiply(mat1, mat2):
    m, k_dim = len(mat1), len(mat1[0])
    n = len(mat2[0])
    res = [[0] * n for _ in range(m)]
    
    for i in range(m):
        for k in range(k_dim):
            if mat1[i][k] != 0:
                for j in range(n):
                    if mat2[k][j] != 0:
                        res[i][j] += mat1[i][k] * mat2[k][j]
                        
    return res