Multiplying two polynomials - Educational Content

Unlike addition and subtraction, which maintained $O(n+m)$ linearity through merging, polynomial multiplication requires a distinct, more computationally intensive approach.

To find the product $S(x) = P(x) \times R(x)$, we must apply the distributive property, multiplying every term in $P(x)$ by every term in $R(x)$.
This requirement translates directly into a nested loop structure.
If $P(x)$ has $n$ terms and $R(x)$ has $m$ terms, the generation of all intermediate terms requires $n \times m$ fundamental operations.
This results in a worst-case time complexity of $O(n \cdot m)$.

Multiplication Procedure

Use the outer pointer `p` to iterate through all $n$ nodes of $P(x)$.
For each node in $P(x)$, use the inner pointer `q` to iterate through all $m$ nodes of $R(x)$.
For the pair $(p, q)$, the resulting Polynomial_Term has:
- New Coefficient: $coeff_p \times coeff_q$
- New Exponent: $exp_p + exp_q$
The final step (not shown here) involves combining all the resulting $O(n \cdot m)$ intermediate terms, which may require sorting and merging.

Polynomial Multiplication (Intermediate Term Generation)

1class PolynomialNode:
2    """Represents a Node_Structure holding a Polynomial_Term."""
3    def __init__(self, coeff, exp, next_node=None):
4        self.term = (coeff, exp) 
5        self.next = next_node
6
7def multiply_polynomials_generate(P_head, R_head):
8    """
9    Generates all intermediate terms from P(x) * R(x) (O(n*m)).
10    This function implements the nested loop structure.
11    """
12    p = P_head # Outer pointer: Traverses P(x)
13    intermediate_terms = [] 
14    
15    # O(n) loop
16    while p is not None:
17        p_coeff, p_exp = p.term
18        
19        q = R_head # Inner pointer: Resets for each term in P(x)
20        
21        # O(m) loop
22        while q is not None:
23            q_coeff, q_exp = q.term
24            
25            # Calculate new term (O(1))
26            new_coeff = p_coeff * q_coeff
27            new_exp = p_exp + q_exp
28            
29            # Store the intermediate (new_term is temporary, not inserted yet)
30            intermediate_terms.append((new_coeff, new_exp))
31            
32            q = q.next
33        p = p.next
34            
35    return intermediate_terms

Knowledge Check

What is the time complexity for generating all intermediate terms when multiplying two polynomials of size $n$ and $m$?

A) $O(n+m)$
B) $O(n \cdot m)$
C) $O(n \log n)$