Finding Degree of a Polynomial - Educational Content

Determining the highest exponent in a sorted polynomial list.

Building upon our defined structure where polynomial terms are sorted by decreasing exponent, calculating the degree becomes an extremely fast operation.
The **degree** of a polynomial is defined as the value of the highest exponent.
Since the list is sorted, the term with the highest exponent is guaranteed to be stored in the first node (the head).
This operation does not require list traversal, executing in $O(1)$ constant time, regardless of the number of terms $n$.

Efficiency Insight: Degree Calculation

Because the polynomial linked list is maintained in descending order of exponents, finding the degree only requires accessing the exponent field of the head node.

Operation	Complexity	Reason
Finding Degree	$O(1)$	Direct access to head node.
Finding $k$-th term	$O(n)$	Requires traversal.

Polynomial Linked List: Finding Degree (Python)

1class PolyNode:
2    def __init__(self, coeff, exp):
3        self.coefficient = coeff
4        self.exponent = exp
5        self.next = None
6
7class Polynomial:
8    def __init__(self, head=None):
9        self.head = head
10        
11    def get_degree(self):
12        """Returns the highest exponent of the polynomial in O(1)."""
13        
14        # Check for an empty polynomial
15        if self.head is None:
16            return -1 # Conventionally indicates an empty/zero polynomial
17
18        # Direct O(1) access to the exponent of the first Polynomial_Term
19        return self.head.exponent
20
21# Example: P(x) = 4x^5 + 7x^2 - 1
22poly_A = Polynomial(PolyNode(4, 5))
23poly_A.head.next = PolyNode(7, 2)
24poly_A.head.next.next = PolyNode(-1, 0)
25
26print(f"Degree: {poly_A.get_degree()}")
27

Knowledge Check

Why is finding the degree of this polynomial list $O(1)$?

A) The list is sorted by exponent, allowing direct access to the head node.
B) The list is short, making any operation fast.
C) We only need to check the coefficient, not the exponent.