Add Two Polynomials Represented as Linked Lists

Add Two Polynomials Represented as Linked Lists

What is this problem about?

The Add Two Polynomials Represented as Linked Lists interview question involves mathematical expression manipulation using data structures. Each node in the linked list represents a term in a polynomial, containing a coefficient and an exponent. The lists are typically sorted by exponents in descending order. Your task is to sum the two polynomials and return a new linked list representing the resulting polynomial, ensuring that terms with zero coefficients are excluded.

Why is this asked in interviews?

Companies like Amazon use this problem to evaluate a candidate's mastery of the linked list interview pattern. It requires more than simple traversal; you must manage pointers across two different lists simultaneously while handling mathematical logic. It also tests edge-case handling, such as merging terms with the same exponent or removing terms that cancel each other out.

Algorithmic pattern used

This problem effectively utilizes the Two Pointers interview pattern. Since the input lists are sorted by their exponents, you can use a pointer for each list and compare the exponents at each step. This is very similar to the "Merge" step in Merge Sort, but with added logic for coefficient addition.

Example explanation

Suppose you have two polynomials:

• $P1 = 3x^2 + 5x^1$ (Linked List: [3, 2] -> [5, 1])
• $P2 = 4x^3 + 2x^2 + 1x^0$ (Linked List: [4, 3] -> [2, 2] -> [1, 0])

1. Compare exponents: $3$ vs $2$. $P2$ has $x^3$, which is higher. Add $[4, 3]$ to result.
2. Next: Exponents are both $2$. Add coefficients $(3 + 2 = 5)$. Add $[5, 2]$ to result.
3. Next: $P1$ has $x^1$, $P2$ has $x^0$. $1$ is higher. Add $[5, 1]$ to result.
4. Final: Add remaining $[1, 0]$ to result. Result: $4x^3 + 5x^2 + 5x^1 + 1x^0$.

Common mistakes candidates make

• Missing Zero Coefficients: Failing to check if the sum of two coefficients equals zero. If $2x^2 + (-2x^2)$ occurs, that term should not appear in the final list.
• Pointer Loss: Accidentally "orphaning" part of the list by not correctly updating the next pointers.
• Memory Leaks: In languages like C++, not properly managing memory when creating the new result list.

Interview preparation tip

Always use a "Dummy Head" node when building a new linked list. This simplifies the logic by allowing you to treat the first term the same as every subsequent term, avoiding messy "if head is null" checks.