Establishing the foundation of Linked Lists by defining the Node structure and analyzing the efficiency of core pointer operations.
- The structural differences we just observed—particularly the use of dynamic pointers—make the Linked List a powerful tool for certain applications where frequent insertion and deletion are necessary. Before diving into complex algorithms, we must first establish a firm foundation in the definition and core mechanics of this structure.
- This lecture segment is dedicated to mastering the Linked List. Our roadmap will guide us through its fundamental concepts and its practical application to a classic data structure problem:
- Goal: Understand why the Linked List is chosen when the size of $n$ is volatile or unknown, and efficiency depends on pointer relinking rather than memory shifting.
- Roadmap Overview:
- 1. Structure & Definition: We will formally define the Node_Structure (item and $next$ pointer) and examine the differences between Singly, Doubly, and Circular Linked Lists.
- 2. Core Operations: Mastering the manipulation of pointers:
- Traversal: Iterating through the list, requiring $O(n)$ time.
- Insertion: Adding a node at a known position $i$, which can be done in efficient $O(1)$ time by adjusting the $next$ pointer using a Pointer_Relinking_Color.
- Deletion: Removing a node and adjusting pointers, also $O(1)$.
- 3. Illustrative Application (Polynomials): We will use the Linked List to store and manipulate algebraic polynomials, where each node holds a Polynomial_Term ($\langle coefficient, exponent \rangle$). This showcases how Linked Lists excel at dynamic structure management, particularly for polynomial addition, which often runs in $O(n+m)$ time.
Linked List Core Operation Complexities
| Operation | Description | Complexity |
|---|---|---|
| Traversal | Finding an element or end of list. | $O(n)$ |
| Insertion (at known $i$) | Adjusting two pointers. | $O(1)$ |
| Deletion (at known $i$) | Adjusting one pointer. | $O(1)$ |