Question

Describe a nonrecursive function for finding, by link hopping, the middle node of a doubly linked list with header and trailer sentinels. (Note: This function must only use link hopping; it cannot use a counter.) What is the running time of this function?

Short answer

Initialize two pointers, 'slow' and 'fast'. Move 'slow' by one step and 'fast' by two steps until 'fast' reaches the trailer sentinel. The 'slow' pointer will be at the middle node. Running time is O(n).

Step-by-step solution

Initialize Pointers

Start by initializing two pointers: one (let's call it 'slow') at the first element (header's next) and the other ('fast') also at the first element (header's next).

Loop through the List

Move the 'fast' pointer two steps forward and the 'slow' pointer one step forward in each iteration, until the 'fast' pointer reaches the trailer sentinel or goes beyond it.

Handle Edge Cases

Check if the doubly linked list is empty by verifying if the header's next is the trailer. If so, return null or an appropriate indication that the list is empty.

Return the Middle Node

After the loop terminates, the 'slow' pointer will point to the middle node of the linked list. Return the 'slow' pointer as the middle node.

Analyze Running Time

Since each step involves moving pointers in constant time, and the 'fast' pointer effectively traverses the list once, the running time of this function is O(n), where n is the number of nodes in the list.

Key concepts

nonrecursive function

In a nonrecursive function, the process of finding a solution does not involve calling the function itself again. Instead, everything is handled within a single call.
For finding the middle node in a doubly linked list, we leverage simple loops and straightforward logic. This approach is preferred in many programming scenarios as it avoids the overhead of multiple function calls and is generally easier to debug and understand.
Here, we use two pointers to traverse the list and locate the middle node without needing any recursive calls.

link hopping

Link hopping is the technique of moving directly from one node to another using pointers. In a doubly linked list, each node contains pointers to both the next and previous nodes.
By initializing two pointers (`slow` and `fast`), both starting at the header's next node, we can traverse the list efficiently.
The `slow` pointer moves one step at a time while the `fast` pointer moves two steps at a time. This double-speed approach ensures that when the `fast` pointer reaches the end, the `slow` pointer is at the middle node.
The algorithm does not count the nodes, relying purely on relative position movement to find the middle.

running time analysis

Running time analysis helps us understand the efficiency of our algorithm in terms of time complexity. For the middle-node-finding algorithm, each operation (moving a pointer) is constant time, denoted as O(1).
Since the `fast` pointer covers the list in a single traversal (moving two steps at a time), the total number of operations is proportional to the number of nodes, n. Thus, the algorithm runs in O(n).
This time complexity ensures that the function performs efficiently even for large lists, as it scales linearly with the size of the list.

doubly linked list

A doubly linked list is a type of data structure where each node contains links to both its previous and next nodes. This bi-directional linking allows traversal in both directions and makes certain operations more efficient.
Sentinel nodes at the beginning (header) and the end (trailer) simplify edge cases by removing the need for null checks at the boundaries.
In the context of finding the middle node, the doubly linked list's structure aids in efficient pointer manipulation, which is critical for the link hopping process.

pointers in data structures

Pointers are variables that store memory addresses. In data structures like linked lists, they are used to connect nodes.
For our middle node algorithm, two pointers (`slow` and `fast`) are employed to traverse the linked list. The `slow` pointer's single-step progression ensures it will be at the middle position when the `fast` pointer reaches the end.
Effective use of pointers allows for efficient navigation and manipulation of the list without needing extra memory or complex calculations. Understanding how pointers work is fundamental to implementing and optimizing linked list operations.