Question

Briefly describe how to perform a new sequence function make First(p) that moves an element of a sequence S at position p to be the first element in S while keeping the relative ordering of the remaining elements in S unchanged. Your function should run in O(1) time if S is implemented with a doubly linked list.

Short answer

Remove the node at position `p`, then reattach it at the beginning of the list.

Step-by-step solution

- Identify the Node at Position p

Locate the node at position `p` in the doubly linked list `S` using a pointer. This pointer will be referred to as `node_p`.

- Remove Node at Position p

Update the pointers of the nodes adjacent to `node_p`. For the node before `node_p` (let's call it `node_before_p`), set its `next` pointer to point to the node after `node_p` (let's call it `node_after_p`). For `node_after_p`, set its `prev` pointer to point to `node_before_p`.

- Insert Node at the Beginning

Update `node_p`'s `next` pointer to point to the current first node in the list (let's call it `first_node`). Set `first_node`'s `prev` pointer to `node_p`. Finally, update the head pointer of the list to point to `node_p`.

Key concepts

Sequence manipulation

Sequence manipulation involves operations that change the order or structure of elements within a sequence such as an array or a list. In the context of a doubly linked list, moving nodes around is a common form of sequence manipulation.

To implement the makeFirst(p) function, we aim to move an element at position `p` to the beginning of the list while preserving the order of other elements. This involves several steps: locating the node, removing it from its current position, and inserting it at the start.

The key here is maintaining the integrity of the list. We need to ensure that all node connections (pointers) are correctly updated to prevent breaking the list structure. This process is more complex than simple arrays due to the bidirectional pointers in a doubly linked list.

Node pointers

Understanding node pointers is crucial for any operation on linked lists. In a doubly linked list, each node contains two pointers: one pointing to the next node and the other pointing to the previous node.

When performing the makeFirst(p) operation, we need to manipulate these pointers carefully:

• Identify the node at position `p` (node_p). Use a traversal method to locate this node efficiently.
• Update the `next` pointer of the node before `node_p` to skip `node_p` and point to the node after `node_p`.
• Update the `prev` pointer of the node after `node_p` to point back to the node before `node_p`.
• Adjust the pointers of `node_p` so it gets placed at the beginning of the list, ensuring it now points to the former first node.

By meticulously handling these pointers, we ensure the list remains doubly linked and free of errors.

Algorithm efficiency

Algorithm efficiency determines how effectively an algorithm performs in terms of time and space complexity. For the makeFirst(p) function, our goal is to achieve O(1) time complexity. This means the operation should complete in constant time, regardless of the list size.

Achieving O(1) time involves direct pointer manipulation without traversing the list multiple times. Here's how our solution manages this:

• Directly identifying `node_p` if we have an index or pointer reference.
• Swiftly updating adjacent node pointers (`next` and `prev`) to exclude `node_p` and re-link the list.
• Placing `node_p` at the list's start through quick pointer adjustments.

Efficient algorithms are critical for operations on large data structures, as they ensure fast performance and responsiveness. By following these principles, our doubly linked list operations maintain high efficiency.