Doubly linked list

In computer science, a doubly linked list is a linked data structure that consists of a set of sequentially linked records called nodes. Each node contains three fields: two link fields (references to the previous and to the next node in the sequence of nodes) and one data field. The beginning and ending nodes' previous and next links, respectively, point to some kind of terminator, typically a sentinel node or null, to facilitate traversal of the list. If there is only one sentinel node, then the list is circularly linked via the sentinel node. It can be conceptualized as two singly linked lists formed from the same data items, but in opposite sequential orders.



The two node links allow traversal of the list in either direction. While adding or removing a node in a doubly linked list requires changing more links than the same operations on a singly linked list, the operations are simpler and potentially more efficient (for nodes other than first nodes) because there is no need to keep track of the previous node during traversal or no need to traverse the list to find the previous node, so that its link can be modified.

Nomenclature and implementation
The first and last nodes of a doubly linked list for all practical applications are immediately accessible (i.e., accessible without traversal, and usually called head and tail) and therefore allow traversal of the list from the beginning or end of the list, respectively: e.g., traversing the list from beginning to end, or from end to beginning, in a search of the list for a node with specific data value. Any node of a doubly linked list, once obtained, can be used to begin a new traversal of the list, in either direction (towards beginning or end), from the given node.

The link fields of a doubly linked list node are often called next and previous or forward and backward. The references stored in the link fields are usually implemented as pointers, but (as in any linked data structure) they may also be address offsets or indices into an array where the nodes live.

Basic algorithms
Consider the following basic algorithms written in Ada:

Open doubly linked lists
record DoublyLinkedNode { next // A reference to the next node prev // A reference to the previous node data // Data or a reference to data }

record DoublyLinkedList { DoublyLinkedNode firstNode  // points to first node of list DoublyLinkedNode lastNode   // points to last node of list }

Traversing the list
Traversal of a doubly linked list can be in either direction. In fact, the direction of traversal can change many times, if desired. Traversal is often called iteration, but that choice of terminology is unfortunate, for iteration has well-defined semantics (e.g., in mathematics) which are not analogous to traversal.

Forwards node := list.firstNode while node ≠ null  node := node.next

Backwards node := list.lastNode while node ≠ null  node := node.prev

Inserting a node
These symmetric functions insert a node either after or before a given node:

function insertAfter(List list, Node node, Node newNode) newNode.prev := node if node.next == null newNode.next := null -- (not always necessary) list.lastNode := newNode else newNode.next := node.next node.next.prev := newNode node.next := newNode

function insertBefore(List list, Node node, Node newNode) newNode.next := node if node.prev == null newNode.prev := null -- (not always necessary) list.firstNode := newNode else newNode.prev := node.prev node.prev.next := newNode node.prev := newNode

We also need a function to insert a node at the beginning of a possibly empty list:

function insertBeginning(List list, Node newNode) if list.firstNode == null list.firstNode := newNode list.lastNode  := newNode newNode.prev := null newNode.next := null else insertBefore(list, list.firstNode, newNode)

A symmetric function inserts at the end:

function insertEnd(List list, Node newNode) if list.lastNode == null insertBeginning(list, newNode) else insertAfter(list, list.lastNode, newNode)

Removing a node
Removal of a node is easier than insertion, but requires special handling if the node to be removed is the firstNode or lastNode:

function remove(List list, Node node) if node.prev == null list.firstNode := node.next else node.prev.next := node.next if node.next == null list.lastNode := node.prev else node.next.prev := node.prev

One subtle consequence of the above procedure is that deleting the last node of a list sets both firstNode and lastNode to null, and so it handles removing the last node from a one-element list correctly. Notice that we also don't need separate "removeBefore" or "removeAfter" methods, because in a doubly linked list we can just use "remove(node.prev)" or "remove(node.next)" where these are valid. This also assumes that the node being removed is guaranteed to exist. If the node does not exist in this list, then some error handling would be required.

Traversing the list
Assuming that someNode is some node in a non-empty list, this code traverses through that list starting with someNode (any node will do):

Forwards node := someNode do do something with node.value node := node.next while node ≠ someNode

Backwards node := someNode do do something with node.value node := node.prev while node ≠ someNode

Notice the postponing of the test to the end of the loop. This is important for the case where the list contains only the single node someNode.

Inserting a node
This simple function inserts a node into a doubly linked circularly linked list after a given element:

function insertAfter(Node node, Node newNode) newNode.next := node.next newNode.prev := node node.next.prev := newNode node.next      := newNode

To do an "insertBefore", we can simply "insertAfter(node.prev, newNode)".

Inserting an element in a possibly empty list requires a special function:

function insertEnd(List list, Node node) if list.lastNode == null node.prev := node node.next := node else insertAfter(list.lastNode, node) list.lastNode := node

To insert at the beginning we simply "insertAfter(list.lastNode, node)".

Finally, removing a node must deal with the case where the list empties:

function remove(List list, Node node); if node.next == node list.lastNode := null else node.next.prev := node.prev node.prev.next := node.next if node == list.lastNode list.lastNode := node.prev; destroy node

Deleting a node
As in doubly linked lists, "removeAfter" and "removeBefore" can be implemented with "remove(list, node.prev)" and "remove(list, node.next)".

Asymmetric doubly linked list
An asymmetric doubly linked list is somewhere between the singly linked list and the regular doubly linked list. It shares some features with the singly linked list (single-direction traversal) and others from the doubly linked list (ease of modification)

It is a list where each node's previous link points not to the previous node, but to the link to itself. While this makes little difference between nodes (it just points to an offset within the previous node), it changes the head of the list: It allows the first node to modify the firstNode link easily.

As long as a node is in a list, its previous link is never null.

Inserting a node
To insert a node before another, we change the link that pointed to the old node, using the prev link; then set the new node's next link to point to the old node, and change that node's prev link accordingly.

function insertBefore(Node node, Node newNode) if node.prev == null error "The node is not in a list" newNode.prev := node.prev atAddress(newNode.prev) := newNode newNode.next := node node.prev = addressOf(newNode.next)

function insertAfter(Node node, Node newNode) newNode.next := node.next if newNode.next != null newNode.next.prev = addressOf(newNode.next) node.next := newNode newNode.prev := addressOf(node.next)

Deleting a node
To remove a node, we simply modify the link pointed by prev, regardless of whether the node was the first one of the list.

function remove(Node node) atAddress(node.prev) := node.next if node.next != null node.next.prev = node.prev destroy node