Linked list

In computer science, a linked list is one of the fundamental data structures used in computer programming. It consists of a sequence of nodes, each containing arbitrary data fields and one or two references ("links") pointing to the next and/or previous nodes. The principle benefit of a linked list over a conventional list is that the order of the linked items may be different to the order that the data items are stored in memory or on disk, allowing the list of items to be traversed in a different order. A linked list is a self-referential datatype because it contains a pointer or link to another data of the same type. Linked lists permit insertion and removal of nodes at any point in the list in constant time, but do not allow random access. Several different types of linked list exist: singly-linked lists, doubly-linked lists, and circularly-linked lists. One of the biggest advantages of linked lists is that nodes may have multiple pointers to other nodes, allowing the same nodes to simultaneously appear in different orders in several linked lists[1].

Linked lists can be implemented in most languages. Languages such as Lisp and Scheme have the data structure built in, along with operations to access the linked list. Procedural or object-oriented languages or such as C, C++, and Java typically rely on mutable references to create linked lists.

History

Linked lists were developed in 1955-56 by Allen Newell, Cliff Shaw and Herbert Simon at RAND Corporation as the primary data structure for their Information Processing Language. IPL was used by the authors to develop several early artificial intelligence programs, including the Logic Theory Machine, the General Problem Solver, and a computer chess program. Reports on their work appeared in IRE Transactions on Information Theory in 1956, and several conference proceedings from 1957-1959, including Proceedings of the Western Joint Computer Conference in 1957 and 1958, and Information Processing (Proceedings of the first UNESCO International Conference on Information Processing) in 1959. The now-classic diagram consisting of blocks representing list nodes with arrows pointing to successive list nodes appears in "Programming the Logic Theory Machine" by Newell and Shaw in Proc. WJCC, February 1957. Newell and Simon were recognized with the ACM Turing Award in 1975 for having "made basic contributions to artificial intelligence, the psychology of human cognition, and list processing".

The problem of machine translation for natural language processing led Victor Yngve at Massachusetts Institute of Technology (MIT) to use linked lists as data structures in his COMIT programming language for computer research in the field of linguistics. A report on this language entitled "A programming language for mechanical translation" appeared in Mechanical Translation in 1958.

LISP, standing for list processor, was created by John McCarthy in 1958 while he was at MIT and in 1960 he published its design in a paper in the Communications of the ACM, entitled "Recursive Functions of Symbolic Expressions and Their Computation by Machine, Part I". One of LISP's major data structures is the linked list.

By the early 1960s, the utility of both linked lists and languages which use these structures as their primary data representation was well established. Bert Green of the MIT Lincoln Laboratory published a review article entitled "Computer languages for symbol manipulation" in IRE Transactions on Human Factors in Electronics in March 1961 which summarized the advantages of the linked list approach. A later review article, "A Comparison of list-processing computer languages" by Bobrow and Raphael, appeared in Communications of the ACM in April 1964.

Several operating systems developed by Technical Systems Consultants (originally of West Lafayette Indiana, and later of Raleigh, North Carolina) used singly linked lists as file structures. A directory entry pointed to the first sector of a file, and succeeding portions of the file were located by traversing pointers. Systems using this technique included Flex (for the Motorola 6800 CPU), mini-Flex (same CPU), and Flex9 (for the Motorola 6809 CPU). A variant developed by TSC for and marketed by Smoke Signal Broadcasting in California, used doubly linked lists in the same manner.

The TSS operating system, developed by IBM for the System 360/370 machines, used a double linked list for their file system catalog. The directory structure was similar to Unix, where a directory could contain files and/or other directories and extend to any depth. A utility flea was created to fix file system problems after a crash, since modified portions of the file catalog were sometimes in memory when a crash occurred. Problems were detected by comparing the forward and backward links for consistency. If a forward link was corrupt, then if a backward link to the infected node was found, the forward link was set to the node with the backward link. A humorous comment in the source code where this utility was invoked stated "Everyone knows a flea caller gets rid of bugs in cats".

Variants

Linearly-linked List

Singly-linked list

The simplest kind of linked list is a singly-linked list (or slist for short), which has one link per node. This link points to the next node in the list, or to a null value or empty list if it is the final node.

A singly linked list containing three integer values

Doubly-linked list

A more sophisticated kind of linked list is a doubly-linked list or two-way linked list. Each node has two links: one points to the previous node, or points to a null value or empty list if it is the first node; and one points to the next, or points to a null value or empty list if it is the final node.

An example of a doubly linked list.

In some very low level languages, Xor-linking offers a way to implement doubly-linked lists using a single word for both links, although the use of this technique is usually discouraged. A similar link storage preserving technique is the subtraction edge.

Circularly-linked list

In a circularly-linked list, the first and final nodes are linked together. This can be done for both singly and doubly linked lists. To traverse a circular linked list, you begin at any node and follow the list in either direction until you return to the original node. Viewed another way, circularly-linked lists can be seen as having no beginning or end. This type of list is most useful for managing buffers for data ingest, and in cases where you have one object in a list and wish to see all other objects in the list.

The pointer pointing to the whole list is usually called the end pointer.

Singly-circularly-linked list

In a singly-circularly-linked list, each node has one link, similar to an ordinary singly-linked list, except that the next link of the last node points back to the first node. As in a singly-linked list, new nodes can only be efficiently inserted after a node we already have a reference to. For this reason, it's usual to retain a reference to only the last element in a singly-circularly-linked list, as this allows quick insertion at the beginning, and also allows access to the first node through the last node's next pointer.

Doubly-circularly-linked list

In a doubly-circularly-linked list, each node has two links, similar to a doubly-linked list, except that the previous link of the first node points to the last node and the next link of the last node points to the first node. As in doubly-linked lists, insertions and removals can be done at any point with access to any nearby node.

Sentinel nodes

Linked lists sometimes have a special dummy or sentinel node at the beginning and/or at the end of the list, which is not used to store data. Its purpose is to simplify or speed up some operations, by ensuring that every data node always has a previous and/or next node, and that every list (even one that contains no data elements) always has a "first" and "last" node. Lisp has such a design - the special value nil is used to mark the end of a 'proper' singly-linked list, or chain of cons cells as they are called. A list does not have to end in nil, but such a list would be termed 'improper'.

Applications of linked lists

Linked lists are used as a building block for many other data structures, such as stacks, queues and their variations.

The "data" field of a node can be another linked list. By this device, one can construct many linked data structures with lists; this practice originated in the Lisp programming language, where linked lists are a primary (though by no means the only) data structure, and is now a common feature of the functional programming style.

Sometimes, linked lists are used to implement associative arrays, and are in this context called association lists. There is very little good to be said about this use of linked lists; they are easily outperformed by other data structures such as self-balancing binary search trees even on small data sets (see the discussion in associative array). However, sometimes a linked list is dynamically created out of a subset of nodes in such a tree, and used to more efficiently traverse that set.

Tradeoffs

As with most choices in computer programming and design, no method is well suited to all circumstances. A linked list data structure might work well in one case, but cause problems in another. This is a list of some of the common tradeoffs involving linked list structures. In general, if you have a dynamic collection, where elements are frequently being added and deleted, and the location of new elements added to the list is significant, then benefits of a linked list increase.

Linked lists vs. arrays

	Array	Linked list
Indexing	O(1)	O(n)
Inserting / Deleting at end	O(1)	O(1)
Inserting / Deleting in middle (with iterator)	O(n)	O(1)
Persistent	No	Singly yes
Locality	Great	Bad

Linked lists have several advantages over arrays. Elements can be inserted into linked lists indefinitely, while an array will eventually either fill up or need to be resized, an expensive operation that may not even be possible if memory is fragmented. Similarly, an array from which many elements are removed may become wastefully empty or need to be made smaller.

Further memory savings can be achieved, in certain cases, by sharing the same "tail" of elements among two or more lists — that is, the lists end in the same sequence of elements. In this way, one can add new elements to the front of the list while keeping a reference to both the new and the old versions — a simple example of a persistent data structure.

On the other hand, arrays allow random access, while linked lists allow only sequential access to elements. Singly-linked lists, in fact, can only be traversed in one direction. This makes linked lists unsuitable for applications where it's useful to look up an element by its index quickly, such as heapsort. Sequential access on arrays is also faster than on linked lists on many machines due to locality of reference and data caches. Linked lists receive almost no benefit from the cache.

Another disadvantage of linked lists is the extra storage needed for references, which often makes them impractical for lists of small data items such as characters or boolean values. It can also be slow, and with a naïve allocator, wasteful, to allocate memory separately for each new element, a problem generally solved using memory pools.

A number of linked list variants exist that aim to ameliorate some of the above problems. Unrolled linked lists store several elements in each list node, increasing cache performance while decreasing memory overhead for references. CDR coding does both these as well, by replacing references with the actual data referenced, which extends off the end of the referencing record.

A good example that highlights the pros and cons of using arrays vs. linked lists is by implementing a program that resolves the Josephus problem. The Josephus problem is an election method that works by having a group of people stand in a circle. Starting at a predetermined person, you count around the circle n times. Once you reach nth person, take them out of the circle and have the members close the circle. Then count around the circle the same n times and repeat the process, until only one person is left. That person wins the election. This shows the strengths and weaknesses of a linked list vs. an array, because if you view the people as connected nodes in a circular linked list then it shows how easily the linked list is able to delete nodes (as it only has to rearrange the links to the different nodes). However, the linked list will be poor at finding the next person to remove and will need to recurse through the list until it finds that person. An array, on the other hand, will be poor at deleting nodes (or elements) as it cannot remove one node without individually shifting all the elements up the list by one. However, it is exceptionally easy to find the nth person in the circle by directly referencing them by their position in the array.

Doubly-linked vs. singly-linked

Double-linked lists require more space per node (unless one uses xor-linking), and their elementary operations are more expensive; but they are often easier to manipulate because they allow sequential access to the list in both directions. In particular, one can insert or delete a node in a constant number of operations given only that node's address. (Compared with singly-linked lists, which require the previous node's address in order to correctly insert or delete.) Some algorithms require access in both directions. On the other hand, they do not allow tail-sharing, and cannot be used as persistent data structures.gh

Circularly-linked vs. linearly-linked

Circular linked lists are most useful for describing naturally circular structures, and have the advantage of regular structure and being able to traverse the list starting at any point. They also allow quick access to the first and last records through a single pointer (the address of the last element). Their main disadvantage is the complexity of iteration, which has subtle special cases.

Sentinel nodes

Sentinel nodes may simplify certain list operations, by ensuring that the next and/or previous nodes exist for every element. However sentinel nodes use up extra space (especially in applications that use many short lists), and they may complicate other operations. To avoid the extra space requirement the sentinel nodes can often be reused as references to the first and/or last node of the list.

Linked list operations

When manipulating linked lists in-place, care must be taken to not use values that you have invalidated in previous assignments. This makes algorithms for inserting or deleting linked list nodes somewhat subtle. This section gives pseudocode for adding or removing nodes from singly, doubly, and circularly linked lists in-place. Throughout we will use null to refer to an end-of-list marker or sentinel, which may be implemented in a number of ways.

Linearly-linked lists

Singly-linked lists

Our node data structure will have two fields. We also keep a variable firstNode which always points to the first node in the list, or is null for an empty list.

 record Node {
    data // The data being stored in the node
    next // A reference to the next node, null for last node
 }

 record List {
     Node firstNode   // points to first node of list; null for empty list
 }

Traversal of a singly-linked list is easy, beginning at the first node and following each next link until we come to the end:

 node := list.firstNode
 while node not null {
     (do something with node.data)
     node := node.next
 }

The following code inserts a node after an existing node in a singly linked list. The diagram shows how it works. Inserting a node before an existing one cannot be done; instead, you have to locate it while keeping track of the previous node.

 function insertAfter(Node node, Node newNode) { // insert newNode after node
     newNode.next := node.next
     node.next    := newNode
 }

Inserting at the beginning of the list requires a separate function. This requires updating firstNode.

 function insertBeginning(List list, Node newNode) { // insert node before current first node
     newNode.next   := list.firstNode
     list.firstNode := newNode
 }

Similarly, we have functions for removing the node after a given node, and for removing a node from the beginning of the list. The diagram demonstrates the former. To find and remove a particular node, one must again keep track of the previous element.

 function removeAfter(Node node) { // remove node past this one
     obsoleteNode := node.next
     node.next := node.next.next
     destroy obsoleteNode
 }

 function removeBeginning(List list) { // remove first node
     obsoleteNode := list.firstNode
     list.firstNode := list.firstNode.next          // point past deleted node
     destroy obsoleteNode
 }

Notice that removeBeginning() sets list.firstNode to null when removing the last node in the list.

Since we can't iterate backwards, efficient "insertBefore" or "removeBefore" operations are not possible.

Appending one linked list to another can be inefficient unless a reference to the tail is kept as part of the List structure, because we must traverse the entire first list in order to find the tail, and then append the second list to this. Thus, if two linearly-linked lists are each of length $n$, list appending has asymptotic time complexity of $O(n)$. In the Lisp family of languages, list appending is provided by the append procedure.

Many of the special cases of linked list operations can be made eliminated by including a dummy element at the front of the list. This ensures that there are no special cases for the beginning of the list and renders both insertBeginning() and removeBeginning() unnecessary. In this case, the first useful data in the list will be found at list.firstNode.next.

Doubly-linked lists

With doubly-linked lists there are even more pointers to update, but also less information is needed, since we can use backwards pointers to observe preceding elements in the list. This enables new operations, and eliminates special-case functions. We will add a prev field to our nodes, pointing to the previous element, and a lastNode field to our list structure which always points to the last node in the list. Both list.firstNode and list.lastNode are null for an empty list.

 record Node {
    data // The data being stored in the node
    next // A reference to the next node; null for last node
    prev // A reference to the previous node; null for first node
 }

 record List {
     Node firstNode   // points to first node of list; null for empty list
     Node lastNode    // points to last node of list; null for empty list
 }

Iterating through a doubly linked list can be done in either direction. In fact, direction can change many times, if desired.

Forwards

 node := list.firstNode
 while node ≠ null
     <do something with node.data>
     node := node.next

Backwards

 node := list.lastNode
 while node ≠ null
     <do something with node.data>
     node := node.prev

These symmetric functions add a node either after or before a given node, with the diagram demonstrating after:

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

 function insertBefore(List list, Node node, Node newNode)
     newNode.prev := node.prev
     newNode.next := node
     if node.prev is null
         list.firstNode := newNode
     else
         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 is easier, only requiring care with the firstNode and 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
   destroy node

One subtle consequence of this procedure is that deleting the last element 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.

Circularly-linked lists

Circularly-linked lists can be either singly or doubly linked. In a circularly linked list, all nodes are linked in a continuous circle, without using null. For lists with a front and a back (such as a queue), one stores a reference to the last node in the list. The next node after the last node is the first node. Elements can be added to the back of the list and removed from the front in constant time.

Both types of circularly-linked lists benefit from the ability to traverse the full list beginning at any given node. This often allows us to avoid storing firstNode and lastNode, although if the list may be empty we need a special representation for the empty list, such as a lastNode variable which points to some node in the list or is null if it's empty; we use such a lastNode here. This representation significantly simplifies adding and removing nodes with a non-empty list, but empty lists are then a special case.

Circular linked list is the favorite data structure of Anders Hejlsberg, the creator of LINQ.

Doubly-circularly-linked lists

Assuming that someNode is some node in a non-empty list, this code iterates 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.

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

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

Linked lists using arrays of nodes

Languages that do not support any type of reference can still create links by replacing pointers with array indices. The approach is to keep an array of records, where each record has integer fields indicating the index of the next (and possibly previous) node in the array. Not all nodes in the array need be used. If records are not supported as well, parallel arrays can often be used instead.

As an example, consider the following linked list record that uses arrays instead of pointers:

 record Entry {
    integer next; // index of next entry in array
    integer prev; // previous entry (if double-linked)
    string name;
    real balance;
 }

By creating an array of these structures, and an integer variable to store the index of the first element, a linked list can be built:

integer listHead;
Entry Records[1000];

Links between elements are formed by placing the array index of the next (or previous) cell into the Next or Prev field within a given element. For example:

Index	Next	Prev	Name	Balance
0	1	4	Jones, John	123.45
1	-1	0	Smith, Joseph	234.56
2 (listHead)	4	-1	Adams, Adam	0.00
3			Ignore, Ignatius	999.99
4	0	2	Another, Anita	876.54
5
6
7

In the above example, ListHead would be set to 2, the location of the first entry in the list. Notice that entry 3 and 5 through 7 are not part of the list. These cells are available for any additions to the list. By creating a ListFree integer variable, a free list could be created to keep track of what cells are available. If all entries are in use, the size of the array would have to be increased or some elements would have to be deleted before new entries could be stored in the list.

The following code would traverse the list and display names and account balance:

i := listHead;
while i >= 0 { '// loop through the list
     print i, Records[i].name, Records[i].balance // print entry
     i = Records[i].next;
}

When faced with a choice, the advantages of this approach include:

• The linked list is relocatable, meaning it can be moved about in memory at will, and it can also be quickly and directly serialized for storage on disk or transfer over a network.
• Especially for a small list, array indexes can occupy significantly less space than a full pointer on many architectures.
• Locality of reference can be improved by keeping the nodes together in memory and by periodically rearranging them, although this can also be done in a general store.
• Naïve dynamic memory allocators can produce an excessive amount of overhead storage for each node allocated; almost no allocation overhead is incurred per node in this approach.
• Seizing an entry from a pre-allocated array is faster than using dynamic memory allocation for each node, since dynamic memory allocation typically requires a search for a free memory block of the desired size.

This approach has one main disadvantage, however: it creates and manages a private memory space for its nodes. This leads to the following issues:

• It increase complexity of the implementation.
• Growing a large array when it is full may be difficult or impossible, whereas finding space for a new linked list node in a large, general memory pool may be easier.
• Adding elements to a dynamic array will occasionally (when it is full) unexpectedly take linear (O(n)) instead of constant time (although it's still an amortized constant).
• Using a general memory pool leaves more memory for other data if the list is smaller than expected or if many nodes are freed.

For these reasons, this approach is mainly used for languages that do not support dynamic memory allocation. These disadvantages are also mitigated if the maximum size of the list is known at the time the array is created.

Language support

Many programming languages such as Lisp and Scheme have singly linked lists built in. In many functional languages, these lists are constructed from nodes, each called a cons or cons cell. The cons has two fields: the car, a reference to the data for that node, and the cdr, a reference to the next node. Although cons cells can be used to build other data structures, this is their primary purpose.

In other languages, linked list are typically built using references together with records. Here is a complete example in C:

#include <stdio.h>   /* for printf */
#include <stdlib.h>  /* for malloc */

typedef struct ns {
	int data;
	struct ns *next;
} node;

node *list_add(node **p, int i) {
    node *n = malloc(sizeof(node)); /* you normally don't cast a return value for malloc */
    n->next = *p;
    *p = n;
    n->data = i;
    return n;
}

void list_remove(node **p) { /* remove head */
    if (*p != NULL) {
        node *n = *p;
	*p = (*p)->next;
	free(n);
    }
}

node **list_search(node **n, int i) {
    while (*n != NULL) {
        if ((*n)->data == i) {
            return n;
        }
        n = &(*n)->next;
    }
    return NULL;
}

void list_print(node *n) {
    if (n == NULL) {
        printf("list is empty\n");
    }
    while (n != NULL) {
        printf("print %p %p %d\n", n, n->next, n->data);
        n = n->next;
    }
}

int main(void) {
    node *n = NULL;

    list_add(&n, 0); /* list: 0 */
    list_add(&n, 1); /* list: 1 0 */
    list_add(&n, 2); /* list: 2 1 0 */
    list_add(&n, 3); /* list: 3 2 1 0 */
    list_add(&n, 4); /* list: 4 3 2 1 0 */
    list_print(n);
    list_remove(&n);            /* remove first (4) */
    list_remove(&n->next);      /* remove new second (2) */
    list_remove(list_search(&n, 1)); /* remove cell containing 1 (first) */
    list_remove(&n->next);      /* remove second to last node (0) */
    list_remove(&n);            /* remove last (3) */
    list_print(n);

    return 0;
}

Internal and external storage

When constructing a linked list, one is faced with the choice of whether to store the data of the list directly in the linked list nodes, called internal storage, or merely to store a reference to the data, called external storage. Internal storage has the advantage of making access to the data more efficient, requiring less storage overall, having better locality of reference, and simplifying memory management for the list (its data is allocated and deallocated at the same time as the list nodes).

External storage, on the other hand, has the advantage of being more generic, in that the same data structure and machine code can be used for a linked list no matter what the size of the data is. It also makes it easy to place the same data in multiple linked lists. Although with internal storage the same data can be placed in multiple lists by including multiple next references in the node data structure, it would then be necessary to create separate routines to add or delete cells based on each field. It is possible to create additional linked lists of elements that use internal storage by using external storage, and having the cells of the additional linked lists store references to the nodes of the linked list containing the data.

In general, if a set of data structures needs to be included in multiple linked lists, external storage is the best approach. If a set of data structures need to be included in only one linked list, then internal storage is slightly better, unless a generic linked list package using external storage is available. Likewise, if different sets of data that can be stored in the same data structure are to be included in a single linked list, then internal storage would be fine.

Another approach that can be used with some languages involves having different data structures, but all have the initial fields, including the next (and prev if double linked list) references in the same location. After defining separate structures for each type of data, a generic structure can be defined that contains the minimum amount of data shared by all the other structures and contained at the top (beginning) of the structures. Then generic routines can be created that use the minimal structure to perform linked list type operations, but separate routines can then handle the specific data. This approach is often used in message parsing routines, where several types of messages are received, but all start with the same set of fields, usually including a field for message type. The generic routines are used to add new messages to a queue when they are received, and remove them from the queue in order to process the message. The message type field is then used to call the correct routine to process the specific type of message.

Example of internal and external storage

Suppose you wanted to create a linked list of families and their members. Using internal storage, the structure might look like the following:

 record member { // member of a family
     member next
     string firstName
     integer age
 }
 record family { // the family itself
     family next
     string lastName
     string address
     member members // head of list of members of this family
 }

To print a complete list of families and their members using internal storage, we could write:

 aFamily := Families // start at head of families list
 while aFamily ≠ null { // loop through list of families
     print information about family
     aMember := aFamily.members // get head of list of this family's members
     while aMember ≠ null { // loop through list of members
         print information about member
         aMember := aMember.next
     }
     aFamily := aFamily.next
 }

Using external storage, we would create the following structures:

 record node { // generic link structure
     node next
     pointer data // generic pointer for data at node
 }
 record member { // structure for family member
     string firstName
     integer age
 }
 record family { // structure for family
     string lastName
     string address
     node members // head of list of members of this family
 }

To print a complete list of families and their members using external storage, we could write:

 famNode := Families // start at head of families list
 while famNode ≠ null { // loop through list of families
     aFamily = (family)famNode.data // extract family from node
     print information about family
     memNode := aFamily.members // get list of family members
     while memNode ≠ null { // loop through list of members
         aMember := (member)memNode.data // extract member from node
         print information about member
         memNode := memNode.next
     }
     famNode := famNode.next
 }

Notice that when using external storage, an extra step is needed to extract the record from the node and cast it into the proper data type. This is because both the list of families and the list of members within the family are stored in two linked lists using the same data structure (node).

As long as a member can only belong to one family, internal storage works fine. If, however, the data collection referenced multiple generations, so that a person can appear as a child in one family, but a parent in another, then external storage would be required.

Speeding up search

Finding a specific element in a linked list, even if it is sorted, normally requires O(n) time (linear search). This is one of the primary disadvantages of linked lists over other data structures. In addition to some of the variants discussed in the above section, there are number of simple ways of improving search time.

In an unordered list, one simple heuristic for decreasing average search time is the move-to-front heuristic, which simply moves an element to the beginning of the list once it is found. This scheme, handy for creating simple caches, ensures that the most recently used items are also the quickest to find again.

Another common approach is to "index" a linked list using a more efficient external data structure. For example, one can build a red-black tree or hash table whose elements are references to the linked list nodes. Multiple such indexes can be built on a single list. The disadvantage is that these indexes may need to be updated each time a node is added or removed (or at least, before that index is used again).

Related data structures

Both stacks and queues are often implemented using linked lists, and simply restrict the type of operations which are supported.

The skip list is a linked list augmented with layers of pointers for quickly jumping over large numbers of elements, and then descending to the next layer. This process continues down to the bottom layer, which is the actual list.

A binary tree can be seen as a type of linked list where the elements are themselves linked lists of the same nature. The result is that each node may include a reference to the first node of one or two other linked lists, which, together with their contents, form the subtrees below that node.

An unrolled linked list is a linked list in which each node contains an array of data values. This leads to improved cache performance, since more list elements are contiguous in memory, and reduced memory overhead, because less metadata needs to be stored for each element of the list.

A hash table may use linked lists to store the chains of items that hash to the same position in the hash table.

References

• National Institute of Standards and Technology (August 16, 2004). Definition of a linked list. Retrieved December 14, 2004.
• Antonakos, James L. and Mansfield, Kenneth C., Jr. Practical Data Structures Using C/C++ (1999). Prentice-Hall. ISBN 0-13-280843-9, pp. 165–190
• Collins, William J. Data Structures and the Java Collections Framework (2002,2005) New York, NY: McGraw Hill. ISBN 0-07-282379-8, pp. 239–303
• Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford Introductions to Algorithms (2003). MIT Press. ISBN 0-262-03293-7, pp. 205–213, 501–505
• Green, Bert F. Jr. (1961). Computer Languages for Symbol Manipulation. IRE Transactions on Human Factors in Electronics. 2 pp. 3-8.
• McCarthy, John (1960). Recursive Functions of Symbolic Expressions and Their Computation by Machine, Part I. Communications of the ACM. [2] HTML DVI PDF PostScript
• Donald Knuth. Fundamental Algorithms, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89683-4. Sections 2.2.3–2.2.5, pp.254–298.
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 10.2: Linked lists, pp.204–209.
• Newell, Allen and Shaw, F. C. (1957). Programming the Logic Theory Machine. Proceedings of the Western Joint Computer Conference. pp. 230-240.
• Parlante, Nick (2001). Linked list basics. Stanford University. PDF
• Sedgewick, Robert Algorithms in C (1998). Addison Wesley. ISBN 0-201-31452-5, pp. 90–109
• Shaffer, Clifford A. A Practical Introduction to Data Structures and Algorithm Analysis (1998). NJ: Prentice Hall. ISBN 0-13-660911-2, pp. 77–102
• Wilkes, Maurice Vincent (1964). An Experiment with a Self-compiling Compiler for a Simple List-Processing Language. Annual Review in Automatic Programming 4, 1. Published by Pergamon Press.
• Wilkes, Maurice Vincent (1964). Lists and Why They are Useful. Proceeds of the ACM National Conference, Philadelphia 1964 (ACM Publication P-64 page F1-1); Also Computer Journal 7, 278 (1965).
• Kulesh Shanmugasundaram (April 4, 2005). Linux Kernel Linked List Explained.

External links

• Description from the Dictionary of Algorithms and Data Structures
• Some linked list materials are available from the Stanford University Computer Science department:
  • Introduction to Linked Lists
  • Linked List Problems
• Citations from CiteSeer
• SimCList, an open source C library for linked lists
• SGLib, an collection of algorithms for operating on linked lists (and other containers)
• Generic List Container Type for ANSI C by Jeff Hay
• shared singly linked list implementations
• Linked Lists Tutorial with diagrams and Java code example
• Patent for the idea of having nodes which are in several linked lists simultaneously (note that this technique was widely used for many decades before the patent was granted)
• Non-patented data structure for having nodes which are in several linked lists simultaneously