6 2 Node Lists 2004 Goodrich Tamassia NodeLists

6. 2 Node Lists © 2004 Goodrich, Tamassia Node-Lists 1

Using an index is not the only means of referring to the place where an element appears in a list (sequence). If we have a list S implemented with a (singly or doubly) linked list, then it’s natural to use a node instead of an index to identify where to access and update S. We’ll define the node list ADT, which abstracts the concrete linked list data structure using a position ADT that abstracts the notion of “place” in a node list. © 2004 Goodrich, Tamassia Node-Lists 2

6. 2. 1 Node-Based Operations Assuming List S be implemented using (singly or doubly) linked list. We would like to define methods for S that take nodes as parameters and provide nodes as return types. For instance, we could define a method remove (v) that removes the element of S at node v of the list. This could be executed in O(1) time, by simply “linking out” this node by updating next and prev links of its neighbors. © 2004 Goodrich, Tamassia Node-Lists 3

Similarly, we could insert, in O(1) time, a new element e into S with an operation such as add. After(v, e), that specifies the node v after which the node of the new element should be inserted. In this case, we simply “linking in” this new node. By defining methods of a list ADT by adding such node-based operations requires much more information about the implementation of our list. Certainly, it’s desirable to be able to use either a singly or doubly linked list, without revealing this detail to a user. To abstract and unify the different ways of storing elements in the various implementations of a list, we introduce the concept of “position”, which formalizes the notion of “place” of an element relative to others in the list. © 2004 Goodrich, Tamassia Node-Lists 4

6. 2. 2 Position ADT So as to safely expand the set of operations for lists, we abstract a notion of “position” that allows us to enjoy the efficiency of doubly or singly linked list implementation, without violating “object-oriented” design principles. We view a list as a “collection of elements” that stores each element at a position and that keeps these positions arranged in a linear order. © 2004 Goodrich, Tamassia Node-Lists 5

Position ADT The Position ADT models the notion of place within a data structure where a single object is stored It supports just one method: n object element(): returns the element stored at this position © 2004 Goodrich, Tamassia Node-Lists 6

Properties of Position ADT A position is always defined relatively, that is in terms of its neighbors. In a list, a position p will always be “after” some position q and “before” some position s (unless p is the first or last position). A position p, which is associated with some element e in a list S, does not change even if the index of e changes in S, unless we explicitly remove e (and, hence, destroy position p). Moreover, the position p does not change even if we replace or swap the element e stored at position p with another element. © 2004 Goodrich, Tamassia Node-Lists 7

6. 2. 3 Node List ADT Using the concept of position to encapsulate the idea of “node” in a list, we can define another type of sequence (list) ADT, called node list ADT. The node list ADT models a sequence of positions storing arbitrary objects. It establishes a before/after relation between positions. 1. Generic methods: n size(), is. Empty() © 2004 Goodrich, Tamassia Node-Lists 8

2. Accessor Methods This Node List ADT supports the following methods for accessing elements in a list S : (All these methods return type position) § first(): Return the position of the first element of S; an error occurs if S is empty. § last(): Return the position of the last element of S; an error occurs if S is empty. 2. prev(p): Return the position of the element of S preceding the one at position p; an error occurs if p is the first position. 3. next(p): Return the position of the element of S following the one at position p; an error occurs if p is the last position. The above methods allow us to refer to relative positions in a list, starting at the start or end, and to move incrementally up or down the list S. © 2004 Goodrich, Tamassia Node-Lists 9

3. Update Methods In addition to the accesor methods and the generic methods, we also include the following update methods for the node list ADT, that take position objects as parameters and/or provide position object in return values: § set(p, e): Replace the element at position p with e, returning the element formerly found at position p. § add. First(e): Insert a new element e into S as the first element. 3. add. Last(e): Insert a new element e into S as the last element. 4. add. Before(p, e): Inserts a new element e into S before position p. 5. add. After(p, e): Inserts a new element e into S after position p. 6. remove(p): Remove and return the element at position p in S, invalidating this position p in S. © 2004 Goodrich, Tamassia Node-Lists 10

The node list ADT allows us to view an ordered collection of objects (list) in terms of their places, without worrying about the exact way those places are represented (or, implemented). See Figure below. Baltimore New York p q Paris r Providence s Figure 6. 1: A node list. The positions in the current order are p, q, r, and s. © 2004 Goodrich, Tamassia Node-Lists 11

No Redundancy They may at first seem to be redundancy in the above set of operations for the node list ADT, since we can perform operation: n add. First(e) with add. Before(first(), e), and n add. Last(e) with add. After(last(), e). But these substitutions can only be done for nonempty list. © 2004 Goodrich, Tamassia Node-Lists 12

Error Conditions Note that an error condition occurs if a position passed as argument to one of the list operations (methods) is invalid. Reasons for a position p to be invalid include: n p = null n p was previously removed from the list n p is a position of a different list n p is the first position of the list and we call prev(p) n p is the last position of the list and we call © 2004 Goodrich, Tamassia Node-Lists next(p). 13

Example 6. 2 We show a series of operations for an initially empty list node S. We use variables p 1 , p 2 and so on, to denote different positions, and we show the object currently stored at such a position in parentheses. Operation Output S is. Empty() true (-) last() error (-) add. First(8) - (8) first() p 1 (8) add. After(p 1 , 5) - (8, 5) next(p 1) p 2 (5) (8, 5) add. Before(p 2 , 3) - (8, 3, 5) prev(p 2) P 3 (3) (8, 3, 5) add. First(9) - (9, 8, 3, 5) last() p 2 (5) (9, 8, 3, 5) remove(first()) 9 (8, 3, 5) Set(P 3, 7) 3 (8, 7, 5) add. After(first(), 2) - (8, 2, 7, 5) Size() 4 (8, 2, 7, 5) © 2004 Goodrich, Tamassia Node-Lists 14

Applications of Node List ADT The node list ADT, with its built-in notion of position, is useful in a number of settings. Following are two examples: a program that simulates a game of cards could model each person’s hand as a node list. Inserting and removing cards from a person’s hand could be implemented using the methods of the node list ADT, with the positions being determined by the ordering of cards in the suit. a simple text editor embeds the notion positional insertion and removal relative to a cursor, that represents the current position in the list of characters of text being edited. © 2004 Goodrich, Tamassia Node-Lists 15

Implementing (Array list) Vector ADT with Node-List ADT Vector ADT: List ADT: n size() and is. Empty() n n get(integer i) n n n first(), last() prev(p), next(p) n set(p, e) set(integer i, object e) n insert. Before(p, e) insert. After(p, e) insert. First(e) insert. Last(e) n remove(p) n n add(integer i, object e) n n n remove(integer i) © 2004 Goodrich, Tamassia Node-Lists size(), is. Empty() 16

Node List ADT Exceptions An interface for the node list ADT uses the following exceptions to indicate error conditions: Boundary. Violation. Exception: Thrown if an attempt is made to access an element whose position is outside the range of positions of the list, (ex. , calling next on the last position). Invalid. Position. Exception: Thrown if a position provided as argument in invalid, (ex. , it is null or it has no associated list). © 2004 Goodrich, Tamassia Node-Lists 17

6. 2. 4 Doubly Linked List Implementation A doubly linked list provides a natural implementation of the List ADT Nodes implement position ADT and store: n element n link to the previous node n link to the next node Special trailer and header sentinel nodes Nodes thus define a method element that return the element stored at the node prev elem nodes/ header next positions node trailer elements © 2004 Goodrich, Tamassia Node-Lists 18

Insertion We visualize operation add. After(p, X), which returns position q p A B C p A q B C X p A © 2004 Goodrich, Tamassia q B Node-Lists X C 19

Insertion Algorithm add. After(p, e): Create a new node v v. set. Element(e) v. set. Prev(p) {link v to its predecessor} v. set. Next(p. get. Next()) {link v to its successor} (p. get. Next()). set. Prev(v) {link p’s old successor to v} p. set. Next(v) {link p to its new successor, v} return v {the position for the element e} © 2004 Goodrich, Tamassia Node-Lists 20

Deletion We visualize remove(p), where p = last() A B C p D A © 2004 Goodrich, Tamassia B Node-Lists C 21

Deletion Algorithm remove(p): t = p. element() {a temporary variable to hold the return value} (p. get. Prev()). set. Next(p. get. Next()) {linking out p} (p. get. Next()). set. Prev(p. get. Prev()) p. set. Prev(null) {invalidating the position p} p. set. Next(null) return t © 2004 Goodrich, Tamassia Node-Lists 22

Performance In the implementation of the Node List ADT by means of a doubly linked list, we conclude the following: n n n The space used by a list with n elements is O(n) The space used by each position of the list is O(1) All the methods of the Node List ADT run in O(1) time Operation element() of the Position ADT runs in O(1) time Thus, a doubly linked list is an efficient implementation of the Node List ADT © 2004 Goodrich, Tamassia Node-Lists 23