Chapter 4

Review of Sequential Representations

• Previously introduced data structures, including array, queue, and stack, they all have the property that successive nodes of data object were stored a fixed distance apart.

• The drawback of sequential mapping for ordered lists is that operations such as insertion and deletion become expensive.

• Also sequential representation tends to have less space efficiency when handling multiple various sizes of ordered lists.

Linked List

• A better solutions to resolve the issues of sequential representations is linked lists.

• Elements in a linked list are not stored in sequential in memory. Instead, they are stored all over the memory. They form a list by recording the address of next element for each element in the list. Therefore, the list is linked together.

• A linked list has a head pointer that points to the first element of the list.

• By following the links, you can traverse the linked list and visit each element in the list one by one.

Ordered List for Three-letter words

Linked List Insertion

• To insert an element (GAT) into the three letter linked list:– Get a node that is currently unused;

let its address be x.– Set the data field of this node to GAT.– Set the link field of x to point to the

node after FAT, which contains HAT.– Set the link field of the node

cotaining FAT to x.

Designing a List in C++

• Design Attempt 1: Use a global variable first which is a pointer of ThreeLetterNode.– Unable to access to private data members: data and

link.

• Design Attempt 2: Make member functions public.– Defeat the purpose of data encapsulation.

• Design Attempt 3: Use of two classes. Create a class that represents the linked list. The class contains the items of another objects of another class.

Class ThreeLetterNode {Private:

char data[3];ThreeLetterNode *link;

};

ThreeLetterNode *first;

Design 1

Design 3

Program 4.1 Composite Classes

Nested Classes• The Three Letter List problem can also use

nested classes to represent its structure.

class ThreeLetterList {public: // List Manipulation operations..private:

class ThreeLetterNode { // nested classpublic: char data[3]; ThreeLetterNode *link;};ThreeLetterNode *first;};

Pointer Manipulation in C++

• Addition of integers to pointer variable is permitted in C++ but sometimes it has no logical meaning.

• Two pointer variables of the same type can be compared.– x == y, x != y, x == 0

ax

by

a

x by

bx

by

x = y *x = * y

Composite Classes for List and ListNode

Class List; // forward declaration

Class ListNode {Friend class List;Private:

int data ;ListNode *link;

};

Class List {Public: // List manipulation operations

Private:ListNode *first:

};

Define A Linked List Template

• A linked list is a container class, so its implementation is a good template candidate.

• Member functions of a linked list should be general that can be applied to all types of objects.

• When some operations are missing in the original linked list definition, users should not be forced to add these into the original class design.

• Users should be shielded from the detailed implementation of a linked list while be able to traverse the linked list.

Solution => Use of ListIterator

List Iterator• A list iterator is an object that is used to

traverse all the elements of a container class.

• ListIterator<Type> is delcared as a friend of both List<Type> and ListNode<Type>.

• A ListIterator<Type> object is initialized with the name of a List<Type> object l with which it will be associated.

• The ListIterator<Type> object contains a private data member current of type ListNode<Type> *. At all times, current points to a node of list l.

• The ListIterator<Type> object defines public member functions NotNull(), NextNotNull(), First(), and Next() to perform various tests on and to retrieve elements of l.

Template of Linked ListsEnum Boolean { FALSE, TRUE};template <class Type> class List;template <class Type> class ListIterator;

template <class Type> class ListNode {friend class List<Type>; friend class ListIterator <Type>;private: Type data; ListNode *link;

};

Template <class Type> class List {friend class ListIterator <Type>;public: List() {first = 0;}; // List manipulation operations . .private: ListNode <Type> *first;

};

Template of Linked Lists (Cont.)

template <class Type> class ListIterator {

public:

ListIterator(const List<Type> &l): list(l), current(l.first) {};

Boolean NotNull();

Boolean NextNotNull();

Type * First();

Type * Next();

Private:

const List<Type>& list; // refers to an existing list

ListNode<Type>* current; // points to a node in list

};

Attaching A Node To The End Of A List

Assume there is a private ListNode<Type>* last in class List<Type>

Template <class Type>Void List<Type>::Attach(Type k){

ListNode<Type>*newnode = new ListNode<Type>(k);if (first == 0) first = last =newnode;else { last->link = newnode; last = newnode;

}};

Inverting a List

Concatenating Two Chains

Template <class Type>void List<Type>:: Concatenate(List<Type> b)// this = (a1, …, am) and b = (b1, …, bn) m, n ≥ ,// produces the new chain z = (a1, …, am, b1, bn) in

this.{

if (!first) { first = b.first; return;}if (b.first) { for (ListNode<Type> *p = first; p->link; p = p->link); // no body p->link = b.first;}

}

When Not To Reuse A Class

• If efficiency becomes a problem when reuse one class to implement another class.

• If the operations required by the application are complex and specialized, and therefore not offered by the class.

Circular Lists

• By having the link of the last node points to the first node, we have a circular list.– Need to make sure when current is pointing

to the last node by checking for current->link == first.

– Insertion and deletion must make sure that the circular structure is not broken, especially the link between last node and first node.

Examples of Circular Lists

Diagram of A Circular List

first

last

Linked Stacks and Queues

front rear

top

0

0

Linked Stack

Linked Queue

Revisit Polynomials

123 814 xxa

143 2 8 1 0 0a.first

148 -3 10

10 6 0b.first

61014 1038 xxxb

Polynomial Class Definition

struct Term// all members of Terms are public by default{

int coef; // coefficientint exp; // exponentvoid Init(int c, int e) {coef = c; exp = e;};

};

class Polynomial{

friend Polynomial operator+(const Polynomial&, const Polynomial&);private: List<Term> poly;

};

Operating On Polynomials

• With linked lists, it is much easier to perform operations on polynomials such as adding and deleting.– E.g., adding two polynomials a and b

a.first 143 2 8 1 0 0

b.first 148 -3 10 10 6 0

p

q(i) p->exp == q->exp

c.first 11 14 0

Operating On Polynomials

a.first 143 2 8 1 0 0

b.first 148 -3 10 10 6 0

p

q

(ii) p->exp < q->exp

c.first 11 14 0 -3 10 0

Operating On Polynomials

a.first 143 2 8 1 0 0

b.first 148 -3 10 10 6 0

p

q

(iii) p->exp > q->exp

c.first 11 14 0 -3 10 2 8 0

Memory Leak

• When polynomials are created for computation and then later on out of the program scope, all the memory occupied by these polynomials is supposed to return to system. But that is not the case. Since ListNode<Term> objects are not physically contained in List<Term> objects, the memory they occupy is lost to the program and is not returned to the system. This is called memory leak.

• Memory leak will eventually occupy all system memory and causes system to crash.

• To handle the memory leak problem, a destructor is needed to properly recycle the memory and return it back to the system.

List Destructor

Template <class Type>List<Type>::~List()// Free all nodes in the chain{

ListNode<Type>* next;for (; first; first = next) {

next = first->link;delete first;

}}

Free Pool

• When items are created and deleted constantly, it is more efficient to have a circular list to contain all available items.

• When an item is needed, the free pool is checked to see if there is any item available. If yes, then an item is retrieved and assigned for use.

• If the list is empty, then either we stop allocating new items or use new to create more items for use.

Using Circular Lists For Polynomials

• By using circular lists for polynomials and free pool mechanism, the deleting of a polynomial can be done in a fixed amount of time independent of the number of terms in the polynomial.

Deleting A Polynomial with a Circular List Structure

a.first 143 2 8 1 0

av

1

second2

3

av

Equivalence Class

• For any polygon x, x ≡ x. Thus, ≡ is reflexive.

• For any two polygons x and y, if x ≡ y, then y ≡ x. Thus, the relation ≡ is symmetric.

• For any three polygons x, y, and z, if x ≡ y and y ≡ z, then x ≡ z. The relation ≡ is transitive.

Equivalence

Definition: A relation ≡ over a set S, is said to be an equivalence relation over S iff it is symmetric, reflexive, and transitive over S.

Example: Supposed 12 polygons 0 ≡ 4, 3 ≡ 1, 6 ≡ 10, 8 ≡ 9, 7 ≡ 4, 6 ≡ 8, 3 ≡ 5, 2 ≡ 11, and 11 ≡ 0. Then they are partitioned into three equivalence classes:{0, 2, 4, 7, 11}; {1 , 3, 5}; {6, 8, 9 , 10}

Equivalence (Cont.)

• Two phases to determine equivalence– In the first phase the equivalence pairs (i,

j) are read in and stored.– In phase two, we begin at 0 and find all

pairs of the form (0, j). Continue until the entire equivalence class containing 0 has been found, marked, and printed.

• Next find another object not yet output, and repeat the above process.

Equivalence

Definition: A relation ≡ over a set S, is said to be an equivalence relation over S iff it is symmetric, reflexive, and transitive over S.

Example: Supposed 12 polygons 0 ≡ 4, 3 ≡ 1, 6 ≡ 10, 8 ≡ 9, 7 ≡ 4, 6 ≡ 8, 3 ≡ 5, 2 ≡ 11, and 11 ≡ 0. Then they are partitioned into three equivalence classes:{0, 2, 4, 7, 11}; {1 , 3, 5}; {6, 8, 9 , 10}

Equivalence (Cont.)

• Two phases to determine equivalence– In the first phase the equivalence pairs (i,

j) are read in and stored.– In phase two, we begin at 0 and find all

pairs of the form (0, j). Continue until the entire equivalence class containing 0 has been found, marked, and printed.

• Next find another object not yet output, and repeat the above process.

Equivalence Classes (Cont.)

• If a Boolean array pairs[n][n] is used to hold the input pairs, then it might waste a lot of space and its initialization requires complexity Θ(n2) .

• The use of linked list is more efficient on the memory usage and has less complexity, Θ(m+n) .

Linked List Representation

Linked List for Sparse Matrix

• Sequential representation of sparse matrix suffered from the same inadequacies as the similar representation of Polynomial.

• Circular linked list representation of a sparse matrix has two types of nodes: – head node: tag, down, right, and next– entry node: tag, down, row, col, right, value

• Head node i is the head node for both row i and column i.

• Each head node is belonged to three lists: a row list, a column list, and a head node list.

• For an nxm sparse matrix with r nonzero terms, the number of nodes needed is max{n, m} + r + 1.

Node Structure for Sparse Matrices

15000

0040

00012

01100

A 4x4 sparse matrix

Linked Representation of A Sparse Matrix

4 4

0 2

1 0

2 1

3 3

-4

12

11

-15

H0

H1

H2

H3

H0 H1 H2 H3Matrix head

Reading In A Sparse Matrix

• Assume the first line consists of the number of rows, the number of columns, and the number of nonzero terms. Then followed by num-terms lines of input, each of which is of the form: row, column, and value.

• Initially, the next field of head node i is to keep track of the last node in column i. Then the column field of head nodes are linked together after all nodes has been read in.

Complexity Analysis

• Input complexity: O(max{n, m} + r) = O(n + m + r)

• Complexity of ~Maxtrix(): Since each node is in only one row list, it is sufficient to return all the row lists of a matrix. Each row is circularly linked, so they can be erased in a constant amount of time. The complexity is O(m+n).

Doubly Linked Lists

• The problem of a singly linked list is that supposed we want to find the node precedes a node ptr, we have to start from the beginning of the list and search until find the node whose link field contains ptr.

• To efficiently delete a node, we need to know its preceding node. Therefore, doubly linked list is useful.

• A node in a doubly linked list has at least three fields: left link field (llink), a data field (item), and a right link field (rlink).

Doubly Linked List

• A head node is also used in a doubly linked list to allow us to implement our operations more easily. llin

kitem rlink

Head Node

llink

item rlink

Empty List

Deletion From A Doubly Linked Circular List