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Chapter 3

Data Representation

Part 1

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Goals• Introduce the different ways in which data may be represented

• Concepts

– Abstract data types

– Formula-based, linked, indirect addressing, and simulated-pointer representations

– Chains, circular lists, and doubly linked lists

• Applications

– Bin sort, radix sort

– Equivalence class

– Convex hull

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Data objectsA set of instances or values

• Boolean = {false, true}

• Digit = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

• Letter = {A, B, C, …, Z, a, b, … z}

• Natural Number = {0, 1, 2, …}

• Integer = {0, ±1, ±2, ±3, …}

• String = {a, b, …, aa, ab, ac, …}

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Primitive and composed data object

• Primitive (or atomic) data objects

• Composed data objects

• Relationships among instances, e.g., order

• Functions associated with data objects

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Data structures• Data structure: a data object together with the

relationships that exist among the instances, usually expressed by functions/operations

• Data structure = data + functions• Standard data types - int, float, bool• User defined data types - using enumeration,

grouping facility such as class, array, and pointers, e.g., char s[MaxSize];

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Linear Lists

• Data object in the form (e1, e2, …, en) where n is a finite natural number

• elements - ei’s

• length - n

• empty list - when n = 0

• Order - e1 precedes e2, e2 precedes e3, and so on

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Examples of linear lists

• An alphabetized list of students in a class

• a list of exam scores in nondecreasing order

• an alphabetized list of members of Congress

• a list of gold-medal winners in the Olympics men’s basketball event

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ADT specification

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Formula-based representation

• Use an array to represent the instances of a linear list

• Cell (or node) holds an instance of the data object

• Locations of each element is given by a mathematical formula, e.g.,location(i) = i-1

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An example

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Class Definition

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Class Definition (Continue)

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What if new fails?

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Constructor

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Find and Search

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Delete

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Insert

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Output

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A client program

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A client program (continue)

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Output of the client program

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Evaluation

• Merits– Fast: Search, Delete, and Insert functions have

a worst complexity that is linear

• Shortcomings– Inefficient use of space

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Enhancement:Represent multiple lists in a single array

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Insertion

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Insertion (continue)Cost: a single insertion may require as many as

MaxSize-1 moves

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Linked representation• Each node keeps an explicit information (link or

pointer) about the the location of other relevant nodes

• Singly linked list, also called chain

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Chain

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Chain (continue)

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Destructor - Θ(n)

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Length - Θ(n)

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Find - O(k)

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Find - O(n)

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Output - Θ(n)

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Delete the fourth node

• Locate the third and fourth nodes

• Link the third node to the fifth

• Free the fourth node

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Delete

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Delete (continue)

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InsertionTwo cases: k=0 and k<>0

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Insert function

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Insert function (continue) - O(k)

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Extensions to the Class ChainErase: delete the nodes in the chain

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Chain

last = 0; }

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Chain (continue)

, *last ;

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Append: add an element to the end of a chain - Θ(1)

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In case we need to visit elements of the whole chain

Chain<int> L; ……..

int len = L.Length();

int x, sum =0;

for (int i = 1; i <= len; i++) {

L.Find(i, x);

sum = sum + x;

}• The complexity of this code is Θ(n2)

• It is necessary to define a function which traverse the chain in Θ(n) time

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Chain Iterator Class

Location

Initialize()

next()

ChainIterator

Chain

first

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Chain Iterator Class

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Traverse the chain using IteratorChain<int> L; …….…..int *p;ChainIterator<int> c; p = c.Initialize(L); // c is iterator of chain Lwhile (p != NULL){

sum = sum + (*p);p = c.Next(); // advance p to point to next

// node in the chain}

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Circular List• singly linked circular list

• add the head node at the front of the list

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Search a circular list - O(n)

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Comparison with formula-based representation

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Doubly linked list

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Class definition

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Class definition (continue)

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The end of Chapter 3Part 1