Introduction to Data Structures

Presented By:Presented By:Anil DuttAnil Dutt

Data Structures

• Outline

• Introduction

• Linked Lists

• Stacks

• Queues

• Trees

• Graphs

Introduction

• Dynamic data structures– Data structures that grow and shrink during execution

• Linked lists– Allow insertions and removals anywhere

• Stacks– Allow insertions and removals only at top of stack

• Queues– Allow insertions at the back and removals from the front

• Binary trees– High-speed searching and sorting of data and efficient

elimination of duplicate data items

Linked Lists

• Linked list – Linear collection of self-referential class objects, called nodes

– Connected by pointer links

– Accessed via a pointer to the first node of the list

– Subsequent nodes are accessed via the link-pointer member of the current node

– Link pointer in the last node is set to null to mark the list’s end

• Use a linked list instead of an array when– You have an unpredictable number of data elements

– Your list needs to be sorted quickly

Linked Lists

• Types of linked lists:– Singly linked list

• Begins with a pointer to the first node

• Terminates with a null pointer

• Only traversed in one direction

– Circular, singly linked• Pointer in the last node points back to the first node

– Doubly linked list• Two “start pointers” – first element and last element

• Each node has a forward pointer and a backward pointer

• Allows traversals both forwards and backwards

– Circular, doubly linked list• Forward pointer of the last node points to the first node and

backward pointer of the first node points to the last node

SINGLY LINKED LIST

DOUBLY LINKED LIST

Previous node Data part Next

node

DATAADDRES

S

Data 1 Data 2 . . . . . . Data n

ADDRESS

ADDRESS

Data 1 Data 2 . . . . . Data n

ADDRESS

ID Name Desig Address of Node2 ID Name Desig Address of Node 3 ID Name Desig NULL

Start

NULL Data 1 Address of Node 2Address of Data 2 Address of Node 1 Node 3Address of Data 3 Node 2 NULL

Start

CIRCULAR LINKED LISTSINGLY

Node 1 Node 2 Node 3

DOUBLY

Node 1 Node 2

Node 3

Start

Data 1 Node 2

Data 2 Node 3

Data 3 Node1

Start

Node3 Data 1 Node 2

Node 1 Data 1 Node 3

Node 2 Data 1 Node1

struct node{

int x; char c[10];struct node *next;}*current;

x c[10] next

create_node(){ current=(struct node *)malloc(sizeof(struct

node)); current->x=10; current->c=“try”; current-

>next=null; return current;}Allocation

Assignment

x c[10] next

10 try NULL

Create a linked list for maintaining the employee details such as Ename,Eid,Edesig.

Steps:1)Identify the node structure2)Allocate space for the node3)Insert the node in the list

EID EName EDesig

struct node

{

int Eid;

char Ename[10],Edesig[10];

struct node *next;

} *current;

next

The basic operations on linked lists are1. Creation 2. Insertion3. Deletion4. Traversing5. Searching6. Concatenation7. Display

Stacks

• Stack – New nodes can be added and removed only at the top

– Similar to a pile of dishes

– Last-in, first-out (LIFO)

– Bottom of stack indicated by a link member to NULL– Constrained version of a linked list

• push– Adds a new node to the top of the stack

• pop– Removes a node from the top

– Stores the popped value

– Returns true if pop was successful

Stack

A

X

R

C

push(M)

A

X

R

C

M

item = pop()item = M

A

X

R

C

Implementing a Stack

• At least three different ways to implement a stack– array– vector– linked list

• Which method to use depends on the application– what advantages and disadvantages does

each implementation have?

Queues

• Queue– Similar to a supermarket checkout line

– First-in, first-out (FIFO)

– Nodes are removed only from the head

– Nodes are inserted only at the tail

• Insert and remove operations – Enqueue (insert) and dequeue (remove)

Queues 18

A QUEUE IS A CONTAINER IN WHICH:

. INSERTIONS ARE MADE ONLY AT

THE BACK;

. DELETIONS, RETRIEVALS, AND

MODIFICATIONS ARE MADE ONLY

AT THE FRONT.

Queues 19

The Queue

A Queue is a FIFO (First in First Out) Data Structure.

Elements are inserted in the Rear of the queue and are removed at the Front.

C

B C

A B C

A

bac kf ro nt

p u s h A

A Bf ro nt bac k

p u s h B

f ro nt bac kp u s h C

f ro nt bac kp o p A

f ro ntbac k

p o p B

Queues 20

PUSH (ALSO CALLED ENQUEUE) -- TOINSERT AN ITEM AT THE BACK

POP (ALSO CALLED DEQUEUE) -- TODELETE THE FRONT ITEM

IN A QUEUE, THE FIRST ITEM

INSERTED WILL BE THE FIRST ITEM

DELETED: FIFO (FIRST-IN, FIRST-OUT)

21

Printing Queue

Now printing A.doc

A.doc is finished. Now printing B.doc

Now still printing B.docD.doc comes

• A.doc B.doc C.doc arrive to printer.ABC

BC

BCD

CD

D

B.doc is finished. Now printing C.doc

C.doc is finished. Now printing D.doc

Trees

• Tree nodes contain two or more links– All other data structures we have discussed only contain one

• Binary trees– All nodes contain two links

• None, one, or both of which may be NULL

– The root node is the first node in a tree.

– Each link in the root node refers to a child

– A node with no children is called a leaf node

Trees

• Diagram of a binary tree

B

A D

C

Trees

• Binary search tree – Values in left subtree less than parent

– Values in right subtree greater than parent

– Facilitates duplicate elimination

– Fast searches - for a balanced tree, maximum of log n comparisons

47

25 77

11 43 65 93

68 7 17 31 44

2

Trees

• Tree traversals:– Inorder traversal – prints the node values in ascending order

1. Traverse the left subtree with an inorder traversal

2. Process the value in the node (i.e., print the node value)

3. Traverse the right subtree with an inorder traversal

– Preorder traversal1. Process the value in the node

2. Traverse the left subtree with a preorder traversal

3. Traverse the right subtree with a preorder traversal

– Postorder traversal1. Traverse the left subtree with a postorder traversal

2. Traverse the right subtree with a postorder traversal

3. Process the value in the node

Trees Data Structures Tree

Nodes Each node can have 0 or more children A node can have at most one parent

Binary tree Tree with 0–2 children per node

Tree Binary Tree

Trees

Terminology Root no parent Leaf no child Interior non-leaf Height distance from root to leaf

Root node

Leaf nodes

Interior nodes Height

Binary Search Trees

Key property Value at node

Smaller values in left subtree Larger values in right subtree

Example X > Y X < Z

Y

X

Z

Binary Search Trees Examples

Binary search trees

Not a binary search tree

5

10

30

2 25 45

5

10

45

2 25 30

5

10

30

2

25

45

Binary Tree Implementation

Class Node {int data; // Could be int, a class, etcNode *left, *right; // null if empty

void insert ( int data ) { … }void delete ( int data ) { … }Node *find ( int data ) { … }

…}

Iterative Search of Binary TreeNode *Find( Node *n, int key) {

while (n != NULL) { if (n->data == key) // Found it

return n;if (n->data > key) // In left subtree n = n->left;else // In right subtree n = n->right;

} return null;

}Node * n = Find( root, 5);

Recursive Search of Binary TreeNode *Find( Node *n, int key) {

if (n == NULL) // Not foundreturn( n );

else if (n->data == key) // Found itreturn( n );

else if (n->data > key) // In left subtreereturn Find( n->left, key );

else // In right subtreereturn Find( n->right, key );

}Node * n = Find( root, 5);

Example Binary Searches Find ( root, 2 )

5

10

30

2 25 45

5

10

30

2

25

45

10 > 2, left

5 > 2, left

2 = 2, found

5 > 2, left

2 = 2, found

root

Example Binary Searches Find (root, 25 )

5

10

30

2 25 45

5

10

30

2

25

45

10 < 25, right

30 > 25, left

25 = 25, found

5 < 25, right

45 > 25, left

30 > 25, left

10 < 25, right

25 = 25, found

Types of Binary Trees Degenerate – only one child Complete – always two children Balanced – “mostly” two children

more formal definitions exist, above are intuitive ideas

Degenerate binary tree

Balanced binary tree

Complete binary tree

Binary Trees Properties Degenerate

Height = O(n) for n nodes

Similar to linked list

Balanced Height = O( log(n) )

for n nodes Useful for searches

Degenerate binary tree

Balanced binary tree

Binary Search Properties

Time of search Proportional to height of tree Balanced binary tree

O( log(n) ) time Degenerate tree

O( n ) time Like searching linked list / unsorted array

Binary Search Tree Construction How to build & maintain binary trees?

Insertion Deletion

Maintain key property (invariant) Smaller values in left subtree Larger values in right subtree

Binary Search Tree – Insertion Algorithm

1. Perform search for value X

2. Search will end at node Y (if X not in tree)

3. If X < Y, insert new leaf X as new left subtree for Y

4. If X > Y, insert new leaf X as new right subtree for Y

Observations O( log(n) ) operation for balanced tree Insertions may unbalance tree

Example Insertion

Insert ( 20 )

5

10

30

2 25 45

10 < 20, right

30 > 20, left

25 > 20, left

Insert 20 on left

20

Binary Search Tree – Deletion Algorithm

1. Perform search for value X

2. If X is a leaf, delete X

3. Else // must delete internal nodea) Replace with largest value Y on left subtree OR smallest value Z on right subtreeb) Delete replacement value (Y or Z) from subtree

Observation O( log(n) ) operation for balanced tree Deletions may unbalance tree

Example Deletion (Leaf)

Delete ( 25 )

5

10

30

2 25 45

10 < 25, right

30 > 25, left

25 = 25, delete

5

10

30

2 45

Example Deletion (Internal Node) Delete ( 10 )

5

10

30

2 25 45

5

5

30

2 25 45

2

5

30

2 25 45

Replacing 10 with largest value in left

subtree

Replacing 5 with largest value in left

subtree

Deleting leaf

Example Deletion (Internal Node) Delete ( 10 )

5

10

30

2 25 45

5

25

30

2 25 45

5

25

30

2 45

Replacing 10 with smallest value in right

subtree

Deleting leaf Resulting tree

Balanced Search Trees

Kinds of balanced binary search trees height balanced vs. weight balanced “Tree rotations” used to maintain balance on insert/delete

Non-binary search trees 2/3 trees

each internal node has 2 or 3 children all leaves at same depth (height balanced)

B-trees Generalization of 2/3 trees Each internal node has between k/2 and k children

Each node has an array of pointers to children Widely used in databases

Other (Non-Search) Trees

Parse trees Convert from textual representation to tree

representation Textual program to tree

Used extensively in compilers Tree representation of data

E.g. HTML data can be represented as a tree called DOM (Document Object Model) tree

XML Like HTML, but used to represent data Tree structured

Parse Trees Expressions, programs, etc can be

represented by tree structures E.g. Arithmetic Expression Tree A-(C/5 * 2) + (D*5 % 4)

+ - %

A * * 4

/ 2 D 5

C 5

Tree Traversal

Goal: visit every node of a tree in-order traversal

void Node::inOrder () { if (left != NULL) { cout << “(“; left->inOrder(); cout << “)”; } cout << data << endl; if (right != NULL) right->inOrder()} Output: A – C / 5 * 2 + D * 5 % 4

To disambiguate: print brackets

+ - %

A * * 4

/ 2 D 5

C 5

Tree Traversal (contd.)

pre-order and post-order:void Node::preOrder () { cout << data << endl; if (left != NULL) left->preOrder (); if (right != NULL) right->preOrder ();}

void Node::postOrder () { if (left != NULL) left->preOrder (); if (right != NULL) right->preOrder (); cout << data << endl;}

Output: + - A * / C 5 2 % * D 5 4

Output: A C 5 / 2 * - D 5 * 4 % +

+ - %

A * * 4

/ 2 D 5

C 5

Graph Data Structures E.g: Airline networks, road networks, electrical circuits Nodes and Edges E.g. representation: class Node

Stores name stores pointers to all adjacent nodes

i,e. edge == pointer To store multiple pointers: use array or linked list

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