data structures and algorithm - gascwkm.org

DATA STRUCTURES AND

ALGORITHM

ARRAYS

• To store a group of data together in one place

that is called array.

• In this data structure all the elements are stored

in contiguous location .

Shown in figure:

Definition

• An array is a finite,ordered and collection of

homogeneous data elements.

• It contain limited no of elements.

• Stored one by one in contiguous location of the

computer memory in linear ordered fashion.

• All the element of an array are in same data

type.

The following some examples:

An array of integers to store the age of the students

in the class.

An array of string to store the names of all villagers

in village.

The data structure is generally build in data types:

Basic : DIMENSIN A[100]

FORTRAN : DIM A[100]

Pascal : A:ARRAY[1….100] of

Integers

C : int A[100]

Terminology

• Size:

The no of elements in an array called the size of the array.

The size also called length and Dimensions.

Type:

It represents the kind of data type

for example:

Array of String, Array of Character ,Array of Integers

Index:

All the elements in an array can be referenced by subscripts like Ai or A[i] this Subscript known as Index.

Range of Indices:

Indices of array elements may change from a lower bound(L) to upper Bound(U).

Lower bound Formula stands as

Index Ai=L + I – 1

Size of the array calculated as

Size (A) = U + L + 1

Word:

It denotes the size of an elements.

In each memory location a computer can store

an element of word size w

The word size varies to the machine to machine

such as 1 byte to 8 bytes.

The size of the elements is doubled then the

word size of machine to store such element

would require two consecutive memory

location.

ONE DIMENSIONAL ARRAY

• If only one subscript/index is required to

reference all the element in an array then the

termed one dimensional array or simply an

array.

Memory Allocation of an Array:

Address (A[i]) = M + ( I - 1)

• Indexing formula

Address (A[i]) = M + (i-L) X W

Operation of Array

This is sub divided into six categories

1. Traversing

2. Sorting

3. Searching

4. Inserting

5. Deleting

6. Merging

Traversing

Searching

Sorting

Insertion

Deletion

Merging

Applications of Arrays

• There are numerous applications of arrays in

computation.

• This is why almost every programming

language Includes this data type as build in

data type.

Example:

• We can store the records as shown in figure:

MULTI DIMENSIONAL ARRAY

• Two dimensional array:

two dimensional arrays (alternatively termed

matrices)are collections of homogeneous

elements.

Where the elements are stored in a number of

rows and columns.

An example of an M x N matrix.

M denotes the number of rows

N denotes the number of column

Memory Representation of a

Matrix

• One dimensional arrays matrices are

also stored in contagious memory

locations.

• There are two conventions of storing

any matrix in the memory

1. Row –Major order

2. Column –Major order

• Row major order:

the elements of a matrix are stored on a row

by row basis.

All the elements in the first row then in the

second row and so on.

Column major order:

The elements are stored in column by column

First column are stored in their order of rows

then the second column, third column and so

on.

Reference of elements in a matrix

• Logically matrix appears as two dimensional

array.

• It is stored in a linear fashion.

• In order to map from logical view to physical

structure.

• We need a index formula.

The indexing formula for different order are

stated below:

Row mjor order:

Address aij =(I - 1)X n + j.

Column major order:

Address (aij)= (j - 1) X m + i.

Sparse Matrices:

It is a two dimensional array .where the majority of the

elements have the value null.

Example:

Sparse matrices can be classified as:

stacks• A stacks is a linear data structure and much

useful in various application of computer

science.

DEFINITION

• A stack is an ordered collection of

homogeneous data elements when the insertion

and deletion operations take place at end only.

• The insertion and deletion operations in the

case of a stack are specially termed PUSH and

POP.

• The position of the stack where those operation

are performed is known as TOP.

REPRESENTATION OF A

STACK

• Stack may be represented in the memory in

various ways.

• There are two main ways:

1. One Dimensional Array.

2. Single Linked List.

Array Representation of Stacks

• First we have to allocate a memory block of

sufficient size to accommodate the full

capacity of the stack.

• Itemi denoted the ith item in the stack l and u

denote the index range of the stack.

• The starting from the first location of the

memory block the item of the stack can be

stored in a sequential fashion.

• The Top is the pointer of the position of the

array.

EMPTY :

TOP < 1

FULL:

TOP >= u

Linked List Representation of

Stacks

• Array representation of stack is very easy and

convenient but it allows the representation of

only fixed sized stacks.

• Inn several application size of the stack may

vary during the program execution.

• The obvious solution to this problem is to

represent a stack using a linked list.

• The linked list representation the first node on

the list is the current item that is the item in

the top of the stack.

• Last node containing the bottom most item.

• The PUSH operation will add a new node in

the front

• POP operation will remove a node from the

list.

• The SIZE of the stack is not important here

this representation allows dynamic stack.

• Static stack as with arrays.

OPERATION ON STACKS

• PUSH:

To insert an item into a stack.

• POP:

To remove an item from the stack.

• STATUS:

To known the present state of the stack.

• PUSH _ARRAY:

• POP_ARRAY

• STATUS _ARRAY

• PUSH_LINKED LIST

POP_LINKED LIST

STATUS _LINKED LIST

APPLICATIONS OF STACK

• various application of stack are known.

• A classical application in a compiler design is the

evaluation of arithmetic expression.

• Here the compiler uses to translate an input

arithmetic expression into its corresponding

object code.

• Build in stack hardware is called stack machine.

• Another important application of a stack is

during execution of recursive programs.

• Some programming language use stack to run

recursive program.

• Important feature of any programming

language is the build in memory variables.

• There are two scope rule :

static scope rule

Dynamic scope rule.

• The implementation of such scope rules is

possible using a stack known as a runtime

stack.

EVALUTION OF ARITHMATIC

EXPRESSION

• An arithmetic expression consist of operands

and operators.

• Operands are variables or constant.

• Operators are of various type

Arithmetic (+,-,*,\,%,^)

Unary

Binary

Boolean (AND,OR,NOT,XOR)

Relatioal

Notation of Arithmetic

Expression

• There are three notations to represent an arithmetic expression.

–Infix

–Post fix

–Pre fix

• INFIX:

Notations

<operands > <operator> <operands>

Example:

A+B,C-D,E*F,G/H,

This is called infix because the operator come in between

the operands

• PRE FIX

NOTATIONS:

<operator> <operands> <operands>

Ex:

+AB,-CD,*EF,/GH..Etc.,

• POST FIX

NOTATIONS:

<OPERANDS>

<OPERANDS><OPERATOR>

EX:

AB+,CD-,EF*,GH/,etc.,

The postfix notation is just reverse of the polish

notation, hence it is also termed reverse polish

notation.

ALGORITHM –INFIX TO

POSTFIX

• EXAMPLE:

ALGORITHM –EVALUATE POSTFIX

• Conversion of a postfix expression to a code:

Code generation for stack

machine

Algorithm postfix to code for stack machine:

Implementation of recursion

• Recursion is an important tool to describe a procedure having several repetitions of same.

• A procedure is termed recursive if the procedure is defined by itself.

• as an example of factorial of integer n number.

• Algorithm factorial _ I

Algorithm Factorial-R

QUEUES

Introduction:

A queue is simple but very powerful data

structure to solve numerous computer applications.

Like stacks.

Queues are also useful to solve various system

programs.

Let us discuss some simple applications of

queues in our every day.

DEFINITIONS:

a queue is linear datastucture like an array,a stacks and a linked list where the ordering of elements is in a linear fashion.

Difference between stacks and queue

STACKS QUEUE

OPERATIONS

PUSH,POP INSERTION,DELETION

At one end only-TOP Two End- rear AND FRONT

LIFO-last in first out FIFO-first in first out

REPRESENTATION OF

QUEUES

There are two ways to represent a queue in memory:

1.Using a Array.

2.Using a Linked List.

This first kind of representation uses a one

dimensional array and it is a better choice where queue

of fixed size is required.

The other representation uses a double linked list

and provides a queue whose size can vary during

processing.

Representatin of a queue using an Array:

One dimensional Array- Q [1…..N]

Pointers- Rear and Front is indicated two ends.

Three states of a Queue with this representation are given

below:

Queue is EmptyFront=0

Rear=0

Queue is FullRear=N

Front=1

Queue contains element>=1Front<=Rear

Number of elements=Rear – Front +1

Algorithm: Enqueue

Algorithm: Dequeue

Representation of a queue using a linked list:

Two states of the queue either empty or containing some

elements, can be judged by the following tests:

Queue is Empty:

FRONT=REAR=HEADER

HEADERRLINK=NULL

Queue contains at aleast one element:

HEADERRLINK= NULL

VARIOUS QUEUE DATA STRUCTURE:

we have discussed two different queue data

structures. That is either using an array or using a

linked list. other than these, there are some more

known queue structures.

Circular queue:

A queue represented using an array when the rear

pointer reaches at the end, insertion will be denied

even if room is available at the front.

One way to avoid the circular array.

The circular array is same as an ordinary array. say

A[1….n]

With this principle the two state of the queue

regarding, that is empty or full ,will be decided as

follows:

Circular Empty:

FRONT=0,REAR=0

Circular is Full:

Front=(Rear mod Length)=1

Algorithm: Enqueue_CQ

Algorithm :Dequeue_CQ

DEQUE:

Another variation of the queue is known as deque.

May be pronounced deck

Unlike a queue in deque, both insertion and deletion

operations can be made at either end of the structure.

Actually, the termed deque has originated from

double ended queue.

Shown in fig.,

There are various ways of representing a deque on the

computer.

Only simpler way to represent it is using a double

linked list.

Another popular representation is using a circular

array

1.Push_DQ(Item): to insert Item at the Front end of a

deque.

2.Pop_DQ():to remove the front item from a deque.

3.Injection(item):to insert item at the rear end of

deque.

4.Eject(): to remove the rear Item from the deque.

Algorithm:PUSH_DQ

Algorithm: Pop_DQ()

Same as the algorithm DEQUEUE_CQ

Algorithm: Eject()

There are however, two known variation of deque:

1.input-restricted deque.

2.Output-restricted deque.

1.Which allows insertion at one end(say Rear end)

only, but allows deletion at both ends.

2.Where deletions take place at one end only(say Front

end),but allows insertion at both ends.

Priority Queue:

A priority queue is another variation of queue

structure. here, each element has been assigned a

value called the priority of the element.

Element can be insertion and deletion not only at the

end but at any position on the queue.

Here,process of means two basic operations namely

insertion or deletion.

There are various ways of implementing the structure

of a priority queue.

1.Using simple/circular array

2.Multi queue implementation

3.Using double linked list

4.Using heap tree.

Priority queue using an array:

Multi queue implementation:

Linked list representation of a priority queue:

Algorithm: insert _PQ

Algorithm: Delete_PQ

APPLICATIONS OF QUEUE:

ONE MAJOR APPLICATION of queue s is in

Simulation.

Another important application of queue is

observed in the various aspects of operating system.

A Multi Programming environment uses the

several queue of to control of various programs.

Simulation:

it is modeling of real life problem.

It is the model of a real life situation in the form

of a computer program.

To study the real life situation under the control of

various parameters which affect the real problem

and is a research interest of system analysis or

operation research scientist.

Based on the result of simulation the actual problem

can be solved in an optimized way.

Another advantage of this is to experiment the

danger area. For example the area such as military

operation are safer to simulate than to field test. it is

being free from any risk as well as inexpensive.

System can be divided into

Discrete system: ticket reservation

Continuous system: water flow through PIPE to

reservoir

Deterministic system's set of initial input but the final

outcome is predicted.

Stochastic system: both deterministic and stochastic

Simulation can be divided into two models

1.Event driven simulation.

2.Time driven simulation.

CPU scheduling Multiprogramming environment:

Single CPU has to serve more than one program

simultaneously.

Multiprogramming environment where the possible

jobs for the CPU are categorized into three groups.

1.Interrupts to be serviced.

2.Interactive user to be serviced.

3.Batch job to be serviced.

ROUND ROBIN ALGORITHM:this algorithm is a

well known scheduling aalgorithm and is designed

especially for time sharing system.

Here we will see how a circular queue can be used to

implement such an algorithm.

First we described the algorithm with illustration.n

no of process p1,p2,p3,…pn

This algorithm first decide a small unit of time called a

time quantum or time slice.(τ)

A time quantum is generally from 10 to 100

milliseconds.

LINKED LIST

The linked lit is called a dynamic structure,

where the amount of memory required can be varied

during its use.

In linked list, the adjacency between the

elements is maintained by means of links or pointer.

Definition:

A linked list is an ordered collection of

finite ,homogeneous data element called nodes

where the linear order is maintained by means of

links or pointer.

The linked list can be classified into three

major groups

1.Single linked list.

2.Circcular linked list.

3.Double linked list.

SINGLE LINKED LIST

In a single linked list each node contains

only one link which points to the subsequent node in

the list.

Here, Header is empty node, and only

pointer to the first node.

Representation of a linked list in memory:

There are two ways to represent a linked list in

memory.

1. Static representation using Array.

2.Dynamic representation using free pool of storage.

Static representation:

In static representation of a single linked list, two arrays

are maintained .

One array for data and other for links.

Two parallel array of equal size are allocated which should

be sufficient to store the entire linked list.

In some programming language for example for ALGOL,

FORTRAN, BASIC.etc., such a representation is the only

represent to mange a linked list.

Dynamic Representation:

the efficient way of representing a linked list is

using the free pool of storage.

Memory bank:

-----which is nothing but a collection of a

free memory spaces.

Memory manager: a program in fact.

Garbage collector:

the memory manger will then search the

memory bank for the block requested and I found,

grant the desired block to the caller.

the mechanism of dynamic representation of single

linked list is illustrated in fig., A and B.

The list of available memory spaces is there whose

pointer is stored in AVAIL.

arequest of a node, the list AVAIL is searched for

the block of right size.

If AVAIL is null or if the block is desired node is not

found, the memory manger will return a message

accordingly.

The memory manager will return the pointer of XY

to the caller in a temporary buffer.

Newly availed node is XY.

Operations on a single linked list:

1.Traversing.

2.Inserting.

3.Deleting.

4.Copying.

5.Merging.

6.Searching.

Traversing a single linked list:

we visit the every node fro starting to end node.

Insert a node in to single linked list:There are various position where node to insert

1.At front

2.At end

3. At any other position

At front:

At end:

At any position:

TREES

Linear- one dimensional array.

non linear representation of data.(two dimensional

representation).

Tree is a non linear data structure.

Family hierarchy of tree:

Algebraic expression:

Basic Terminology:

Node:

this is main component of any tree structure. the

concept of the node is that same as the linked list.

Parent:

the parent of a node is the immediate predecessor

of a node.

Child:

if the predecessor of the node is the parent of the

node then all immediate successor of the node is

successor is the child.

Left side of the node is LEFT CHILD

Right side of the node is RIGHT CHILD

Link:

This is the pointer to a node in a tree. Left Child and Right Child are two links

of a node.

Root:

This is specifically designated node which has no parent.

Leaf:

The node which is at the end node and does not have any child is called leaf.

Level:

level is the rank in hierarchy.

Height:

The maximum number of node that is possible in a path starting from the root

node to a leaf node is called the height.

Degree:

the maximum number of childs that is possible for a node is kknown as the

degree node.

Sibling:

the node which have the same parent are called sibling.

Definition and concept :

A tree is a finite set of one or more nodes such that:

• There is a specially designated node called the root.

• The remaining nodes are partitioned into n(n>0) disjoint set

T1,T2,T3,T4……TN.

In a sample tree T, there are set of 12 nodes,

A is a root node.

Remaining node are portioned into 3 sets T1,T2 and T3.

By definition each sub tree is also a tree.

Observe that a tree is defined recursively.

The same tree can be expressed in a string notation as shown

below:

BINARY TREE:

A Binary tree is a special form of a tree.

Binary tree is more important and frequently used in

various applications of computer science.

A binary tree can also be defined as a finite set of

nodes such that:

• T is empty.

• T contain a specially designated node called the root

of T.

• T form a disjoint binary trees T1 and T2 which are

called the left sub tree and right sub tree.

Difference between Tree and Binary Tree:

TREE BINARY TREE

Never empty. May be empty.

Many number of children. At most of the two children.

It can be divied into two special situations

1.FULL BINARY TREE.

2.COMPLETE BINARY TREE.

FULL BINARY TREE:

It contains the maximum possible number of nodes at all levels.

Complete binary tree:

if its level ,except the possible the last

level have the maximum number of possible nodes,

and all the nodes in the last level appear as far left

as possible.

REPRESENTATION OF BINARY TREE:

it must represent a hierachical relationship

between a parent node and child nodes.

There are two common methods used for

representing this conceptual structure.

1.linear(sequential) representation:-it is using an

array we do not require the overhead of maintaining

pointers(links).

2.Linked representation:- it uses a pointers, the main

objective is that one should have direct access to the

root node of the tree, and for an given node, one

should direct access to the children of it.

LINEAR REPRESENTATION OF A BINARY

Tree:

• this type of representation is static in the sense of

memory for an array is allocated before storing the

actual tree, once the memory is allocated the size of

the tree is restricted as permitted by the memory.

• The nodes are stored level by level, start from 0

level.

• The root node is stored in first memory location.

• The root node is at location 1.

• For any node with index i,1<i<=n

How can the size of an array estimated?

The value can be obtained easily if binary tree is full

binary tree.

Height is h. can most 2h-1 nodes.

So the size of the array to fit such a binary tree is 2h-1.

Size max= 2n-1

Size min=2[log2(n+1)]-1

ADVANTAGES:

• Any node can accessed from any other node by calculating the index

and this is efficient from execution point of view.

• Only data are stored without any pointers to their successor or

ancestor which are mentioned implicitly.

• Programming languages, where dynamic memory location is not

possible (SUCH AS BASIC,FORTRAN)array representation is only

means to store a tree.

DISADVANTAGES:

• The majority of array entries may be empty.

• It allows only static representation, the array size is limited.

• Insert a new node, delete a node are inefficient with this

representation. data movement up and down the array which

demand excessive amount of processing time.

LINKED REPRESENTATIN OF BINARY TREE:

• It is easy to implementation and simplicity.

• It have number of over heads.

Structure diagram :

• Here the LC and RC are stored in memory address.

• DATA contain information content of node.

• This representation ,if one knows the address of the

root node then from it any other node can be

accessed.

ADVANTAGES:

• It allows the dynamic memory location.

• The size of the tree can be changed as and when the

need arises without any limitation except.

OPERATIONS ON A BINARY TREE:

Operations are as follows:

Insertion:-

To include into an existing binary tree(may be empty)

Deletion:-

To delete a node from a non empty binary tree.

Traversal:-

To visit all the nodes in a binary tree.

Merge:-

To merge two binary trees into a large one.

INSERTION:

INSERT BINARY TREE IN SEQUENTIAL(LINEAR):

SEARCH-SEQUENTIAL(LINEAR)

INSERT BINARY TREE IN LINK:

SEARCH-LINK:

TYPES OF BINARY TREE:

1.Expression tree

2.Binary search tree

3.Heap tree

4.Threaded binary tree

5.Huffman tree

6.Height balanced tree

7.Red black tree

8.Splay tree

9.Decision tree.

Expression tree:

BINARYSEARCH TREE:

OPEREATIONS;

• search

• insert

• deleting

• Traversal

HEAP SORT:

H is complete binary tree, it will be termed Heap Tree.

Properties:

• For each node N in H, the value at N is greater than or equal

to the value of each of the children of N

• N has a value which is greater than or equal to the value of

every successor of N

Representation of heap tree:

operations:

• Insert

• Delete

• Merge

• Sort

RED BLOCK TREE:

Tree balanced called the red block tree.

OPERATIONS:

• INSERT

• DELETE

• SEARCH

GRAPHS:

It is no linear data structure,

It have many parent and many children

Examples:

AIRLINES:

Cities are connected through airlines. this can be

represented through a graph structure which is

shown in fig., here airports are connected solid dots

and airline s by lines.

SOURCE DESTINATION NETWORK:

Shown in fig., represent network connections of

three commodities: electricity, gas and water among

three distant destinations D1,d2 and d3

KONIGSBRIDGE’S BRIDGES:

In eastern prussia, there is city named

konigsbridge. this city is surrounded be four lands,

A,B,Cand D which are divide by the river pregal.

Seven Bridges connected the Four lands. this

geographical description can be easily represented

through a graph structure which is shown in fig

.,(c).

FLOW CHART PROGRAM:

The flow chart of a program is in fact the graphical

representation of an algorithm of problem. shown in

fig.,(d).

GRAPH TERMINOLOGIES:

GRAPH:

a graph G consist of two sets:

A set of v called the set of all vertices.(nodes)

A set E called the set of all edges(arc).this set E is the

set of all pair of elements from V.

For example the figure g1:

V={v1,v2,v3,v4}

E={(v1,v2)(v1,v3)(v1,v4)(v2,v3)(v3,v4)}

Digraph:

Weighted graph:

The edges in it are labeled with some weights, for

example g3 and g4.

Adjacent vertices: Fig., g3 and g4

Self loop: Fig., g5

Parallel loop: Fig., g5

Simple graph: Fig., g5 and g10

Complete graph: Fig., g6 and g9

Acyclic graph: Fig g4,g7

Isolated vertex: fig., G8

Degree vertex: fig., g6

Pendant vertex: fig., G7

Connected vertex: fig., g1,g3 and g6

REPRESENTATION OF GRAPH:

1.Set representation

2.Linked representation

3.Sequential representation(matrix)

Set Representation:

It is one of the straight forward method of representing

a graph. with this method two sets are maintained:

1.V the set of vertices.

2.E the set of Edges.

Which is the subset of V X V.

But if the graph is weighted the set E is the ordered

collection of three tuples, that is E=W X V X V

where W is the Set of weight.

Linked Representation:

Matrix Representation:

OPERATIONS OF GRAPHS:

INSERT:

• To insert a vertex and hence establish connectivity

with other vertices in the existing graph.

• To insert an edge between two vertices in the graph.

Deletion:

• Delete vertex from graph

• Delete edge fro graph.

Merging:

To merge two graph G1 and G@ in to simple graph.

Traversal:

To visit all the vertices in the graphs.

Operations on linked list Representations of Graph:

Insertion:

Deleteion;

Graph traversal: visit all the vertices in graph exactly

once.

Methods:

DFS-Dept first Search

BFS- Breadth first Search

DFS:

Traversal is similar to the inorder traversal of a binary tree,starting from a given node this can visit all the node up to the deepeset level.

the two Graphs G1 nad G2

Strating from vertex v1,the path traversal are indicated in the thick lines,the sequence of visiting of the vertices can be obtained as:

DFS(G1)=v1-v2-v5-v7-v4-v8-v6-v3

DFS(G2)=v1-v2-v5-v7-v4-v8-v3-v6

BFS

This traversal is very similar to the level by level

traversal of a tree.

BFS(G1)=v1-v2-v8-v3-v5-v4-v6-v7

BFS(G2)=v1-v2-v3-v5-v4-v6-v7-v8