DATA STRUCTURE BY SIVASANKARI

DATA
STRUCTURE
A. SIVASANKARI MSc., M.Phil., DCP.,
HEAD OF THE DEPARTMENT,
DEPT OF COMPUTER SCIENCE,
D.K.M. COLLEGE FOR WOMEN (AUTONOMOUS)
SAINATHAPURAM, VELLORE-1.

DATASTRUCTURE (CS &CA)
SIVASANKARI A Page 2
ACKNOWLEDGEMENTS
It is my immense pleasure to offer my first and foremost gratitude to the
Almighty God for giving me the power to believe in myself and pursue my
dreams.
I would like to express my gratitude to my Father B.M.Arunachalam,
Mother A.Kalaiarasi and Brother Dr. A. Dineshkarthik for their love,
encouragement and support.
I offer my humble gratitude to our honourable Secretary
Mr.D.Maninathan, BE., for permitting me to bring out this book and for their
valuable advice.
I would like to extend my heartfelt gratitude to our beloved Principal
Dr.P.N.Sudha Ph.D., for permitting me to bring out this book and for their
valuable appreciation advice and blessings.
I wish to express my profound thanks to my Colleagues and my well-
wishers Dr.M.Sobia and R.Rajeshbabu for their moral support and
encouragement in making this book a reality.
I also thank all my Friends and Students who supported directly or indirectly
to bring out this book in good direction.
I am happy to express my thanks to the publisher and the entire team of
Publications, who have taken immense pain to get this book.
I wish our readers all the very best in their all endeavours.
SIVASANKARI

UNIT-I
DATA STRUCTURE
Data are represented by data values held temporarily within programs data area or
recorded permanently on a file often the different data values are related to each other. To enable
programs to make use of these relationship. These data values must be in an organized form. The
organized collection of data is called a data structure. The program have to follow certain values
to access and process.
An Organized collection of data is called data structure. Data may be organized in many
different ways, the logical or mathematical model of a particular organization of the data is
called structure.
Data Structure :
Organized data allowed operator alternative definition is
d=[DFA]->D means –Set of Domains
F Means-Set of Functions
A Means –Set of Axioms
A Triple set of DFA is called data structure which is denoted by Symbol d.
DIFFERENT TYPES OF DATA STRUCTURE (OR)
CLASSIFICATION OF DATA STRUCTURE
1.We may classify the data structures as Linear and Non Linear Data Structure. In Linear data
structure the data item are arranged in a linear sequence line an array. In a Non-Linear the data
item are not in a sequence an example of a Non-Linear data structure is TREE.
2.Data structure may also classified as Homogeneous(Ex: Array) and Non-Homogeneous(Ex:
Record) Data Structure).An Array is a Homogeneous Structure in which all elements are of same
type records are a common example of Non-Homogeneous data Structure.
3.Another way of Classifying data structure is a static or dynamic data structure .A Static
structure are one whose sizes and structures are associated memory locations are fixed at compile
time. Dynamic structure are one which expand or shrink required during the program execution
and their associated memory location change. Records are a common example of non-
Homogeneous data structure.
4. Another way of classifying data structure is primitive and non-Primitive. Another name of
Primitive data structure is standard data structure. Non-Primitive data structure is called as
composite.

DATA STRUCTURE
PRIMITIVE[STANDARDDATA TYPE] NON-PRIMITIVE[COMPOSITE DATA TYPE]
EXAMPLE EXAMPLE
1. INT 1. FILE
2. REAL 2. LIST[LINEAR,NON-LINEAR]
3. LONG INT 3. STACK
4. STRING 4. ARRAY
5.QUEUE
PRIMITIVE DATA STRUCTURE
NUMBER SYSTEM [BINARY,OCTAL,DECIMAL,HEXA-DECIMAL]
Numbers are used symbolically to represent quantities of objects.
Number system such as Roman Number system, Positional Number System.
ROMAN NUMBER SYSTEM ------ I,II,III,IV
POSITIONAL NUMBER SYSTEM ------- Decimal, Binary number system
Example: 329-Decimal ------300+20+9
3*102
+2*101
+9*100
0,1,2 are Positional weights.
Example: 10110
1*24
+0*23
+1*22
+1*21
+0*20
COMPLIMENTS
 1’s ,2’s,9’s,10’s with examples.
INT
 Long int
 Short Int
 Signed
 UnSigned
FLOATING POINT
 Real Number

CHARCTER INFORMATION
 BCDIC and ASCII
LOGICAL IMMPLEMENTATION INFORMATION
 And
 OR
 Not
NON-PRIMITIVE DATA STRUCTURE
Non-Primitive Data structure can be classified as Arrays, List and Files.
ARRAY
1. An Array is an ordered set which consist of a fixed number or objects.
2. No deletion or insertion operations are performed on arrays. At best elements can be
changed to a value which represents an element to be ignored.
List
1. A List is an ordered set consisting of a variable number of elements to which insertion
and deletion can be made.
2. A List which displays the relationship of adjacency between elements is said to be linear.
And other List is said to Non-Linear.
Data
Data is a element such as number,character etc…
[information raw data fact]
Field
A collection of data is called as Field
[name, sex..]
Record
A Collection of field is called Record
File
A Collection of record is called as file
Stack
Last in First out
Queue
First in first out

OPERATION ON DATA STRUCTURE
1. CREATION
An operation frequently used in conjunction with data structure is one which create a data
structure. This operation will be called CREATION Operation.
2. DESTROY
One which destroys the operation, the operation is called DESTROY Operation.
3. SELECTION [access]
The most frequently used operation associated with data structure is one that the
programmer uses to access data within a data structure. This type of operation is
known as SELECTION Operation.
4. UPDATE [Changing the Data]
Another operation used in Conjunction with data structure is one which changes
data structure.
This operation will be called an UPDATE.
ASYMTOTIC NOTATION [0,Ω,Θ]
1. [Big ‘oh’] The function f(n)=o(g(n)) (Read as f of n is big oh of g of n) if(if and only
if)their exist positive constants c and n0 .Such that f(n)<=c*g(n)) for all n, n>=n0.
2. [Omega] The function f(n)= Ω(g(n)) (Read as f of n is omega of g of n)if their exist
positive constant c and n0.such that f(n)>=c*g(n) for all n, n>=n0.
3. [Theta] The function f(n)= Θ(g(n)) (Read as f of n is Theta of g of n)if there exist +ve
constant c1,c2 and n0 .such that c1,g(n)<=f(n)<=c2g(n) forn,n>=n0.
4. [Little ‘oh’]The function f(n)=o(g(n)) (Read as of n is Little oh of g of n)if ,
𝑛 → ∞ 𝑓(𝑛)/g(n)=n->∞
𝑓(𝑛)
𝑔(𝑛)
= 0
5. [Little Omega]The function f(n)=w(g(n)) (Read as ‘F of n is little omega of g of n’)if,
lim n → ∞
g(n)
f(n)
= 0
ARRAY
1. Array is an ordered set which consists of a fixed number of objects.
2. No deletion or insertion operations are performed on array. At best elements can be
changed to a value which represent an element to be ignored.
3. Array can be classified as one dimensional, two dimensional and Multi-dimensional.

ONE –DIMENSIONAL ARRAY
The simplest form of an array is a one dimensional array or vector the various element of
an array are distinguished by giving each piece of data separate index or subscript. The subscript
of an element designate its position in arrays ordering. An array named which consists on N
elements can be depicted as shown in figure.
Representation of a Vector
LO-1 1
Where Lo=> is starting Position
C=> Mention the type of array
.. i=> Finding Position Member
Lo+c(i-1)-->A[i]---- A[i]
Another Representation
A(1)
A(2)…………………………………………………………………….A(n)
If we want to find A[10]
Loc a[5] = L0 + c * (I-1)
= 1 + 1* (5-1)
5 = 1 + 4
5 = 5
TWO DIMENSIONAL ARRAY
Any Array Defined To Have More Than One Dimension Is Considered To Be Multi
Dimensional Array. An array can be 2,3,4 or N dimensional array.
Two dimensional arrays, sometimes called matrices. This array is to visualize a table
column and rows. The first dimension in the array refer to the row, and the second dimension
refers to the columns.

Example: A Collection of data about the grades of students in a class in the four different
exams can be represented using a two dimensional array. If we have 10 students and each given
grades in 4 exams, we can depict it as in the table(fig 2)
Grade
Student Number
Each cell in this table contains a grade value for the student number (given by the
corresponding row number) and exam number)(g and exam number)(given by the corresponding
column number). An array A of order [0*4.A[I,j] represent an element of A. Where I runs from
to 10 and J runs from 1 to 4 A[3,4]]will have the grade value of 8th
Student in first exam and so
on.
[Loc A[I,j]=L0 + (i-1) *(j-1) ]
In General an array of the order M*N consist of M rows, N Columns and MN elements.
It may be depicted as
1 , 2 , ……………………………………………….N
1
2
.
.
M
OPERATIONS AN ARRAY
The array is homogeneous structure (I,e) the element of an array are the same type .
Following set of operations are defined for this structure.
1. Creating an array
2. Initializing an array
3. Storing an element
4. Ret riving an element
5. Inserting an element
1 2 3 4
1
2
3
4
5
6
7
8
9
10

6. Deleting an element
7. Searching for an element
8. Sorting element
9. Printing an array
STORAGE OF ARRAYS IN MAIN MEMORY
The data represented in an array is an array is actually stored in the Memory cells of the
machine Because computer’s memory is linear, a one dimension array can be mapped. Onto
the memory cells in a rather straight forward manner. Storage for element A[I+1] will be
adjacent to storage for elements BCD for I=1 to n to find the actual address of an elements .
One Merely needs to subtract one from the position of the desired entry. And then add the
result to the address of the first cell in the sequence.
Two dimensional array, we think of data being arranged in rows and columns. However
machine’s memory is arranged as a row of memory cells. Then the rectangular structure of a
two dimensional array must be stimulated. We first calculate the amount of storage area
needed and allocate a block of contiguous memory cells of that size on way to score the data
in the cells is row by row. That is we store first, the first row of that array and the second row
of the array and then the next and soon.
For Example :- The array defined by a which logically appears as given in figure.
Int [3][4]
A[0][0] A[0][1] A[0][2] A[0][3]
A[1][0] A[1][1] A[1][2] A[1][3]
A[2][0] A[2][1] A[2][2] A[2][3]
A[0][0] A[0][1] A[0][2] A[0][3]
A[1][0]  A[1][1] A[1][2] A[1][3]
A[2][0] A[2][1] A[2][2] A[2][3]
Appears physically as given in row major representation. Such a storage scheme is called
row major order. The other alternative is to store the array column by column, it is called
column, it is called column major order.
A[0][0] A[0][1] A[0][2] A[0][3]
A[1][0] A[1][1] A[1][2] A[1][3]
A[2][0] A[2][1] A[2][2] A[2][3]

OPERATION ON ARRAY
1. To find the location
 Single Dimensional
 Two Dimensional
SINGLE DIMENSIONAL
EXAMPLE: FORMULA: Loc A[i]= L0 + c *I – 1
A[10]=A[0],A[1],----------------------a[9]
To Find the Location of a[5]
Loc A[i] = L0 + C * I – 1
Where I = Position of the Location
L0 = First Position of Array
C = Mention the dimensional type
Loc A[5]= L0 + c * (i-1)
A[5]= 1 +1 * [5-1]
A[5] = 1 + 1 * 4
A[5] = 1 + 4
A[5] = 5
TWO DIMENSIONAL ARRAY
Row Major Order : Loc A[I,j]=L0 + [i-1] *n + (j-1)
Column Major Order : Loc A[I,j]=L0 + [j-1] *m+(i-1)
Where i = Row Position
J= Column Position
m=row of the given matrix
n=Column of the given matrix
Lo= 1 st
Position of the given matrix

The given matrix is [6,4]
Representation of given matrix = A[1][1]A[1][2]A[1][3]A[1][4]
A[2][1]A[2][2]A[2][3]A[2][4]
A[3][1]A[3][2]A[3][3]A[3][4]
A[4][1]A[4][2]A[4][3]A[4][4]
A[5][1]A[5][2]A[5][3]A[5][4]
A[6][1]A[6][2]A[6][3]A[6][4]
ROW MAJOR ORDER
Loc A[I,j]=L0 + [i-1] *n + [j-1]
Loc A[3,2] = 1 + [3-1] * 4 +[2-1]
= 1+ 2 * 4 + 1
= 1 + 8 + 1
Loc A[3,2] = 10
COLUMN MAJOR ORDER
The Given matrix is [6,4]
Representation of Given matrix = A[1][1] A[1][2]A[1][3]A[1][4]
= A[2][1]A[2][2]A[2][3]A[2][4]
= A[3][1]A[3][2]A[3][3]A[3][4]
= A[4][1]A[4][2]A[4][3]A[4][4]
= A[5][1]A[5][2]A[5][3]A[5][4]
= A[6][1]A[6][2]A[6][3]A[6][4]
Loc A[I,j] = Lo + [j-1]m + [i-1]
Loc A[3,2]= 1 + (2-1) 6 + (3-1)
Loc A[3,2]= 1 + 6 + 2
Loc A[3,2] = 9

UNIT – II
STACK AND QUEUE
NON-PRIMITIVE DATA TYPE
STACK
One of the most important linear data structure of variable size is the stack.
Example : Pile of Tray (pile-stack)
DEFINITION
In the most general form of a linear list we are allowed to delete an element from and
insert on to any portion in the list. An important subclass of lists permits the insertion on deletion
of an element to occur only at any end.
A Linear list belonging to the subclass is called a stack.
This insertion operation is referred to as PUSH and the deletion Operation as POP. The
most and the least accessible elements in a stack are known as top and bottom of the stack
respectively.
Since insertion and deletion operations are performed at one end of the stack, the
elements can only be removed in the opposite order from that in they were added to the stack.
A Linear list is frequently referred to as a LIFO List, Which means “LAST IN FIRST OUT”.
Example: A common example of a stack phenomenon which permits the selection of only its end
element is a pile of trays in Cafeteria.
DIAGRAM REPRESENTATION
TOP TRAY OF FILE
Stacked Trays
Spring

Another example of a stack is a railway system for shunting cars as shown in the
diagram.
In this system the last railway can to be placed on the stack is the first to leave. Using
repeatedly the insertion and deletion operation permits that cars, to be arranged on the output
railway line in various orders.
OUTPUT DELETION
STACK
The operation on a stack are stimulated by a vector consisting of some large number of
elements which should be sufficient in number to handle all possible insertions likely to be made
to stack. A Pointer top keeps track of the top elements in the stack. Initially when the stack is
empty top has a value ‘0’ and when the stack contains a single element top has the value ‘1’ and
soon. Each time a new elements is inserted in the stack, the pointer is incremented by 1, before
the element is placed on the stack.
The pointer is decremented by 1,each time the deletion is made from the stack.
Bottom of stack
Vector of elements
representing a stack Top of Stack
Unused Stack
A more suitable representation of a stack is given in the stack represent the top element.
The left most occupied element of the stack represent its bottom element.

ALTERNATIVE REPRESENTATION OF STACK
DELETION
INSERTION
BOTTOM TOP UNUSED STACK
OPERATIONS ON A STACK
The algorithm for insertion and deleting an element from a stack are given below
1. PROCEDURE PUSH(S,TOP,X)
This procedure inserts an element a to the top of a stack which is represented by a vector
s containing N elements with a pointer top denoting the top elements in the stack.
1. [Check for stack overflow]
If top>=N
Then write (Stack Overflow)
Return
2. [Increment Top]
TopTop +1
3. [Insert element]
S[Top] x
4. [Finished]
Return
2. FUNCTION POP(S,TOP)
This function removes the top element from a stack which is represented b a vector a and
returns this element. Top is a pointer to the top element of the sack.
1. [Check for underflow on stack]
If Top=0
Then write(STACK UNDERFLOW ON POP)
Take action in response to underflow
Exit

2. [Decrement pointer]
Top  Top-1
3. [Return for met top element of stack]
Return (S[top+1])
4. [Finished]
3. FUNCTION PEEP(S,TOP,I)
Given a vector S (Consisting of N elements S) represent a sequentially allocated stack
and a pointer top denoting the top elements of the stak. This function return the value of
the I th element from the top of the stack. The element is not deleted by this function
1. [Check for stack underflow]
If Top- I + 1 <=0
Then write (STACK UNDERFLOW ON PEEP)
Take action in response to underflow
Exit
2. [Return I th
element from top of stack]
Return (S[Top – I + 1])
4. PROCEDURE CHANGE (S,TOP,X,I)
As before a vector S(containing of N elements)represent a sequentially allocated
stack and a pointer top denotes the Top element of the stack. This procedure changes the
value of the Ith
element from the top of the stack to the value contained in X.
1. [Check for stack underflow]
If Top – I + 1 <=0
Then write (STACK UNDERFLOW ON CHANGE)
Return
2. [Change I th
element from Top to Stack]
S[TOP – I + 1] X
3. [Finished]
Return.
APPLICATION OF STACK
The first application deals with recursion. The second application of a stack is classical, it
deals with the compilation of infix expressions in to object code
1. RECURSION
A property or a set P can be specified by an inductive definition. An inductive definition
of a set can be realized by using a given finite set of elements. A and the following three classes
:Basis, inductive and external class.

A Vector Representation Of n Stack
S[1]---------------------------------------------------------------------------------------------S[M]
B[1]+[1] B[2]+[2] B[N]+B[N] B[N+1]
The factorial function, whose domain is natural number, can be recursively defined as,
1 IF N = 0
FACTORIAL = N * FACTORIAL(N-1) OTHERWISE
Here FACTORIAL(N) is defined in terms of FACTORIAL(N-1) which in turn is defined
in terms of FACTORIAL(N-2) etc. Until finally FACTORIAL(0) is reached, whose value is
given as “ONE”. A recursive definition of a set or process must contain an explicit definition for
particular vales(s) of the argument(s).Otherwise the definition could never change.
An important facility available to the program is the procedure (function or subroutine).A
procedure that contains a procedure call to itself, or a procedure call to a second procedure which
eventually causes the first procedure to be called is known as “Recursive’ procedure.
There are two conditions that must be satisfied by and recursive procedure.
1. Each time a procedure calls itself (either directly or indirectly) it must be nearer in
some sense to a solution. In the case of the factorial function, each time that the
function calls itself its argument is decremented by ‘ONE’. So the argument of the
function is getting smaller.
2. Second , there must be a decision criterion for stopping, the process or computation.
In the factorial function the value of n must be zero.
There re essentially two types of recursion. The first type concerns recursively
defined function.(or primitive recursive function)an example of this kind is the
factorial function. The second type is recursion). A typical example of this kind is
Ackerman’s function which is defined as,

N+1, if m=0
A(M,N) = A(M-1 , 1 ) , if n= 0
A (M -1), A(M, N-1), OTHERWISE
Primitive recursive function can be solved iteratively.
A Iterative process can be illustrated with the aid of the flow chart given below:
There are four parts in this process,
1. Initialization
2. Decision
3. Computation (Calculation)
4. Updation (Change)
Initialization
The parameters of the function and a decision parameter in this part are set to there initial
values. The decision parameter is used to determine when to exit from loop.
Decision
The Decision parameter decides whether to remain in the loop.
Computation
The Required computation is performed in this part.
Updation
The decision parameter is updated and a transfer to the next iteration results. Recursion is
becoming increasingly important in symbol manipulation and non-numeric applications.
Done
Not
Not Done
ENTRY
INITIALIZATION
DECISION
COMPUTATION
RETURN
UPDATE

The general algorithm model for any recursive procedures contain the following steps:
1. [PROLOGUE] Save the parameters, local variables and return address.
2. [BODY]I f the base criterion has been reached then perform the final
computation and goto step3,Otherwise perform the partial computation and goto
Step1.
3. [EPILOGUE]Restore the most recently saved parameters local variable and
return address, go to this return address.
A Flow chart model for this algorithm is given the model consists of a prologue,
a body and an epilogue. The purpose of the prologue is to save the formal
parameters, local variable and return address and that of epilogue is to restore
them. Note that the parameters, local variables and return address that are restored
are those which were most recently(i.e) the last saved are the first to be restored
(LIFO). The body of the procedure contains a procedure call to itself. In fact,
They may be more than one call to itself in certain procedures.
The keybox contained in the body of the procedure is the one which invokes a call
to itself. The dotted-line exit from this box indicated that a call to itself is being
initiated within the same procedure. Each time a procedure saves all necessary
information required for its proper functioning.
4. The procedure body contains two computation boxes namely, the partial and final
computation boxes frequently the partial computation box is combined with the
Values of the arguments. The procedure call box(Updation box).
(This is case for computation of the factorial function).
The final computation box gives the explicit definition of the process for same
values or values of the arguments. The test box determines whether the arguments
values(s) in that for which explicit definition of the procedure is
given,[Characteristic associated with each call to(entry into a recursive procedure
is a level number. The entry in to the procedure due to the initial call from the
main program is given the level number ‘one’ as the main program is assumed to
have level number ‘Zero’. Each subsequent entry in to the procedure has an
associated level number ‘one’, higher than the level number of the procedure
from which the call was made. Another characteristic of the recursive procedure,
in the depth of recursion, which is the number of times the procedure is called
recursively in the process of evaluating a given argument or arguments, usually
the quantity is not obvious, except in the case of extremely simple recursive
functions. Such as factorial(N) for which the depth is N.
5. The last -in and first-out characteristic of a recursive procedure suggest that a
stack is the most obvious data structure to use to implement step1 and three of this
procedure. At each time procedure call (or level of recursion) the stack is pushed
to save the necessary values, upon exit from that level, the stack is pushed to save
the ‘popped’ to restore the saves values of the proceeding (or calling level).

The recursive mechanism is best described by an example. Consider as algorithm
to calculate factorial(N)recursively which explicitly shows the recursive frame
work.
FACTORIAL
Given an integer N, this algorithm computers N. The stack A is used to store an
activation record associated with each recursive call. Each activation records contains the current
value of N & the current return address (RET-ADDRESS). TEMP-REC is also a record which
contains two variables (PARM and ADDRESS). This temporary recorders required to simulate
the proper transfer of control from one activation record of algorithm FACTORIAL to another.
Whenever a temp-rec is placed on the stack A, copies of PARM and ADDRESS are
pushed on to A and assigned to N and RET-ADDRESS respectively. Top points to the top
element of stack A and its value is initially 0.
Initially ,the return address is set to the main calling address.(i.e. ADDRESSMain address)
PARM 2 is set to the initial values of N2.
1. [save N and return address]
Call PUSH[A,TOP,TEMP-REC]
2. [Is base criterion found]
If N=0 then
Go to step 4
Else
PARM N-1
ADDRESS step 3
Go to step 1
3. [Calculate N]
FACTORIAL N * FACTORIAL
4. [Restore previous N and return address]
TEMP-RECPOP(A,TOP)
(i.e., PARM N)
ADDRESS RET-ADDR
Go to ADDRESS

EXAMPLE
LEVEL NO DESCRIPTION STACK A CONTAINS
1 1.PUSH[A,0 (2,MAIN ADDRESS)]
2.If n≠0
PARMN-12-1==>1
ADDRESS STEP 3
TOP VALUE
2 1.PUSH [A,1(1,STEP 3)]
IF N≠ 0
PARMN-11-1==> 0
2. ADDRESS STEP 3 TOP VALUE
Top
3 1. PUSH[A,2(0,STEP 3)]
2.N=O
FACTORIAL1 TOP
Top
RETURN 4. TEMP.RECPOP(A,STEP 3)
ENTER LEVEL 2 3.FACTORIAL1 *1
4.TEMPPOP(A,STEP 3)
GO TO STEP 3
Top
RETURN
ENTER LEVEL 1 5.FACT2*1==>2
TEMP REC-POP(A,MAINADD)
GO TO MAIN ADDRESS
Top
2 1
Main
Add
Step 3
2 1 0
Main
Add
Step 3 Step 3
2 1
Main
Add
Step 3
2
Main
Add
2
MAIN ADDRESS

PROGRAM FOR FACTORIAL
# INCLUDE <IOSTREAM.H>
VOID MAIN()
INT FACT=1,N,I;
CIN>>N;
FOR(I=1,I<=N;I++)
FACT=FACT*I;
}
COUT<<FACT;
}
2. TOWER OF HANOI
Another complex recursive problem is that of the tower of Hanoi has a historical basis in the
virtual of the ancient tower of Brahma. The Problem is as follows
Given N discs of decreasing size stack on one needle and two empty needles, it is
required to stack all the discs onto a second needle in decreasing order of size. The third needle
may be used as Temporary Storage.
The movements of disc is restricted by the following rules,
i) Only one drive may be moved at a time.
ii) A Disc may be moved from any needle to any other
iii) At no time may a larger discs rest upon a smaller disc.
PICTURE OF HANOI
(A) (B) (C)

STEPS ARE
1. A TO C
2. A TO B
3. C TO B
4. A TO C
5. B TO A
6. B TO C
7. A TO C
The solution on this problem is most clearly seen with the aid of induction. To move one
disc, merely move it from needle A to needle C. To move two disc, move the first disc to needle
B, move the second from needle A to needle C, then move the disc from needle B to C.
In general , the solution of the problem of moving discs from needle A to needle C has
three types,
1. Move N-1 disc from A to B
2. Move discs from A to C
3. Move N-1 discs from B to C.
A close examination of the first and third steps will reveal that they are recursive in nature,
A General algorithm for solving the tower of Hanoi problem is now given,
i) Push parameters and return address on Stack.
ii) If the stopping value has been reached then pop the stack to return to previous
level else move all expect the final disc from starting to intermediate needle.
iii) Move final disc from start to destination needle .
iv) Move remaining discs from intermediately to destination needle.
v) Return to previous level by popping stack.
ALGORITHM
1. [SAVE PARAMETER AND RETURN ADDRESS]
CALL PUSH (ST,TOP,TEMP-REC)
THE EFFECT OF THE PUSH OPERATION IS AS FOLLOWS
TOPTOP+1,N[TOP]N-VALUE,SN[TOP]SN-VALUE,IN[TOP]IN-
VALUE,DN[TOP]DN-VALUE,RETADDR[TOP]ADDRESS)
2. [TEST FOR STOPPING VALUE OF N, IF NOT REACHED MOVE N-1 DISCS FROM
START NEEDLE TO INTERMEDIATE NEEDLE]
IF N[TOP]=0
THEN GO TO RET-ADDR[TOP]
ELSE

N-VALUE N[TOP]-1
SN-VALUESN[TOP]
IN-VALUEDN[TOP]
DN-VALUEIN[TOP]
ADDRESSSTEP 3
GO TO STEP 1
3. [MOVE N TH DISC FROM START TO DESTINATION NEEDLE MOVE N-1 DISCS
FROM INTERMEDIATE TO DESTINATION NEEDLE]
TEMP-RECPOP(ST,TOP)
WRITE (‘DISC’,N,’FROM NEEDLE’,SN,’TO NEEDLE,DN)
N-VALUEN[TOP]
SN-VALUEIN[TOP]
IN-VALUESN[TOP]
LEVEL NO DESCRIPTION STACK CONTAINS
1. STEP1:PUSH[ST,0(2,A,B,C,MAINADDS)
STEP2: IF N≠0
N-VALUEN[TOP]-1
SN-VALUESN[TOP] A
IN-VALUEDN[TOP] C
DN-VALUEIN[TOP] B
RET-ADDRESS STEP 3
GO TO STEP 1
TOP AC
2. STEP 1:PUSH[ST,1(1,A,C,B,STEP 3)]
STEP 2:IF N≠0
N-VALUEN[TOP]-1
SN-VALUESN[TOP] A
IN-VALUEDN[TOP] B
DN-VALUEIN[TOP] C
RET-ADDRESS STEP 3
GO TO STEP 1 TOP AB
N 2
SN A
IN B
DN C
RE Main
address
2 1
A A
B C
C B
Main
address
STEP 3

3. STEP 1:PUSH[ST,2(0,A,B,C,STEP3)]
STEP 2: IF N=0
POP(ST,TOP)
N-VALUEN[TOP]-1
SN-VALUEIN[TOP] B
IN-VALUESN[TOP] A
DN-VALUEDN[TOP] C
RET-ADDRESS[TOP] TOP AC
4. POP(ST,TOP)
WRITE(DISC 1 A TO B)
N-VALUEN[TOP]-1=0
SN-VALUEIN[TOP] C
IN-VALUESN[TOP] A
DN-VALUEDN[TOP] B
RET-ADDRESSSTEP 4
TOP CB
STEP 4: POP(ST,TOP)3
2(ST,TOP)
GO TO RET-ADDR
WRITE (DISC 2 FROM A TO C)
N-VALUE2-1
SN-VALUEB
IN-VALUEA
DN-VALUEC
ADDRESSSTEP 4
GO TO STEP 1 TOP AC &BC
5. STEP 1:PUSH[ST,1(1,B,A,C,STEP 4)]
STEP 2:IF N≠0
N-VALUE1-1
SN-VALUEB
IN-VALUEC
DN-VALUEA
ADDRESSST3P 3
GO TO STEP 1
STEP 3: DN-VALUEDN[TOP] TOP BA
ADDRESSSTEP 4
GO TO STEP 4
2 1 0
A A A
B C B
C B C
Main
address
Step
3
Step3
2 1 0
A A C
B B A
C C B
Main
address
Step
3
Step4
N 2 1
SN A B
IN B A
DN C C
RE Main
address
Step
4
2 1 0
A B B
B A C
C C A
Main
address
STEP 3 STEP
3

STEP 4: [RETURN TO PREVIOUS LEVEL]
TEMP-RECPOP(ST,TOP)
GO TO REC-ADDR
3.INFIX, POSTFIX AND PREFIX NOTATIONS
Every expression can be denoted in 3 types and they are
i) Infix Notation
ii) Postfix or polish or reverse polish notation
iii) Prefix notation
INFIX
The expressions which are in normal format is known as infix notation. In this type the
operation lies between the two variables.
Ex: A+B
PREFIX
In prefix notation the operator comes first followed by two operands.
Ex:+AB
POSTFIX
In Postfix notation the operands comes first followed by the operator
Ex: AB+
ALGORITHM : REVERSE POLISH
REV-POL Given an input string infix containing an infix expression which has been P
added on the right with ‘(‘ and whose symbols have precedence values given by tables, a vector
‘S’ used as a stack and function ‘NEXTCHAR’ which when invoked returns the next character
of its arguments, this algorithm converts INFIX TO REVERSE POLISH and places the results in
string polish.
The integer variable Top denotes the top of the stack algorithm. PUSH & POP are used
for stack manipulation. The integer variables rank accumulates the rank of the expression.
Finally, the string variable TEMP is used for temporary storage purpose.

1. [Initialize stack]
Top 1
S[TOP]’(‘
2. [INITIALIZE OUTPUT STRING AND RANK COUNT]
POLISH”0” OR ” “
RANK0
3. [GET 1ST
INPUT SYMBOL]
NEXTNEXT CHAR (INFIX)
4. [TRANSLATE THE INFIX EXPRESSION]
REPEAT THROUGH STEP-7 WHILE NEXT ≠” “
5. [REMOVE SYMBOLS WITH GREATE PRECEDENCE FROM STACK]
IF TOP< 1 THEN
WRITE (‘INVALID’)
EXIT
REPEAT WHILE F(NEXT)<G(S[TOP])
TEMPPOP(S,TOP)
POLISHPOLISH ‘0’ TEMP
RANK<-RANK + R[TEMP]
IF RANK< 1 THEN
EXIT
6. [ARE THERE MATCHING PARANTHESIS]
IF F(NEXT) ≠G (S[TOP]) THEN
CALL PUSH (S,TOP,NEXT)
ELSE POP (S,TOP)
7. [GET NEXT INPUT SYMBOL]
NEXTNEXT CHAR(INFIX)
IF TOP≠0 OR RANK≠1 THEN
ELSE WRITE(‘VALID’)
EXIT
A trace of the stack contents and the polish string ‘POLISH’ for the infix expression.
Q=A+(B*C-(D/E^/F)*G)*H
P=ABC*DEF^/G*-H*+

PRECEDENCE TABLE
SYMBOL INPUT PRECEDENCE
FUNCTION OF F
STACK PRECEDENCE
FUNCTIONING G
RANK
FUNCTION
R
+,- 1 2 -1
*,/ 3 4 -1
6 5 -1
Variable 7 8 1
( 9 0 -
) 0 - -
SYMBOL SCANNED STACK EXPRESSION P
(
A ( A
+ (+ A
( (+( A
B (+(* AB
* (+(* AB
C (+( ABC
- (+(- ABC*
( (+(-( ABC*
D (+(-( ABC*D
/ (+(-(/ ABC*D
E (+(-(/ ABC*DE
^ (+(-(/^ ABC*DE
F (+(-(/^ ABC*DEF
) (+(- ABC*DEF^/
* (+(-* ABC*DEF^/
G (+(-* ABC*DEF^/G
) (+ ABC*DEF^/G*-
* (+* ABC*DEF^/G*-
H (+* ABC*DEF^/G*-H
) ABC*DEF^/G*-H*+
QUEUES
Another important subclass of lists permits deletion to be performed at one end of a list
and insertion at the other. The information in such as list is processed in the same order as it was
received that is on a FIRST IN FIRST OUT (FIFO)Or a FIRST COME FIRST SERVED(FCFS)
basis. This type of list is frequently referred to as a queue.

Following figure is a representation of a queue illustrating how an insertion is made to
the right of the most element in the queue and how a deletion consists of deleting the left most
element in the queue.
REPRESENTATION OF QUEUE
FRONT REAR
INSERTION
DELETION
The familiar and traditional chain of a queue in a check outline at a super mark cash
register. The first person in line is usually the first to be checked out.
A final example of a queue is the line of cars waiting to proceed in some fixed direction at
an insertion of streets. The deletion of a car corresponds to the first car in the line passing
through the intersection, while an insertion to the queue consists waiting the end of the line of
existing cars waiting to proceed through the intersection.
PROCEDURE QUEUE INSERT(Q,F,R,N,Y)
Given F and R pointers to the front and rear elements of a queue ,a queue consisting of N
elements and y elements, this procedure inserts Y at the rear of a Queue. Prior to the first
invocation of the procedure F and R have been set to Zero.
1. [OVERFLOW]
IF R>=N
THEN WRITE (‘OVER FLOW’)
RETURN
2. [INCREMENT REAR POINTER]
RR+1
3. [INSERT ELEMENT]
Q [R]Y
4. [IS FRONT POINTER PROPERLY SET?]
IF F=0
THEN F1
RETURN
The following algorithm deletes an elements from a queue.
FUNCTION Q DELETE(Q,F,R)

Given F and R the pointers to the front and rear elements of a queue respectively and the queue
Q to which they correspond this function deletes and returns the last elements of the queue y is a
temporary variable.
1.[UNDERFLOW]
IF F=O THEN WRITE (‘UNDERFLOW’)
RETURN(O)
0DENOTES AN EMPTY QUEUE
2. [DELETE ELEMENTS]
YQ[F] (Y IS ITEM)
3. [QUEUE EMPTY]
IF F=R THEN FR0
ELSE
FF+1 (INCREMENT FRONT POINTER)
4. [RETURN ELEMENT]
RETURN [Y]
CIRCULAR QUEUE
Circular queue are the ones implemented in circular from rather than a straight line. Circular
queues overcome the problem of unutilized space in the linear queues. The circular queue
considers at opposite end, that is why, if the space is there in the beginning , the rear shifts back
to beginning after reaching the max, possible storage.
REAR
FRONT
PROCEDURE FOR INSERTION
1. [RESET REAR POINTER]
IF R=N THEN R1
ELSE RR+1
2. [OVERFLOW]
IF F=R THEN WRITE[‘OVERFLOW’]

RETURN
3. [INSERT ELEMENT]
Q[R]Y
4. [IF FRONT POINTER PROPERLY SET]
IF F=0 THEN F1
RETURN
PROCEDURE FOR DELETION
1. [UNDERFLOW]
IF F=0 THEN WRITE (‘UNDERFLOW’)
RETURN (0)
2. [DELETE ELEMENT]
YQ[F]
3. [QUEUE EMPTY]
IF F=R THEN FR0
RETURN [Y]
DEQUEUE
DEQUEUE (DOUBLE ENDED QUEUE) is a linear list in which insertion and deletion are
made to or from either end of the structure. Such a structure can be represented by the following
diagram.
DELETION INSERTION
DELETION
REAR

INSERTION FRONT
It is clear that a dequeue is more general than a stack or a queue.
There are two variation of a dequeue namely
1. The input restricted dequeue
2. The output restricted dequeue.
The input restricted dequeue allows insertion at only one end, while an output restricted dequeue
permits deletions from only one end.
PRIORITY : Q
Adding and Deleting is according to the priority.

UNIT III
LINKED LISTS
DEFINITION
A list has been defined to consists of an ordered set of elements which may vary in number.
A simple way to represent a linear list in to expand each mode to contain a link or pointer to the
next node. This representation is called a one way chain or singly linked list and it can displayed
simply as shown below.
A)
  ……… 
B)
  
2000 2010 2002 2012
C) INSERT NODE X
 
2000 2010 2002 2012
2100
D) DELETE NODE C
  
2000 2010 2002 2012
A 2010 B 2002 C 2012 E
A 2100 B 2002 C 2012 E
X 2010
A 2010 B 2012 E

In figure (A) the variable first contains the address or pointer which gives the location of the
first node of the list. Each node is divided in to two parts. The first part represent the information
of the element and the second part contains the address of the next node. The last node of the list
does not have a successor node and consequently no actual address is stored in the pointer field.
In such case, a null value is stored as address. For Example, the linked list in fig(B) represents a
4 nodes units whose elements are located in memory locations 2000,2010,2002 and 2012
respectively . The linked address of NULL in the last node signals the end of the list.
OPERATIONS
1. INSERTION
2. DELETION
An insertion is performed in a straight forward manner. If a new element is to be
inserted following the first element, this can be accomplished by merely interchanging
pointers.
For example, this can be shown in fig(C) for a 4 th element.
The deletion of the fourth element from the original list can be performed by using the
pointer change as shown in figure(D). It is clear that the insertion and deletion operations are
more efficient when performed on linked list than an sequentially allocated list.
OPERATIONS ON LINEAR LIST USING SINGLY LINKED LIST STORAGE
A typical element or node consists of two fields namely an information field called
INFO and a pointer field denoted by LINK. The name of a typical element is denoted by
NODE.
Pictorially , the NODE structure is as follows
INFO LINK
NODE
It is further assumed that an available area of storage for this node structure consists of a
linked state of available nodes as shown in Fig(A).
TYPES OF LINKED LIST
1. INTERNAL ADDRESS
2. EXTERNAL ADDRESS
3. NULL ADDRESS

FUNCTION INSERT (X,FIRST)
Given X, a new element and first a pointer to the first element of a linked linear list whose
typical node contains INFO & LINK field as previously discussed. This function inserts X.
AVAIL is a pointer to the top element of the availability stack. NEW is a temporary pointer
variable. It is required that X proceed the node whose address is given by FIRST.
1. [UNDERFLOW]
IF AVAIL= NULL THEN WRITE (AVAILIABILITY STACK UNDERFLOW)
RETURN[FIRST]
2. [OBTAIN ADDRESS OF NEXT FREE NODE]
NEW AVAIL
3. [REMOVE FREE NODE FROM AVAILIABILITY STACK]
AVAIL LINK(AVAIL)
4. [INITIALIZE FIELDS OF NEW NODE AND ITS LINK TO LIST]
INFO(NEW) X
LINK(NEW)FIRST
5. [RETURN ADDRESS OF NEW NODE]
RETURN (NEW)
FUNCTION INSERT END (X, FIRST)
1. [UNDERFLOW]
IF AVAIL=NULL THEN WRITE (AVILABILITY STACK UNDERFLOW)
RETURN (FIRST)
2. [OBTAIN ADDRESS OF NEXT FREE NODE]
NEW AVAIL
3. [REMOVE FREE NODE FROM AVAILABILITY STACK]
AVAILLINK(AVAIL)
4. [INITIALIZE FIELDS OF NEW NODE]
INFO(NEW)X
LINK(NEW)NULL
5. [IS THE LIST EMPTY]
IF FIRST =NULL THEN RETURN (NEW)
6. [INITIATE SEARCH FOR LAST NODE]
SAVE FIRST
7. [SEARCH FOR END OF LIST]
REPEAT WHILE LINK(SAVE)≠ NULL
SAVE LINK(SAVE)
8. [SET LINK FIELD OF LIST NODEE TO END]
LINK(SAVE)NEW

9. [RETURN FIRST NODE POINTER]
RETURN (FIRST)
DELETING A NODE
FUNCTION DELETE(X,FIRST)
1. [EMPTY LIST]
IF FIRST= NULL THEN WRITE (UNDERFLOW)
RETURN
2. [INITIALIZS SEARCH FOR X]
TEMP FIRST
3. [FIND X]
REPEAT THROUGH STEP-5 WHILE
TEMP≠ X AND LINK(TEMP) ≠ NULL
4. [UPDATE PREDECESSOR MARKER]
PRED TEMP
5. [MOVE TO NEXT NODE]
TEMPLINK[TEMP]
6. [END OF THE LIST]
IF TEMP ≠ X THEN WRITE(NODE NOT FOUND)
RETURN
7. [DELETE X]
IF X= FIRST [IS X THE FIRST NODE] THEN
FIRST<LINK(FIRST)
ELSE LINK(PRED)LINK(X)
8. [RETURN NODE TO AVAILABILITY AREA]
LINK(X)AVAIL
AVAILX
RETURN
COPY A NODE
A General algorithm to copy a linked list is now presented.
1. [EMPTY LIST]
IF FIRST=NULL THEN RETURN [NULL]
2. [COPY FIRST NODE]
IF AVAIL =NULL THEN WRITE (AVAILABILITY STACK UNDERFLOW)
RETURN (0)
ELSE
NEWAVAIL
AVAILLINK(AVAIL)

FIELDINFO(FIRST)
BEGINNEW
3. [INITIALIZE TRAVERSAL]
SAVEFIRST
4. [MOVE TO NEXT NODE IF NO AT THE END OF LIST]
REPEAT THROUGH STEP-6 WHILE LINK[SAVE] ≠NULL
5. [UPDATE PREDECESSOR AND SAVE POINTERS]
PREDNEW
SAVELINK(SAVE)
6. [COPY NODE]
IF AVAIL= NULL THEN WRITE(AVAILABILITY STACK UNDERFLOW)
RETURN (0)
ELSE
NEWAVAIL
AVAILLINK(AVAIL)
FIELD(NEW)INFO(SAVE)
PTR(PRED)NEW
7. [SET LINK OF LAST NODE AND RETURN]
PTR(NEW)NULL
RETURN(BEGIN)
POLYNOMIAL
A node consists of number of fields each of which can represent an int , a real number
etc..except for one field called the pointer which contains the location of the next node in the list.
Consider the example of representing a term of a polynomial in the variables x,y,z a
typical node is represented as follows.
Power-x Power-y Power-z Co-efficien LINK
TERM
Which consists of 5 sequentially allocated fields, that we collectively refer to ‘TERM’. The
first three fields represent the power of the variables x,y and z respectively. The 4 th and 5 th
field represent the co efficient of the ‘TERM’ in the polynomial and the address of the next in the
polynomial respectively.
For example the term 3xy would be represented in the figure,
1 1 0 3

This selection of a particular field with in a node for a polynomial example in an easy
matter.Our algorithm notation allows the referring of any field of a node given the pointer p to
the node.
Co-eff(p) denotes the coefficient field of the node pointed to by P. similarly the exponent of
x,y,z are given by power x(p),power y(p) and power z(p) respectively and the pointer to the next
node is given by LINK(P).
Consider as an ex, the representation of the polynomial 2x2
+5xy+y2
+yz as a linked list.
Assuming that the nodes in the list are to be stored such the term pointer to by p, proceeds
another term indicated by Q, if power x(P) is greater than power x(Q) or if the powers of x are
equal, then power y(p) ,must be equal than power y(Q) or if the power of y are equal then power
z(p) must be greater than power z(Q).
For example, the list is represented by the following figure,( 2x2
+5xy+y2
+yz)
X y z co-eff
2 0 0 2
2x2
1 1 0 5
5xy
0 2 0 1
Y2
0 1 1 1
Yz
OPERATION ON POLYNOMIAL
INSERTION
FUNCTION POLY FRONT (NX,NY,NZ,NCOEFF,POLY)
Given the definition of the node structure TERM under availability area from which we can
obtain such nodes, it is required to insert a node in the linked list so that it immediately preceeds
the node whose address is designated by the pointer POLY. The fields of the new term are
denoted by NX,NY,NZ,NCOEFF which corresponds to the exponents for x,y,z and the co-
efficient value of the term respectively, NEW is a pointer variable which contains the address of
the new node.
1. [OBTAIN A NODE FROM AVAILABLE STORAGE]
NEWTERM
2. [INITIALIZE NUMERIC FIELDS]

POWER-X (NEW)NX
POWER-Y(NEW)NY
POWER-Z(NEW)NZ
COEFF(NEW)NCOEFF
3. [SET LINK TO THE LIST]
LINK[NEW]POLY
RETURN [NEW]
EXAMPLE :YZ+Y2
+5XY+2X2
1. POLY
0 1 1 1
2. POLY
0 2 0 1
0 1 1 1
3. POLY
1 1 0 5
0 2 0 1
0 1 1 1 -
4. POLY
2 0 0 2
1 1 0 5
0 2 0 1
0 1 1 1 -
FUNCTION POLYEND (NX,NY,NZ,N,COEFF,POLY)
obtain such node is referred to insert a node in the , list so that it immediately proceeds the node
whose address is designated by the pointer POLY. The fields of the new TERM are denoted by
NX,NY,NZ,NCOEFF Which corresponds to the exponents for x,y,z and the co efficient value of
the term respectively. NEW is a pointer variable which contains the address of new node. SAVE
is a temporary pointer variable.
1. [OBTAIN A NODE FROM AVALIABLE STORAGE]
NEW TERM
POWER X(NEW)NX
POWER Y(NEW)NY
POWER Z(NEW)NZ

IF POLY=NULL THEN RETURN[NEW]
4. [INITIATE SEARCH FOR LAST NODE]
SAVEPOLY
5. [SEARCH FOR END OF LIST]
REPEAT WHILE LINK(SAVE) ≠ NULL
SAVELINK(SAVE)
6. [SET LINK FIELD OF LAST NODE TO NEW]
LINK(SAVE)NEW
8. RETURN(POLY)
EXAMPLE :2X2
+5XY+Y+YZ
1. POLY
2 0 0 2
2. POLY
2 0 0 2
1 1 0 5
3. POLY
2 0 0 2
1 1 0 5
0 2 0 1 -
4. POLY
2 0 0 2
1 1 0 5
0 2 0 1
0 1 1 1 -
POLYLAST(NX,NY,NZ,NCOEFF,POLY)
obtain such nodes, it is required to insert a node in a linked list. So that it immediately proceeds
the node whose address is designated by the pointer POLY.
The global pointer variable LAST denotes the address of the last node in the list. The fields of
the new term are denoted by NX,NY,NZ,NCOEFF, which corresponds to the term respectively
NEW is a pointer variable which contains the address of the new node.
1. [OBTAIN A NODE FROM AVAILABLE STORAGE]
NEWTERM

POWER X(NEW)NX
POWER Y(NEW)NY
POWER Z(NEW)NZ
CO-EFF(NEW)NCOEFF
LINK(NEW)NULL
IF POLY = NULL THEN LASTNEW
RETURN(NEW)
4. [INSERT NODE IN NON EMPTY LIST]
LINK(LAST)NEW
LASTNEW
POLYNOMIAL ADDTION
A general algorithm for adding two polynomial is presented.
FUNCTION POLY ADD(P,Q)
Given two polynomial whose first terms are referred by pointer variables P & Q respectively,
and insertion function as described in function copy, is required to add their polynomials and
score their sum in a third polynomial which is referenced by the pointer variable R. The original
polynomials are to remain unchanged. PSAVE & QSAVE are pointer variables.
A1,A2,B1,B2,C1,C2,D1 & D2 are temporary variables of global variables.
1. [INITIALIZE]
RNULL
PSAVEP
QSAVEQ
2. [END OF ANY POLYNOMIAL]
REPEAT THROUGH STEP4 WHILE P ≠ NULL AND Q ≠ NULL
3. [GET VALUES FOR EACH THEM]
A1POWER X(P)
A2POWER X(Q)
B1POWER Y(P)
B2POWER Y(Q)
C1POWER Z(P)
C2POWER Z(Q)
D1 COEFF(P)
D2COEFF(Q)
4. [COMPARE TERMS]

IF (A1=A2)AND(B1=B2)AND(C1=C2) THEN
IF D1+D2 ≠0 THEN
RPOLYLAST(A1,B1,C1,D1+D2,R)
PLINK(P)
QLINK(Q)
ELSEIF[(A1>A2)OR(A1=A2)AND(B1>B2)] OR[(A1=A2)AND(B1=B2)AND(C1>C2)]
THEN RPOLYLAST(A2,B2,C1,D1,R)
PLINK(P)
ELSE
RPOLYLAST(A2,B2,C2,D2,R)
QLINK(Q)
5. [NOT AT END OF ONE OF THE POLYNOMIAL]
IF P ≠ NULL THEN LINK(LAST)COPY(P)
ELSE IF Q≠NULL THEN LINK(LAST)COPY(Q)
6. [RESTORE INITIAL POINTER VALUES FOR P & Q]
PPSAVE
QQSAVE
7. [RETURN POINTER TO SUM POLYNOMIAL]
RETURN(R)
DOUBLY LINKED LINEAR LIST
Linked linear list implies that each node must contain two link fields instead of the usual
one. The links are used to denote the predecessor and successor of a node. The link denoting the
predecessor of a node is called the left link containing this type of node is called doubly linked
linear list or a two way chain.
L R
Where L and R are pointer variables denoting the left most and right most nodes in the list
respectively. The left link of the leftmost and the right link of the rightmost node are both NULL,
indicating the end of the list for each direction. The left and right link of a node are denoted by
the variables LPTR and RPTR respectively.
PROCEDURE FOR DOUB INS(L,R,M,X)
Given a doubly linked linear list whose leftmost and rightmost node address are given by the
pointer variable L and R respectively. It is required to insert a node whose address is given by
pointer variable P. The right and left link of node are denoted by LPTR and RPTR. The

information field of a node denoted by the variable M. The information is to entered in the node
is contained in X.
CREATION OF LINK
1. PROCEDURE DOUBLE INSERT (L,R,M,X)
2. GETNODE (P)
3. INFO(P)X
4. IF R=NULL THEN LPTR(P)RPTR(P)NULL
5. RETURN
INSERTION OF FIRST NODE
1. GETNODE(P)
2. INFO (P)X
3. IF M=L THEN
LPTR (P)NULL
RPTR(P)M
LPTR(M)P
RETURN
INSERTION TO MIDDLE
1. GETNODE (P)
2. INFO(P)
3. LPTR(P)LPTR(M)
4. RPTR(P)M
5. LPTR(M)P
INSERTION TO LAST NODE
1. GETNODE(P)
2. INFO(P)X
3. RPTR(M)P
4. LPTR(P)M
5. RPTR(P)NULL
LPTR INFO RPTR LPTR INFO RPTR LPTR INFO

PROCEDURE DOUB DEL(L,R,OLD)
Given a doubly linked list, get the address of the leftmost and right most nodes given by the
pointer variable L &R respectively. It is required to delete the node whose address is containedin
the variable OLD. Nodes contain left and right links with names LPTR and RPTR respectively.
PROCEDURE DOUB DEL(L,R,OLD)
1. UNDERFLOW
IF R=NULL THEN WRITE(UNDERFLOW)
RETURN
2. DELETE NODE (ONE NODE,L,R)
IF L=R THEN
LRNULL
RETURN
3. DELETE NODE(FIRST NODE]
IF OLD=L THEN LRPTR(L)
LPTRNULL (OR)RETURN(OLD)
FREENODE(OLD)
4. LAST NODE
IF OLD=R THEN
RLPTR (R)
RPTR (R )NULL
FREENODE(OLD)

UNIT-IV
GRAPHS
DEFINITION OF GRAPH
A graph G consist of a non-empty set V called the set of nodes(points, vertices) of the graph, a
set E which is the set of edges of the graph and a mapping from the set of edges to a set of pair of
elements of v. A Graph defined as G=(V,E).
V2
E1 E2
V1 V3
E4 E3
V4
In this V1,V2,V3,V4 are set of nodes and E1,E2,E3,E4 are set of edges.
ADJACENT NODE
Any two nodes which are connected by an edge in a graph are called adjacent node.
V1 E V2
V1 and V2 are adjacent node which are connected by an edge E in a graph.
DIRECTED EDGE
Graph G is equal to (V,E) an edge which is directed from one node to another is called a directed
Edge.
V1 E V2

UNDIRECTED EDGE
An edge which has no specific direction is called an undirected edge.
V1 E V2
There is an no specific direction. So it is undirected graph.
TYPE OF GRAPHS
DIRECTED GRAPH
A Graph In Which Every Edge Is Directed is called a directed graph (or) Dia graph (or) Di
graph.
V2
E1 E2
V1 V3
E4 E3
V4
UNDIRECTED GRAPH
A Graph in which every edge is undirected is called an undirected graph.
V2
E1 E2
V1 V3
E4 E3
V4

MIXED GRAPH
Some of the edges are directed and some of the edges are undirected in a graph, then the graph
is a mixed graph.
V2
E1 E2
V1 V3
E4 E3
V4
INITIATING AND TERMINATING
Pair of node (U,V) then the edge x is said to be initiating is the node U and terminating Or
ending is the node V. The nodes U and V are also called the initial and terminal nodes of the
edge x.
U V
LOOP
An edge of a graph, which join a node itself is called a loop.
V2
E1 E2
V1 V3
E4 E3
V4

PARALLEL GRAPH
Certain pair of nodes joined by more than one edge, such edges are called parallel graph.
V2
E1 E2
V1 V3
E4 E3
V4
MULTI GRAPH
Any graph which contain some parallel edges is called a multi graph.
V2
E1 E2
V1 V3
E4 E3
V4
SIMPLE GRAPH
No more than one directed edge in the case of a directed graph such a graph is called a
simple graph.

WEIGHTED GRAPH
A Graph in which weights are assigned to every edge is called a weighted graph.
5 20
10
V3
30 40
ISOLATED GRAPH
A node which is not adjacent to any other node is called an isolated node
2
1 5
6 6
3 4
HERE 6 IS AN ISOLATED GRAPH
NULL GRAPH
A Graph containing only isolated nodes is called a null graph.
V2
V1 V3
V4

IN DEGREE, OUT DEGREE AND TOTAL DEGREE
In a directed graph for any node, the number of edges which have U as their initial node is
called the Out Degree of the nodes. The number of edges which have V as their terminal node is
called IN DEGREE of V and the sum of the out degree and in degree of a node is called TOTAL
DEGREE.
EXAMPLE 1:
1
3 2
IN DEGREE=1 , OUT DEGREE=2, TOTAL DEGREE=3
EXAMPLE 2:
Here e2 and e4 are the out degree, e1 is the in degree.
PATH
A Path is said to traverse through the nodes appearing in the sequence, originating in the
initial node of the first edge and ending in the terminal node of the last edge in the sequence.
V1 V2
V4 V3

V1V2V3V4
V1V2V4
V1V3V4
V1V4
LENGTH
The number of edges appearing in the sequence of a path is called the length of the path.
V1 V2
V4 V3
V1V2V3V4 =3
V1V2V4 =2
V1V3V4 =2
V1V4 =1
CYCLIC
A path which originate and ends in the same node is called a cyclic.
ACYCLIC
A Simple diagraph which does not have any cyclic is called Acyclic.
DIRECTED TREE
A Directed tree is an acyclic diagraph which has one node called its root. An isolated
node is called a directed tree. In a directed tree any node which has out degree 0 is called a
terminal node or leaf. An other node are called branch node.

STRONGLY CONNECTED GRAPH
A Diagraph is called strongly connected if there is a directed path from any vertex to any
other vertex.
WEEKLY CONNECTED GRAPH
There does not exist a directed path from vertex, one to vertex 4, also from vertex 5 to
other vertex. It is called a weekly connected graph.
A special type of graph is called tree. A graph is a tree if it has two properties.
1. It is connected
2. There are no cycles in a graph.
EXAMPLES
1
2
3
4 5
A
B
C
D
1
2
3
4 5
1
2 3
4 5 6 7

GRAPH REPRESENTATION
Two such representations are commonly used there are,
1. Adjacent matrix
2. Adjacent list representation
The choice of representation depends on the application and function to be performed
on the graph.
ADJACENT MATRIX
A Adjacent matrix A for a graph g=(V,E) with n vertices, is an n cross n matrix of bits,
Such that A(i,j)=1, iff there is no such edge.
The figure shows the adjacent matrix the graph shown in figure.
The total number of 1’s accounts for the number of edges in the diagraph. The
number of 1’s in each row tells the out degree of corresponding vertex.
The adjacent matrix is a simple way to represent a graph but it has two disadvantages.
1. It take1s O(n2
) space to represent a graph with n vertices even for sparse graph and
2. It takes (n2
) time to save most of the graph problems.
Vertex 1 2 3 4 5
1 0 1 1 0 0
2 0 0 1 0 0
3 0 0 0 1 0
4 0 0 0 0 0
5 1 0 1 1 0
1
5
4
3
2

ADJACENCY LIST REPRESENTAION
In this representation we score a graph as a linked structure, We score all the vertices
in a list and then for each vertex, we have a linked list of its adjacent vertices.
EXAMPLE
V6
V5
V4
V2
V3
V1
V1 V1

UNIT – V
TREES
DEFINITION OF TREE
Usually, draw trees with a root at the top. Each node has exactly one node about it which is
called its parent. A node directly below a node is called a children.
TREE RELATED TERMS
A Tree is a non-empty collection of vertices and edges that satisfy certain requirements.
A vertex is a simple object (node) that can have a name and carry other associated
information. An edge is a connection between two vertices. A tree may, therefore, be defined as
a finite set of one or more vertices such that,
1. There is a one specially designated vertex called root.
2. The remaining vertices are partitioned in to collection of subtree, with each of which is
also a tree.
FOREST TREE
A Set of trees is called a forest tree.
EXAMPLE
A
B C D
G
F
E
H I J
K L M

PROPERTIES OF A TREE
1. Any node can be root of the tree and each node in the tree has the property that there is
exactly one path connecting that node with every other node in the tree.
2. Each node, except the root has a unique parent and every edge connects a node to its
parent. A tree worth N nodes has N-1 edges.
BINARY TREES
A Binary tree is a tree which is either empty or consists of a root node and two disjoint
binary trees called left sub tree and right sub tree. In figure a binary tree is depicted with a left
sub tree L(T) and the right sub tree is denoted by R(T).In a binary tree no node can have more
than two children, or two node.
PROPERTIES OF BINARY TREE
1. A Binary tree with N internal node have maximum of N+1 external node. Root is
considered as an external node.
2. The external path, length of any binary tree with a internal node is 2n
>the internal path
length.
3. The height of full binary tree with N internal node is above log2 N.
FULL BINARY TREE
A Full binary tree or a complete binary tree is a binary tree in which all internal nodes have
degree and all leaves are at the same level.
EXAMPLE
LEVEL 1
LEVEL2
LEVEL3
1
2 3
4 5 6 7
1
2 3
4 5 6 7

TREE REPRESENTATION
1. VENN DIAGRAM
2. NESTING PARANTHESIS
3. TEBLE OF CONTENTS
4. LEVEL NUMBER FORMAT
EXAMPLE
VENN DIAGRAM
NESTING PARANTHESIS
(V0(V7(V8)(V9(V10)))(V1(V2(V5)(V6))(V3)(V4)))
V0
V1
V7
V3
V9
V8 V4
V10 V5
V2
V6

TABLE OF CONTENTS
VO --------------------------------------------
V1--------------------------------------------
V2-----------------------------------------
V5-------------------------------------
V6-----------------------------------
V3-------------------------------------------
V4--------------------------------------------
V7----------------------------------------------
V8------------------------------------------
V9-------------------------------------------
V10---------------------------------------
LEVEL NUMBER FORMAT
1. VO --------------------------------------------
2. V1--------------------------------------------
3. V2-----------------------------------------
4. V5-------------------------------------
4. V6-----------------------------------
3. V3-------------------------------------------
3. V4--------------------------------------------
2. V7----------------------------------------------
3. V8------------------------------------------
3. V9-------------------------------------------
4. V10---------------------------------------

CONVERSION OF BINARY TREE
A forest tree can also be represented by a binary tree. The two stages to obtain a binary
tree which represents a given ordered tree is as follows. As a first step, we delete all the left most
branch. Also we draw edges from a node to the node on the immediate right, if any, which is
situated at the same level.
Once this is done, then for any particular node we choose its left and right children in the
following manner. The left child is the node and the right child is the node to the immediate right
of the given node on the same horizontal line. Such a binary tree will not have a right subtree.
GENERAL TREE TO BINARY TREE
GIVEN DIRECTED TREE
STAGE 1
a
b f
c d g j k
e h i

STAGE 2
FOREST TO BINARY TREE
STAGE 1

STAGE 2
OPERATION S ON BINARY TREE [TREE TRAVERSAL]
One of the most common operations performed on tree structure is that of traversal. This is a
procedure by which each node in the tree is processed exactly once in a systematic manner.
There are three main ways of traversing a binary tree.
1. PREORDER
2. INORDER
3. POSTORDER
The preorder traversal of a binary tree is defined as follows,
1. Process the root node
2. Traverse the left subtree in preorder
3. Traverse the right subtree in preorder
EXAMPLE
A
B D
C E G
F
ANS: ABCDEFG

The inorder traversal of a binary tree is given by the following steps,
EXAMPLE
A
B D
C E G
F
ANS: CBAEFDG
Finally, we define the post order traversal of a binary tree as follows,
EXAMPLE
A
B D
C E G
F
ANS: CBFEGDA

NON-RECURSIVE MANNER
PROCEDURE FOR PREORDER
1. [INITIALIZE]
IF T=NULL THEN WRITE(EMPTY TREE)
RETURN
ELSE
TOP0
CALL PUSH(S,TOP,T)
2. [PROCESS EACH STACKED BRANCH ADDRESS]
REPEAT STEP 3 WHILE TOP >0
3. [GET STORED ADDRESS AND BRANCH LEFT]
PPOP(S,TOP)
REPEAT WHILE P≠ NULL
WRITE (DATA(P))
IF RPTR(P) ≠NULL
THEN CALL PUSH(S,TOP,RPTR(P))
PLPTR(P)
4. [FINISHED]
RETURN
PROCEDURE FOR POST ORDER
1. [INITIALIZE]
IF T=NULL
THEN WRITE(EMPTY TREE)
RETURN
ELSE PT
TOP0
2. [TRAVERSA IN POSTORDER]
REPEAT THROUGH STEP 5 WHILE TRUE
3. [DESCEND LEFT]
REPEAT WHILE P≠NULL
CALL PUSH(S,TOP,P)
PLPTR(P)
4. [PROCESS A NODE WHOSE LEFT AND RIGHT SUBTREE HAVE BEEN
TRAVERSED]
REPEAT WHILE S[TOP]<0
PPOP(S,TOP)
WRITE(DATA(P))
IF TOP=0

THEN
RETURN
5. [BRANCH RIGHT AND THEN MARK NODE FROM WHICH WE BRANCHED]
PRPTR(S[TOP])
S[TOP]S[TOP]
RECURSIVELY MANNER
PROCEDURE FOR INORDER
1. [CHECK FOR EMPTY TREE]
IF T=NULL THEN WRITE(EMPTY TREE)
RETURN
2. [PROCESS THE LEFT SUBTREE]
IF LPTR(T) ≠NULL THEN
CALL POSTORDER(LPTR(T))
3. [PROCESS THE RIGHT SUB TREE]
IF RPTR(T) ≠NULL THEN
CALL POSTORDER(RPTR(T))
4. [PROCESS THE ROOT NODE]
WRITE(DATA(T))
5. [FINISHED]
RETURN
GRAPH TRAVERSAL
1. BFS: QUEUEBREADTH FIRST SEARCH
2. DFS:STACKDEPTH FIRST SEARCH
PROCEDURE BFS(INDEX,LINK,QUEUE)
REACHspecifies whether the node has been reached in the traversal and its initial value is
false.
NODE NO Identifies the node number
DATAcontain a information of control
DISTANCEIt is the variable which contains a distance from the start node.
LIST PTRIt is a pointer to a list of adjacent edge for the node.
DEST INContains the number of terminal nodes for this edge.
EDGEPPTRPoints to the next edge in the list.

ALGORITHM
1. [INITIALIZE]
REACH[INDEX]TRUE
DIST[INDEX]0
CALL QINSERT(QUEUE,INDEX)
2. [REPEAT UNTILL ALL NODE HAVE BEEN EXAMINED]
REPEAT THROUGH STEP-5 WHILE QUEUE IS NOT EMPTY.
3. [REMOVE THE CURRENT NODE TO THE EXAMINE FROM THE QUEUE]
CALL QDELETE(QUEUE,INDEX)
4. [FIND ALL UNABLE NODES ADJACENT TO CURRENT NODE]
LINKLIST PTR(INDEX)
REPEAT STEP-5 WHILE LINK ≠ NULL
5. [IS THIS A UNVISITED NODE LABEL IT AND ADDING IT IN A QUEUE]
IF NOT REACHED[DEST IN[LINK]] THEN
REACH [DEST IN[LINK]]DIST[INDEX]
CALL QINSERT(QUEUE,DEST IN (LINK))
LINKEDGE PTR(LINK)
6. [FINISHED]
RETURN
NOTE
INDEXCurrent node being processed
LINKPoint to the edge being examined
QUEUEDenote the name of the QUEUE
200 Mumbai
1300
Chennai 900 780
Delhi
Hyderabad
1400 600
160 Luck now
800 Calcutta
Bangalore
1
2
5 4
3
7
6

REACH NODE DATA DIST LINK(P) DEST IN EDGE PTR
True 1 Chennai 0 2 6
True 2 Bombay 200 5 4
True 3 Delhi 600 2 7
True 4 Hyderabad 780 7 5
True 5 Luck now 900 4 1
True 6 Bangalore 160 7 1
True 7 Calcutta 800 6 3
DEPTH FIRST SEARCH
PROCEDURE DFS(INDEX,COUNT)
Given the structure as described before, this recursive procedure calculates the depth first
search numbers for a graph. INDEX is the current index into the node table directory table and is
assumed to be initialized to one outside the procedure COUNT is used to keep track of the
current DFN node number and is initially set to zero outside the procedure. Finally, it is assumed
the DFN field as initialized to zero when the adjacency list structure was created.
ALGORITHM
1. [UPDATE THE DEPTH FIRST SEARCH NUMBER SET AND MARK CURRENT
NODE]
COUNTCOUNT+1
DFS[INDEX[COUNT
REACH[INDEX]TRUE
2. [SETUP LOOP TO EXAMINE EACH NEIGHBOUR OF CURRENT NODE]
LINKLIST PTR[INDEX]
REPEAT STEP 3 WHILE LINK ≠ NULL
3. [IF NODE HAS NOT BEEN MARKED,LABEL IT AND MAKE RECURSION CALL]
IF NOT REACH[DESIGN (LINK)]
THEN CALL DFS(DESTIN(LINK), COUNT)
LINK EDGE PTR(LINK)
4. [RETURN TO POINT OF CALL]
RETURN

1
3 2 6
5
4
HASHING
A function is a small task, that execute particular statement block. The function has two uses.
1. The result of the function is used as a kind of key in which the list is stored (i.e) the
function is used to determine where is the array to store the record.
2. Function is used as a method of ascending the record. This function is called HASH
function.
Hash Function = key mod 100
The remainder is used as an index in to array of employee records.
Example:
(00)
(01)
Key (02)
459250704 key mod 10024 (99)
COLLISIONS
The same address will occur that is called collisions.
Resolving collision techniques
1. Hash and search
2. Rehashing
3. Chaining
C
A
B D E
F
Hash function

1. HASH AND SEARCH
A Simple approach to resolving collision is to store the colliding element to next
available space.
Key 556677003 00
01
02
03
04
Try index 3
REHASHING
Another common technique for resolving collision is rehashing. If the first computation of
the hash function produce a collision, we can use hash address as the input to the rehash function
and computes new address.
Original hash function = key MOD 100
Rehash function = (key+1) MOD 100
Until find the empty address.
CHAINING OR BUCKET OR LIST METHOD
A third technique for handling collisions was the hash address not as the actual location of
the record, but as the index in to an array of pointers called buckets. Each bucket points to a
linked list or chain of records that share the same hash address.
00
01
02
09
H.P
Empty
02
12345003
04

HASHING RECHNIQUES
1. MID SQUARE METHOD
The key is multiplied by itself , and the central digits of the square are taken and
adjusted to fit the range of addresses.
Thus , if the record have 6-digits key and 1000 buckets are used , the key may be
squared to form a 12-digits field of which digits 5 to 8 are used. Thus if the key is
172148, the square is 02963493304. The central four digits are multiplied by 0.7 : 3493 *
0.7 = 2445. The 2445 is used as the bucket address.This is close to roulette-wheel
randomization and the result are usually found to the close to the theoretical result of
figure.
2 24
2 12 - 0
2 6 - 0
2 3 - 0
2 1 - 1
ADDRESS = 00011
2. DIVIDING
It is possible to find a method which gives better results than a random number
generator. A simple division method is such, the key is divided by a number
approximately equal to the number of available addresses, and the remainder is taken as
the relative bucket address, as in fig. A prime number or number with small factors is
used.
Thus if the key is 172148 again, and there are 7000 buckets, 172148 might be divided
by 6997. The remainder is 4220, and this is taken as the relative bucket address. One
reason division tends to give fewer overflows than a randomizing algorithm is that many
key set have tons of consecutive numbers, there by distributing.
3. SHIFTING
The outer digits of the key at both ends are record to overlap by an amount equal to
the address length, as shown in figure. The digits are then added, and the result is
adjusted to fit the range of bucket address.

EXAMPLE
1 7 2 0
7 3 5 9
4. FOLDING
Digits in the key are folded inward like folding papers as shown in figure . The
digits are then added and adjusted as before. Folding tends to be more appropriate for
large keys.
EXAMPLE
1 7 2 0
7 3 5 9
5. DIGIT ANALYSIS
Some attempts at achieving an even spread of bucket addresses have analyzed the
distribution of values of each digit or character in the key. Those positions having the
most skewed distribution are deleted from the key in the hope that any transform applied
to the other digits will have a better chance of giving a uniform spread.
6. RADIX CONVERSION
The radix of a number may be converted for example to radix 11. The excess high
order digits may then be truncated. The key 172148 is converted to 1 *
115
+7*114
+2*113
+1*112
+4*111
+3*110
= 266373 and the digits 6763 are multiplied by
0.7 to give the relative bucket address 4461. Radix 11 conversion can be performed more
quickly in a computer y a series of shifts and additions.

7. LIN’S METHOD[Q]
In this method a key is expressed in radix p, and the result is taken modulo qm
,
where p and q are prime number (or numbers without small prime factors) and m is
positive integer.
The key 172148 would first be written as a binary string:0001 0111 0010 0001
0100 1000. Grouping the string in to group of three bits, we obtain 000 101 110 010 000
101 001 000 = 05610510. This is expressed as a decimal number and divided by a
constant qm
. The remainder is used to obtain the relative bucket address.
8. POLYNOMIAL DIVISION
Each digit of the key is regarded as a polynomial co-efficient, Thus the key
172148 is regarded as x5
+7x+4
+2x3
+x2
+4x1
+8x+0
. The polynomial so obtained is divided
by another unchanging polynomial. The co efficient in the remainder forms the basis of
the relative bucket address(x=2 or x=3).
SHORTEST PATH PROBLEM
DIJIKSTRA ALGORITHM
We have seen in the graph traversals that we can travel through edges of the graph. It is
very much likely in applications that these edges have some weights attached to it. They weight
may reflect distance time or some other quantity that corresponds to the cost we incur when we
travel through that edge. For example, in the graph in figure, we can go from Delhi to Andaman
Nicobar through Madras of a cost of 7 or through Calcutta at a cost of 5. In these and many other
applications we are often required to find a shortest path, i.e a path having the minimum weights
between of finding shortest path for directed graph in which every edge has a non negative
weight attached.
1
2
4
3
Let us at this stage recall how do we define a path . A path in a graph is sequences of
vertices such that there is an edge that we can follow between each consecutive pair of vertices
length of the path is the sum of weights of the edges of the path . The starting vertex of the path
DELHI
CALCUTTA
MADRAS
ANDAMAN

is called the source vertex and the last vertex of the path is called the destination vertex. Shortest
path from vertex V to vertex W is a path for which the sum of the weights of the arcs on edges
on the path is minimum. Here we must note that the path that may lock longer if we see the
number of edges and vertices visited, at times may be actually shorter cost wise. Also we have
two kinds of problem in finding shortest path. One could be that we have single source vertex
and we seek a shortest path from that source vertex V to every other vertex of the graph. It is
called single source shortest path problem.
Consider the weighted graph in figure with 8 nodes.
WEIGHTED GRAPH
2 1
2 2 4 3
4 6
1 3 3 7
5
There are many paths from A to H
Length of path AFDEH = 1+3+4+6= 14
Another path from A to H is ABCEH
It’s length is 2+2+3+6= 13
********************************ALL THE BEST*******************************
A
B
D
F
C
E
G
H
A
B
D
F
C
E
G
H

DATA STRUCTURE BY SIVASANKARI

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