1.2Array String Recursion

8/10/2019 1.2Array String Recursion

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Array

Array

Group of consecutive memory locations

Same name and type

To refer to an element, specify Array name

Position number

Format:

arrayname[position number]

First element at position 0

nelement array named c:

c[ 0 ], c[ 1 ]...c[ n 1 ]


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Array

Arrays is a list of a finite number of homogeneous

data element.

The element of array are referenced by index set.

The element of the array are stored in successive

memory location.

Length of the array obtained by formula

Length=UB-LB+1


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Representation of Array in memory

A be the Linear array then

LOC(A[I])=address of the element A[I] of the array A.

Base(A)=base address of A or the address of first

element of array.

For the address of any element of A-

A[K]=B+W*K

B= Base address of the array.W=no of words per memory cell for the array A.

K= Index of element.


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Declaring Array

When declaring arrays, specify Name

Type of array

Number of elements

arrayType arrayName[ numberOfElements ];

Examples:

int c[ 10 ];

float myArray[ 3284 ];

Char name[20]; /String

Declaring multiple arrays of same type

Format similar to regular variables

Example: int b[ 100 ], x[ 27 ];


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Examples Using Array

Initializersint n[ 5 ] = { 1, 2, 3, 4, 5 };

If not enough initializers, rightmost elements

become 0int n[ 5 ] = { 0 }

All elements 0

If too many a syntax error is produced syntax error

C arrays have no bounds checking

If size omitted, initializers determine itint n[ ] = { 1, 2, 3, 4, 5 };

5 initializers, therefore 5 element array


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Types of Array

Single Dimensional array

syntax:- [size];

Exaple:- intiarr[3] = {2, 3, 4};

charcarr[20] = "c4learn" ;floatfarr[3] = {12.5,13.5,14.5} ;

Multi Dimensional array

syntax:- [r_sub][c_sub];

Example : - inta[3][3] = { 1,2,3 5,6,7 8,9,0 };


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Passing Array to Function

Passing arrays

To pass an array argument to a function, specify the name of the array

without any brackets

int myArray[24];

myFunction(myArray,24); Array size usually passed to function

Arrays passed call-by-reference

Name of array is address of first element

Function knows where the array is stored

Modifies original memory locations

Passing array elements

Passed by call-by-value

Pass subscripted name (i.e.,myArray[3]) to function


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Passing Array to Function

Function prototypevoid modifyArray( int b[], int

arraySize );

Parameter names optional in prototype int b[]could be written int []

int arraySizecould be simply int


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Operations on Array

Traversal-Processing each element in the list.

Search-finding the location of element with a

given value or key.

Insertion-Adding a new element to the list.

Deletion-Removing an element from list

Sorting-Arranging the element in some order.

Merging-Combining two list to create new list.


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Traversing the Array

TRAVERSE (A, UB,LB)

Ais a linear array,UB is upper bound and LB is lower bound.

1. set i:= LB [Initialize counter]

2. Repeat for i= LB to UB [visit element Display A[i]

[End of loop]

3. EXIT.


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Insertion into Array

INSERT (A, len, pos, num)Ais a linear array,len is total no. of elements within array, posis the position at

which numwill be inserted.

1.check if (pos > len), if it is greater, then display message

Position is greater than array length.

Else:[Initialize counter] Set i: = len.

2. Repeat Steps 3 for i=len down to pos (Decrement loop from len to pos)

3. [Move jth element downward.] Set A [i + 1]: =A [i].

[End of step 2 loop]

4. [Insert element at required position]

Set A [pos]:=num.

6. [Reset LEN] Set LEN:=LEN+1

7. Display the new list of array.

8. EXIT.


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Deletion from Array

DELETE (A ,len, pos)

Ais a linear array,len is total no. of elements within array, posis the position

from where the element should be deleted.

1.check if (pos > len), if it is greater, then display message

Position is greater than array length.

Else:

Set ITEM: = A [pos].

2. Repeat for J = pos to LEN1. (shift element one position upward)

Set A [J]: = A [J +1]. [End of loop]

3. [Reset the number N of elements in LA] Set LEN: = LEN - 1

4. EXIT


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Matrix Addition

IfAand Bare matrices of the same size, then they can be

added. (This is similar to the restriction on adding vectors,

namely, only vectors from the same space Rncan be added;

you cannot add a 2vector to a 3vector, for example.) IfA=[aij] and B= [bij] are both mx nmatrices, then their sum, C=

A+ B, is also an mx nmatrix, and its entries are given by the

formula

Cij= Aij+ Bij


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Matrix Addition

int main()

{

int a[3][3],b[3][3],c[3][3],i,j;

printf("Enter the First matrix->");

for(i=0;i


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Matrix Additionfor(i=0;i


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Matrix Multiplication

In order to form the productAB, the number of columns of

A must match the number of rows of B; if this condition does

not hold, then the productABis not defined. This criterion

follows from the restriction stated above for multiplying a row

matrix rby a column matrix c, namely that the number of

entries in rmust match the number of entries in c. IfAis mx n

and Bis nxp, then the productABis defined, and the size of

the product matrixABwill be mxp.


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Matrix Multiplication

for(i=0;i


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Polynomial

Representation of a Polynomial: A polynomial is an

expression that contains more than two terms. A term is

made up of coefficient and exponent. An example of

polynomial is

P(x) = 4x3+6x2+7x+9

A polynomial thus may be represented using arrays or linked

lists. Array representation assumes that the exponents of

the given expression are arranged from 0 to the highest

value (degree), which is represented by the subscript of thearray beginning with 0. The coefficients of the respective

exponent are placed at an appropriate index in the array.


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How to implement this?

There are different ways of

implementing the polynomial ADT:

Array (not recommended)

Double Array (inefficient)

Linked List (preferred and

recommended)


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Array Implementation:

p1(x) = 8x3

+ 3x2

+ 2x + 6p2(x) = 23x4+ 18x - 3

6 2 3 8

0 2

Index

represents

exponents

-3 18 0 0 23

0 42

p1(x) p2(x)


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This is why arrays arent good to

represent polynomials:

p3(x) = 16x21- 3x5+ 2x + 6

6 2 0 0 -3 0 0 16

WASTE OF SPACE!


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Linked List

Linked list Implementation:

p1(x) = 23x9+ 18x7+ 41x6+ 163x4+ 3

p2(x) = 4x6+ 10x4+ 12x + 8

23 9 18 7 41 6 18 7 3 0

4 6 10 4 12 1 8 0

P1

P2

NODE (contains coefficient & exponent)

TAIL (contains pointer)


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Addition of two Polynomials:

For adding two polynomials using arrays is straightforward

method, since both the arrays may be added up element

wise beginning from 0 to n-1, resulting in addition of two

polynomials. Addition of two polynomials using linked list

requires comparing the exponents, and wherever the

exponents are found to be same, the coefficients are added

up. For terms with different exponents, the complete term

is simply added to the result thereby making it a part of

addition result. The complete program to add twopolynomials is given in subsequent section.


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ALGORITHM:

Step 1: Start

Step 2: Declare structure polynomial

struct polynomial

{

int expo

int coef

}

Step 3: Read number of terms of the first polynomial, no of terms

Step 4: i=0, avail=0

Step 5: if i


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Step 13: startD=avail

Step 14: while(startA


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for(;startB=MAXTERMS) go to step2.else go tostep3

Step 2: Print Too many terms

Step 3: Assign coefficient to terms[avail].coef and exponent to

terms[avail].expo


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Multiplication of two Polynomials:

Multiplication of two polynomials however requires

manipulation of each node such that the exponents are

added up and the coefficients are multiplied. After each

term of first polynomial is operated upon with each term of

the second polynomial, then the result has to be added up

by comparing the exponents and adding the coefficients for

similar exponents and including terms as such with dissimilar

exponents in the result.


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Matrices

Vector and Matrices refer to the collection ofnumbers

Vector list of n number can be given in this form-

V=(V1,V2,V3..VN)

Matrix is the Array of m.n numbers arranged inrows and columns.

Scalar

Diagonal

SquareSparseUpper triangular

Lower triangular

Tridiagonal


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Sparse Matrices

Matrices with high no of zero valuescalled Sparse Matrices.

Nonzero entries above the main diagonalis called upper triangular matrices

Nonzero entries below the main diagonal

is called lower triangular matrices


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Recursion

When a function call itself, it is calledrecursion. Any recursive procedure must

have two properties:

1) There must be certain criteria, called

base criteria, for which the procedure

does not call itself.

2) Each time the procedure call itself, itmust be closer to the base criteria.


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Types of recursion

Recursion is of two types:1) Direct recursion :- In this there is only

one function and it call itself.

2) Indirect recursion :- In this there

involved more than one function. Eg.

A() B()

{ {

B(); A();} }


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Tail recursionTail recursion is important because it can be implemented more

efficiently than general recursion. When we make a normal

recursive call, we have to push the return address onto the call

stack then jump to the called function. It means that we need a

call stack whose size is linear in the depth of the recursive calls.When we have tail recursion we know that as soon as we return

from the recursive call we're going to immediately return as well,

so we can skip the entire chain of recursive functions returning

and return straight to the original caller. That means we don't

need a call stack at all for all of the recursive calls, and can

implement the final call as a simple jump, which saves us

space.


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recursion

int fact(int x)

{

if (x == 0)

Return 1;

else

return (fact* fact(x-1));

}


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Tail recursionfactorial(n)

{

return factorial1(n, 1);

}

factorial1(n, accumulator)

{

if (n == 0)

return accumulator;

elsereturn factorial1(n - 1, n * accumulator);

}


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Efficiency of recursion

The fact is that recursion is rarely the most efficient approach to solvinga problem, and iteration is almost always more efficient. This is because

there is usually more overhead associated with making recursive calls

due to the fact that the call stack is so heavily used during recursion.

This means that many computer programming languages will spend

more time maintaining the call stack then they will actually performingthe necessary calculations. Recursion is generally used because of the

fact that it is simpler to implement, and it is usually more elegantthan

iterative solutions. Remember that anything thats done in recursion

can also be done iteratively, but with recursion there is generally a

performance drawback. But, depending on the problem that you are

trying to solve, that performance drawback can be very insignificantin

which case it makes sense to use recursion. With recursion, you also get

the added benefit that other programmers can more easily understand

d hi h i l d thi t h

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1.2Array String Recursion