Chap 2 Recursion

8/2/2019 Chap 2 Recursion

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IntroductionAfunction call itselfis called a recursive function.

Recursion is a repetitive process in which an

algorithm or function calls itself repeatedly untilsome specified condition has been satisfied.

The recursive function must include a stoppingcondition (anchor condition) or else the function

would never terminate (indefinite loop).Example of the use of recursive algorithms:Factorial, Fibonacci sequence (number) and Towerof Hanoi.


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Iteration vs. RecursionThere are two approaches to writing repetitivealgorithm: iteration & recursion.

Recursive algorithm take more storage and time thanthe iterative version.

This is because a recursive function must store andretrieve the value of each recursive call.

Although many function can be computed easily

without using recursion, there are others that areeasier to compute with recursion than without it.Example: Tower of Hanoi game problem.


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When to used Recursion?If the algorithm or data structure naturally suited torecursion. Example Tower of Hanoi and Quick Sort.

If the recursive solution shorter and moreunderstandable. Example Merge Sort.

If the recursive solution run within acceptable timeand space limit.


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Problem 1: FactorialThe product of the integral positive integers from 1 ton, is called nfactorial.

Denoted by n!Factorial (n) = 1 for n=0

Factorial (n) = n* (n 1) * (n 2) *. * 1 for n>0

= n* factorial (n-1) for n>0


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Example of factorial3! = 3 * 2 * 1 = 6

4! = 4 * 3 * 2 * 1 = 245! = 5 * 4 * 3 * 2 * 1 = 120

6! = 6 * 5 * 4 * 3 * 2 * 1 = 720


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Iterative Algorithm to calculate n

factorial1. Set fact= 1

2. Set i=number

3. Loop (i>1)3.1. Begin

3.2. fact= fact* i

3.3. i - -

3.4. End

4. Return fact


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Programming/*Factorial (n) computation using

iteration*/

#include #include

int fact (int )void main(){ int fact (int);

int number, factnum;

printf(\nEnter a number: ");scanf( %d,&number);factnum=fact(number);

printf("\nThe factorial for );printf(%d! = %d",number,factnum);

}

/*Iterative factorial computation*/

int fact (int number)

{

int i, fact=1;for(i=number;i>1;i--)

{

fact=fact*i;

}

return fact;

}

Output:

Enter a number: 4

The factorial of 4! = 24


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Recursive Algorithm to calculate

n factorial1. If (n== 0)

1.1.Return11. Else

2.1. Return (num * fact(num-1))


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//Recursive factorial

int fact (int num)

{

if(number==0)

{

return 1;

}

else

{return (num*fact(num-1));

}

}

Programming//Factorial (n) computation using

recursion

#include

#include

void main(){ int fact (int);

int num,factnum;

printf(\nEnter a number: ");

scanf( %d, &num);factnum=fact(num);

printf("\nThe factorial of );

printf(" %d!=%d", num, factnum);

} Output:Enter a number: 4

The factorial of 4! = 24


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Traces the executionn = 4Since n!


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Problem 2: FibonacciThe Fibonacci sequence number isdefined as

0, 1, 1, 2, 3, 5, 8, 13, 21, . . .

Each number after the first two is thesum of the two preceding number.

F0 = 0

F1 = 1

Fn = Fn-1 + Fn-2 for n>=2


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QuestionWhat is the next 2 terms ofthe following Fibonacci

sequence number?

0, 1, 1, 2, 3, 5, 8, 13, 21, ?, ?


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Example of fibonacciF(0) = 0

F(1) = 1F(2) = 0 + 1 = 1

F(3) = 1 + 1 = 2

F(4) = 1 + 2 = 3F(5) = 2 + 3 = 5

F(6) = 3 + 5 = 8


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Fibonacci algorithm1. If (num= 0 or num= 1)

Stopping Condition (for which the function does not

call/refer to itself)1.1 Return num

2. Return (fib(num-1) + fib(num-2)


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int fib(int n){

if (num==0||num==1)return num;

return (fib(num-1)+fib(num-2));}

Programming//Fibbonaci sequence using recursion#include #include

void main(){int fib(int);int num;printf("\nEnter num for fibonaccisequence: ");

scanf("%d",&num);

printf("\nFibonacci %d =%d",n,fib(num));

getch();}

Output:

Enter n for fibonacci sequence: 4

Fibonacci 4 = 3


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Problem 3: Towers of HanoiSuppose 3 needles labeled A, B, C and a setof disks (varying sizes)

Arranged the biggest disk at the bottom tothe smallest disk at the top

Move the disk, one by one from needles A toC, using needles B as an intermediate.

Each needles must always be arranged formthe biggest at the bottom to the smallest atthe top

http://www.cut-the-knot.org/recurrence/hanoi.shtml

http://www.mazeworks.com/hanoi/


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Illustration: Initial state for

Towers of Hanoi

A B C


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Solution to the Towers of Hanoi

for disks = 3

A B C

Initial

A B C

Step 1: A -> C


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Continue

A B C

Step 2: A -> B

A B C

Step 3: C -> B


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Continue

A B C

Step 4: A -> C

A B C

Step 5: B -> A


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Continue

A B C

Step 6: B -> C

A B C

Step 7: A -> C


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Algorithm for disks movements

Towers of Hanoi1. If (numDisks = 1)

1.1 Print (Move disk 1 from, from, to destination)

1. Else2.1 Begin

2.2 Move(numDisks-1, from, intermediate, destination)

2.3 Print (Move, numDisks, from, from to, destination)

2.4 Move(numDisks-1,intermediate, destination , from)

2.5 End


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Programming//Tower of Hanoi#include

#include

void main()

{

void move(int, char,char,char);

int numDisk;

printf("\nEnter number of disks: ");

scanf(" %d", &numDisk);

if(numDisk>0)

move(numDisk , 'A','C','B');

getch();

}

void move(int numDisk, char from, char destination , char

intermediate){

if (numDisk==1)

printf("\nMove disk 1 from needle %c to needle %c",from,to);

else

{

move(numDisk-1, from, intermediate, destination);

printf("\nMove disk %d from needle %c to needle%c",numDisk,from,to);

move(numDisk-1, intermediate, destination , from);

}

}

Enter number of disks: 3Move disk 1 from needle A to needle C

Move disk 2 from needle A to needle B

Move disk 1 from needle C to needle B

Move disk 3 from needle A to needle C

Move disk 1 from needle B to needle A

Move disk 2 from needle B to needle C

Move disk 1 from needle A to needle C


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Illustration of solution to Towers

of Hanoi problem for n = 3

Move(3,A,C,B)

Move(2,A,B,C)

Move(2,B,C,A)

A ->C

Move(1,A,C,B)

Move(1,C,B,A)

A ->B

A ->C

C ->B

Move(1,B,A,C)

Move(1,A,C,B)

B ->C

B ->A

A ->C


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ExerciseLab sheet 1: Recursion

Chap 2 Recursion

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