Fundamentals of Programmingce.sharif.edu/courses/92-93/1/ce153-1/resources... · 1 Sharif University of Technology Fundamentals of Programming Session 13 These slides have been created

Fall 2013

Instructor: Reza Entezari-Maleki

Email: [email protected]

Sharif University of Technology 1

Fundamentals of Programming Session 13

These slides have been created using Deitel’s slides

Outlines

Recursion

Example Using Recursion: Fibonacci Series

Recursion vs. Iteration

Tower of Hanoi

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A recursive function is a function that calls itself either directly or

indirectly through another function.

A recursive function is called to solve a problem.

The function actually knows how to solve only the simplest case(s), or

so-called base case(s).

If the function is called with a base case, the function simply returns a

result.

If the function is called with a more complex problem, the function

divides the problem into two conceptual pieces: a piece that the function

knows how to do and a piece that it does not know how to do.

Because this new problem looks like the original problem, the function

launches (calls) a fresh copy of itself to go to work on the smaller

problem—this is referred to as a recursive call and is also called the

recursion step.3

Recursion

The factorial of a nonnegative integer n, written n! (and

pronounced “n factorial”), is the product n · (n –1) · (n – 2) · … · 1

with 1! equal to 1, and 0! defined to be 1.

The factorial of an integer, number, greater than or equal

to 0 can be calculated iteratively (nonrecursively) using a

for statement as follows:factorial = 1;

for ( counter = number; counter >= 1; counter-- )factorial *= counter;

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Recursion …

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Recursion …

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Recursion …

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Recursion …

The conversion specifier %ld is used to print long values.

Unfortunately, the factorial function produces large

values so quickly that even long int does not help us print

many factorial values before the size of a long intvariable is exceeded.

As we’ll explore in the exercises, double may ultimately

be needed by the user desiring to calculate factorials of

larger numbers.

This points to a weakness in C (and most other procedural

programming languages), namely that the language is not

easily extended to handle the unique requirements of various

applications.8

Recursion …

The Fibonacci series 0, 1, 1, 2, 3, 5, 8, 13, 21, …

begins with 0 and 1 and has the property that each

subsequent Fibonacci number is the sum of the previous

two Fibonacci numbers.

The Fibonacci series may be defined recursively as

follows:

fibonacci(0) = 0fibonacci(1) = 1fibonacci(n) = fibonacci(n – 1) + fibonacci(n – 2)

Figure 5.15 calculates the nth Fibonacci number

recursively using function fibonacci.9

Example Using Recursion: Fibonacci Series

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Example Using Recursion: Fibonacci Series …

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Example Using Recursion: Fibonacci Series …

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Example Using Recursion: Fibonacci Series …

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Example Using Recursion: Fibonacci Series …

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Example Using Recursion: Fibonacci Series …

Each level of recursion in the fibonacci function has a

doubling effect on the number of calls; i.e., the number of

recursive calls that will be executed to calculate the nth

Fibonacci number is on the order of 2n.

This rapidly gets out of hand.

Computer scientists refer to this as exponential complexity.

Problems of this nature humble even the world’s most

powerful computers!

Complexity issues in general, and exponential complexity in

particular, are discussed in detail in the upper-level computer

science curriculum course generally called “Algorithms”.15

Example Using Recursion: Fibonacci Series …

A string of binary digits is good if there is no two

consecutive 1s in that string. Write a program to

input the integer number n, and then count the

number of distinct good strings with length of n.

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Question

Answer (Iteration)

int main(){

int n;

scanf("%d",&n);

int a=3, b=2, c=0;

if(n==1)

c=2;

if(n==2)

c=3;

for(int i=3; i<=n; i++){

c=a+b;

b=a;

a=c;

}

printf("%d",c);

return 0;

}17

Question …

Answer (Recursion)

int f1(int);

int main(){

int n;

scanf("%d",&n);

printf("%d",f1(n));

return 0;

}

int f1(int n){

if(n==1)

return (2);

else if(n==2)

return (3);

else

return ( f1(n-1) + f1(n-2) );

}

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Question …

In the previous sections, we studied two functions that can easily be

implemented either recursively or iteratively.

Both iteration and recursion are based on a control structure: Iteration

uses a repetition structure; recursion uses a selection structure.

Both iteration and recursion involve repetition: Iteration explicitly

uses a repetition structure; recursion achieves repetition through

repeated function calls.

Iteration and recursion each involve a termination test: Iteration

terminates when the loop-continuation condition fails; recursion

terminates when a base case is recognized.

Both iteration and recursion can occur infinitely: An infinite loop

occurs with iteration if the loop-continuation test never becomes

false; infinite recursion occurs if the recursion step does not reduce

the problem each time in a manner that converges on the base case.19

Recursion vs. Iteration

Recursion has many negatives.

It repeatedly invokes the mechanism, and consequently the

overhead of function calls.

This can be expensive in both processor time and memory space.

Each recursive call causes another copy of the function (actually

only the function’s variables) to be created; this can consume

considerable memory.

Iteration normally occurs within a function, so the overhead of

repeated function calls and extra memory assignment is omitted.

So why choose recursion?

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Recursion vs. Iteration …

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Recursion vs. Iteration …

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Recursion vs. Iteration …

The Tower of Hanoi or Towers of Hanoi , also called the Tower of

Brahma or Towers of Brahma, is a mathematical game or puzzle.

It consists of three rods, and a number of disks of different sizes

which can slide onto any rod. The puzzle starts with the disks in a

neat stack in ascending order of size on one rod, the smallest at the

top, thus making a conical shape.

Only one disk may be moved at a time.

Each move consists of taking the upper disk from one of the rods and

sliding it onto another rod, on top of the other disks that may already

be present on that rod.

No disk may be placed on top of a smaller disk.

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Tower of Hanoi

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Tower of Hanoi …

At the top level, we will want to move the entire tower, so we

want to move n disks from rod A (source) to rod C (destination).

We can break this into three basic steps.

1. Move n-1 disks from rod A (source) to rod B (spare), using rod

C (destination) as a spare. How do we do this? By recursively

using the same procedure. After finishing this, we'll have all the

disks smaller than disk n on rod B.

2. Now, with all the smaller disks on the spare rod, we can move

disk n from rod A (source) to rod C (destination).

3. Finally, we want move n-1 disks from rod B (spare) to rod C

(destination). We do this recursively using the same procedure

again.25

Tower of Hanoi …

Pseudocode:

FUNCTION MoveTower(disk, source, dest, spare)

IF disk == 1, THEN:

move disk from source to dest

ELSE:

MoveTower(disk - 1, source, spare, dest) // Step 1

move disk from source to dest // Step 2

MoveTower(disk - 1, spare, dest, source) // Step 3

END IF26

Tower of Hanoi …

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Tower of Hanoi …

void hanoi(int x, char from, char to, char aux)

{

if(x==1)

printf ("Move Disk From %c to %c\n", from, to);

else

{

hanoi (x-1, from, aux, to);

printf ("Move Disk From %c to %c\n", from, to);

hanoi (x-1, aux, to, from);

}

}

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Tower of Hanoi …

int main(void)

{

int disk;

int moves;

printf ("Enter the number of disks you want to play with:");

scanf ("%d", &disk);

moves = pow(2, disk) - 1;

printf ("\nThe No. of moves required is=%d \n”, moves);

hanoi (disk, 'A', 'C', 'B');

}