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ESc101: Fundamentals of Com utin

2011-12-Monsoon Semester

Lecture #17, September 5, 2011

Please switch off your mobile phones.

Announcements

uiz on lab da s from 5th to th Se tember in Lab at 2:00 PM.

Please ensure that you have moodle access and Computer Center

access. If you have forgotten either password, please make sure

that you get the problem sorted out before the quiz. No makeup for the quiz.

th

Lec-17

- , .

1Dheeraj Sanghi, CSE Dept., IIT Kanpur

ESc101, 2011-12-Monsoon

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Recap

Recursion

Tower of Hanoi

Power function

Similarity with Mathematical Induction

Local variables are local to each invocation of function

Lec-17 2Dheeraj Sanghi, CSE Dept., IIT Kanpur


Recap: Recursion: Principle of Induction

If a statement is true for N = 1

And if the assumption that the statement is true for N = x

implies that the statement is true for N = x+1 Then the statement is true for all natural numbers.

This is exactly what we have used in recusrion.

own a we can so ve e pro em o s ze = ase case

Shown that if we can solve the problem of size N-1 then the

problem of size N can be solved (recursive case)

Hence problem can be solved for all values of N

Lec-17 Dheeraj Sanghi, CSE Dept., IIT Kanpur


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Recap: Recursion: Power (Example)

following.

power (x, n) = ?

1, if n == 0

x * square (power(x, n/2)), if n is odd

square (power(x, n/2)), if n is even



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Recap: Power function

#include

int power (int x, int n)

{ int t;

if (n == 0) return 1;

t = power (x, n/2);

if (n%2 != 0) return x * t * t;

return t * t;

}

int main ()

nt n, x;

scanf (%d %d, &n, &x);

printf (power is %d\n, power(x, n));

}



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Mutual Recursion: An example

.

Definitions:

Number 0 is even

Number n is even ifn-1 is odd

Number n is odd ifn-1 is even

Input: n a natural number

Output: true ifn is even, false otherwise.



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-

odd (n)

if n is zero, return FALSE

else return even (n-1)

even (n)

if n is zero, return TRUE

else return odd (n-1)



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#include

int odd (int n)

{ if (n == 0) return 0;

e se re urn even n- ;

}

int even (int n)

{ if (n == 0) return 1;

else return odd (n-1);

}

int main ()

nt number;scanf(%d, &number);

if(odd (number) == 1) printf(%d is odd\n, number);

else printf(%d is even\n, number);

}



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Double Recursion: An example

,

C (n, k) = C (n-1, k-1) + C (n-1, k)

C (n, 0) = 1

C (0, k) = 0

n and k are natural numbers.

To make computation more efficient, we can see that

n, n =



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Double Recursion: An example

#include

int choose (int n, int r)

{ if n == 0) return 0;

if (k == 0) return 1;

if (n == k) return 1;

return choose (n-1, k-1) + choose (n-1, k);

}

{ int n, r;

scanf(%d %d, &n, &r);

printf(The answer is %d\n, choose (n, r));

}



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Recursive calls to choose

The following calls are made for n = 4, r =2:

choose (4, 2)

,

choose (2, 0)

choose (2, 1)

choose (1, 0) choose (1, 1)

choose (3, 2)

choose 2, 1

choose (1, 0)

choose (1, 1)

choose (2, 2)



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Iterative Algorithm for Binomial Coefficients

Initialize the matrix elements (i, 0) to be 1 for all rows (i).

Initialize all diagonal elements (i, i) to be 1.

Now compute row wise. For row i,

If j > i, the value of matrix element is 0.

If j < i, the value of matrix element is computed as

C[i, j] = C [i-1, j-1] + C [i-1, j]



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Any Questions?

Lec-17 15Dheeraj Sanghi, CSE Dept., IIT Kanpur


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