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QUESTION: 1

f{n) = 0 (g (n)) if and only if

Solution:

QUESTION: 2

Look at the following algorithms

int n;

int A[100];

void (x){

int i;

for (i=n/2;i > = 1; i -- )

y(i);

}

}

void y(int i)

{

.........;

.........;

.........;

}

Let complexity of y is O(log_{2}n). Then the complexity of x will be

Solution:

In this programme for loop run n/2 times and in this ‘for loop’ y(i) have (log_{2} n) time complexity. In the above we will select maximum time complexity i.e., (n/2) log_{2}n.

Then the total time complexity for the ‘for loop’ will O(n logn).

QUESTION: 3

The time complexity of the following C function

is (assume n > 0)

int recursive (int n)

{

if (n == 1)

return(1);

else

return(recursive(n- 1) + recursive(n - 1));

}

Solution:

The given C function is recursive. The best way to find the time complexity of recursive function is that convert the code (algorithm) into recursion equation and solution of the recursion equation is the time complexity of given algorithm.

1. int recursive (int i){

2 . if ( n == 1) return (1);

3. else

4. return (recursive(n-1) + recursive(n-1));

5. }

Let recursive(n) - T(n)

According to line 2 if (n = = 1) return(1) then the recursive equation is

T(n) = 1, n = 1

According to line 4 the recursion equation is

T(n) = T ( n - 1 ) + T(n-1), n > 1

So the complete recursion equation is

QUESTION: 4

The recurrence equation:

Evaluates to

Solution:

QUESTION: 5

Let T(n) be a function defined by the recurrence

T(n) - 2T(n/2) + √n for n≥2 and T(1) = 1

Which of the following statements is TRUE?

Solution:

By substitution method

QUESTION: 6

Suppose T(n) = 2T(n/2) + n, T(0) = T(1) = 1 Which one of the following is FALSE?

Solution:

T(0)= T(1) = 1

T(n) = 2T(n/2) + n

T(n) be computed as follows:

[T(n) can also proved by Master Theorem]

if then it is also 0(n logn) and O(n^{2}) but it is not Ω(n^{2}).

QUESTION: 7

Consider the following segment of C code

The number of comparisons made in the execution of the loop for any n > 0 is

Solution:

Let the increment of j is 2°, 2^{1}...., 2^{i} for some value of i so according to the question for while loop:

One extra comparison required for the termination of while loop. So total number of comparisons

QUESTION: 8

What is the time complexity of the following recursive function:

int DoSomething ( int n)

{

if {n< = 2)

return 1;

else

return DoSomething (floor (sqrt (n)))+n;

}

Solution:

The given function is recursive so the equivalent recursion equation is

All the level sums are equal to n. The problem size at level k of the recursion tree is n^{2-k} and we stop recursing when this value is a constant. Setting n^{2-k} = 2 and solving for k gives us

QUESTION: 9

Using the standard algorithm, what is the time required to determine that a number n is prime?

Solution:

It takes not more than n/2 comparison to check whether the given number n is prime or not.

QUESTION: 10

The running time of an algorithm T(n), where ‘n’ is the input size is given by

T(n) = 8T(n/2) + qn. if n> 1

p, if n = 1

where p, q are constants. The order of this algorithm is

Solution:

Since q is constant, so our recurrence relation look like

QUESTION: 11

Let m, n be positive integers. Define Q(m, n) as

Then Q(m, 3) is (a div b, gives the quotient when a is divided by b)

Solution:

Two cases: Q(m, 3)

(i) m ≥ n : here m ≥ 3

For

∵ Since 3 is divide by 3 only 1 time = 0 + P= P.

So this recursive program generate P, m/n times until m < n.

So, Q(m, 3) = Px(m div 3)

QUESTION: 12

T(n) = T(2n/3) + 1 then T(n) is equal to

Solution:

By using recursive tree method:

So, height of tree is log_{(3/2)}n

So, T(n) = Height x Work at each level.

QUESTION: 13

An algorithm is made up of 2 m odules m1 and m2. If order of M1 is f(n) and M2 is g(n) then the order of the algorithm is

Solution:

So, order of algorithm

QUESTION: 14

The recurrence relation T(n) = mT(n/2) + an^{2} is satisfied by

Solution:

T(n) = mT (n/2) + an^{2}

Since,

m ≥ 5

So by applying Master Theorem:

QUESTION: 15

The running time T(n) where ‘n' is the input size of a recursive algorithm is given as follow

The order of this algorithm is

Solution:

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