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QUESTION: 1

Each of the function 2^{√n} and n^{logn} has a growth rate .... than that of any polynomial.

Solution:

2^{√n} and n^{logn} grows exponentially which have growth rate greater than any polynomial.

QUESTION: 2

Time complexity of an algorithm T(n), where n is the input size is given by

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

= 1, otherwise

The order of this algorithm is

Solution:

⇒ O(log n)

QUESTION: 3

The concept of order (Big O) is important because

Solution:

Big O notation gives worst case limit for a given problem: Also find out the least upper bound of problem.

QUESTION: 4

Consider the following functions:

Which of the following is true?

Solution:

Here f(n) is O(g(n))

f(n) is O(h(n))

If f(n) is O(g(n))

f(n) is O(g(n)) and g(n) is O (f(n))

QUESTION: 5

f(n) = 3n^{2} + 4n + 2

Which will be the exact value for f(n)?

Solution:

f(n) = 3 n^{2} + 4n + 2

So, f(n) = n^{2} + n^{2} + n^{2} = O (n^{2})

QUESTION: 6

Which of the following is the average number of key comparisons done by sequential reach in the successful case?

Solution:

in linear or sequential search the maximum number of comparisons are (n) for V elements hence the number of comparisons when the element to be found is at the middle of the array

= n/2 [If n is even]

In general average case =

QUESTION: 7

Which sort will operate in quadratic time relative to the number of elements in the array (on the average)?

Solution:

Bubbles sort have time complexity of O(n^{2}) and all other have times complexity 0(nlogn).

QUESTION: 8

If one uses straight two-way merge sort algorithm to sort the following elements in ascending order:

20,47,15,8,9,4,40,30,12,17

Then the order of these elements after the 2^{nd} pass of the algorithm is

Solution:

QUESTION: 9

The recurrence relation:

T(1) = 2

, has the solution T(n) equal to

Solution:

Let, 4^{k} = n

≌ O(n)

QUESTION: 10

Consider the following two functions:

Which of the following is true?

Solution:

Therefore;

n^{2} ≤ n^{3} for N ≥ 10000

g_{1}(n) = O(g_{2}(n))

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