Question

2. 3- What is the growth of the below function: (What is the most accurate answer?)

   ?(?) = 2^(????^3) + ?√? + 7???^6 ? + ?^2????
   options: a) Θ(n) b) Θ (n3) c) Θ (n2logn) d) Θ (n√?) e) Θ (log6n)

3. What is the growth of the below function: (What is the most accurate answer?) ?(?) = ??????? + 4???^2? + ????^2 options: a) O (logn) b) O (loglogn) c) O (log2n) d) O(logn2) e) Neither

4. 5- Assume you want to write a code to calculate the multiplication of two numbers. Provide the running time for your algorithm, assuming the inputs are two n-digit numbers. Explain your answer.

5. 6- Suppose a machine on average takes 10-6 seconds to execute a single algorithm step. What should be the largest input size to finish in 1s ?

   for(i=1; i <= n*n; i++)
   linear_search(a , key); //size(a) = n, and key is the last element in a

6. 7- What is the largest value of n such that an algorithm whose running time is 5nlog2n runs faster than an algorithm whose running time is 35log2n on the same machine?

7. Prove that ?(?) = 2????^2? + 2????^3 + 2^(????) is O(n^2logn), provide the appropriate C and k constants.

8. 9- Compare the growth of f(n) = ???? + 2^???? and g(n) = ?????

9. 10- Prove transitivity of big-O: if f(n) = O(g(n)), then g(n) = Ω(f(n)).

10. 11- Prove that if ??? (?→∞) ?(?)/?(?) = 4, then f(n)= Θ(g(n)).

11. 12- What is the growth of n^2 + 2n^2 + 3n^2 + · · · + n^4?

12. Prove that if f(n) is monotonically decreasing, then

    ?
    ∑ ?(?) = ?(∫(1 to n) ?(?)??)

    ?=1

13. 14- Suppose g(n) ≥ 1 for all n, and that f(n) ≤ g(n) + L, for some constant L and all n. Prove that f(n) = O(g(n)).

14. 15- Given a sorted array with n integers, provide an algorithm with the running time of O(logn) that checks if there is an i for which a[i]= 5i. (e.g. a = [1 4 10 12 17 21] >> true because a[2] = 2*5 = 10) (Explain your answer)

15. 16- Prove or disprove:
    (√?)! = ?((√????)^?^2)

Answer 1

`Hey,

Note: Brother in case of any queries, just comment in box I would be very happy to assist all your queries

3) It is theta(n^3) So, option B s correct

4) f=O(log^2(n)). So, OPTION C IS CORRECT

5) We can use kasturba algorithm to multiply 2 numbers

Algorithm is

Using Divide and Conquer, we can multiply two integers in less time complexity. We divide the given numbers in two halves. Let the given numbers be X and Y.

X =  Xl*2n/2 + Xr    [Xl and Xr contain leftmost and rightmost n/2 bits of X]
Y =  Yl*2n/2 + Yr    [Yl and Yr contain leftmost and rightmost n/2 bits of Y]

XlYr + XrYl = (Xl + Xr)(Yl + Yr) - XlYl- XrYr

So the final value of XY becomes

XY = 2n XlYl + 2n/2 * [(Xl + Xr)(Yl + Yr) - XlYl - XrYr] + XrYr

With above trick, the recurrence becomes T(n) = 3T(n/2) + O(n) and solution of this recurrence is O(n^1.59).

Note: Brother according to HomeworkLib's policy we are only allowed to answer first 3 part if there are many. So, I request you to post other as separate posts.

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