This set of Data Structure Multiple Choice Questions & Answers (MCQs) focuses on “Fibonacci using Dynamic Programming”.

1. The following sequence is a fibonacci sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21,…..

Which technique can be used to get the nth fibonacci term?

a) Recursion

b) Dynamic programming

c) A single for loop

d) All of the mentioned

View Answer

Explanation: Each of the above mentioned methods can be used to find the nth fibonacci term.

2. Consider the recursive implementation to find the nth fibonacci number:

int fibo(int n) if n <= 1 return n return __________

Which line would make the implementation complete?

a) fibo(n) + fibo(n)

b) fibo(n) + fibo(n – 1)

c) fibo(n – 1) + fibo(n + 1)

d) fibo(n – 1) + fibo(n – 2)

View Answer

Explanation: Consider the first five terms of the fibonacci sequence: 0,1,1,2,3. The 6th term can be found by adding the two previous terms, i.e. fibo(6) = fibo(5) + fibo(4) = 3 + 2 = 5. Therefore,the nth term of a fibonacci sequence would be given by:

fibo(n) = fibo(n-1) + fibo(n-2).

3. What is the time complexity of the recursive implementation used to find the nth fibonacci term?

a) O(1)

b) O(n^{2})

c) O(n!)

d) Exponential

View Answer

Explanation: The recurrence relation is given by fibo(n) = fibo(n – 1) + fibo(n – 2). So, the time complexity is given by:

T(n) = T(n – 1) + T(n – 2)

Approximately,

T(n) = 2 * T(n – 1)

= 4 * T(n – 2)

= 8 * T(n – 3)

:

:

:

= 2

^{k}* T(n – k)

This recurrence will stop when n – k = 0

i.e. n = k

Therefore, T(n) = 2

^{n}* O(0) = 2

^{n}

Hence, it takes exponential time.

It can also be proved by drawing the recursion tree and counting the number of leaves.

4. Suppose we find the 8th term using the recursive implementation. The arguments passed to the function calls will be as follows:

fibonacci(8) fibonacci(7) + fibonacci(6) fibonacci(6) + fibonacci(5) + fibonacci(5) + fibonacci(4) fibonacci(5) + fibonacci(4) + fibonacci(4) + fibonacci(3) + fibonacci(4) + fibonacci(3) + fibonacci(3) + fibonacci(2) : : :

Which property is shown by the above function calls?

a) Memoization

b) Optimal substructure

c) Overlapping subproblems

d) Greedy

View Answer

Explanation: From the function calls, we can see that fibonacci(4) is calculated twice and fibonacci(3) is calculated thrice. Thus, the same subproblem is solved many times and hence the function calls show the overlapping subproblems property.

5. What is the output of the following program?

#include<stdio.h> int fibo(int n) { if(n<=1) return n; return fibo(n-1) + fibo(n-2); } int main() { int r = fibo(50000); printf("%d",r); return 0; }

a) 1253556389

b) 5635632456

c) Garbage value

d) Runtime error

View Answer

Explanation: The value of n is 50000. The function is recursive and it’s time complexity is exponential. So, the function will be called almost 2

^{50000}times. Now, even though NO variables are stored by the function, the space required to store the addresses of these function calls will be enormous. Stack memory is utilized to store these addresses and only a particular amount of stack memory can be used by any program. So, after a certain function call, no more stack space will be available and it will lead to stack overflow causing runtime error.

6. What is the space complexity of the recursive implementation used to find the nth fibonacci term?

a) O(1)

b) O(n)

c) O(n^{2})

d) O(n^{3})

View Answer

Explanation: The recursive implementation doesn’t store any values and calculates every value from scratch. So, the space complexity is O(1).

7. Consider the following code to find the nth fibonacci term:

int fibo(int n) if n == 0 return 0 else prevFib = 0 curFib = 1 for i : 1 to n-1 nextFib = prevFib + curFib __________ __________ return curFib

Complete the above code.

a) prevFib = curFib

curFib = curFib

b) prevFib = nextFib

curFib = prevFib

c) prevFib = curFib

curFib = nextFib

d) none of the mentioned

View Answer

Explanation: The lines, prevFib = curFib and curFib = nextFib, make the code complete.

8. What is the time complexity of the ABOVE for loop method used to compute the nth fibonacci term ?

a) O(1)

b) O(n)

c) O(n^{2})

d) Exponential

View Answer

Explanation: To calculate the nth term, the for loop runs (n – 1) times and each time a for loop is run, it takes a constant time. Therefore, the time complexity is of the order of n.

9. What is the space complexity of the ABOVE for loop method used to compute the nth fibonacci term?

a) O(1)

b) O(n)

c) O(n^{2})

d) Exponential

View Answer

Explanation: To calculate the nth term, we just store the previous term and the current term and then calculate the next term using these two terms. It takes a constant space to store these two terms and hence O(1) is the answer.

10. What will be the output when the following code is executed?

#include<stdio.h> int fibo(int n) { if(n==0) return 0; int i; int prevFib=0,curFib=1; for(i=1;i<=n-1;i++) { int nextFib = prevFib + curFib; prevFib = curFib; curFib = nextFib; } return curFib; } int main() { int r = fibo(10); printf("%d",r); return 0; }

a) 34

b) 55

c) Compile error

d) Runtime error

View Answer

Explanation: The output is the 10th fibonacci number, which is 55.

11. Consider the following code to find the nth fibonacci term using dynamic programming:

1. int fibo(int n) 2. int fibo_terms[100000] //arr to store the fibonacci numbers 3. fibo_terms[0] = 0 4. fibo_terms[1] = 1 5. 6. for i: 2 to n 7. fibo_terms[i] = fibo_terms[i - 1] + fibo_terms[i - 2] 8. 9. return fibo_terms[n]

Which property is shown by line 7 of the above code?

a) Optimal substructure

b) Overlapping subproblems

c) Both overlapping subproblems and optimal substructure

d) None of the mentioned

View Answer

Explanation: We find the nth fibonacci term by finding previous fibonacci terms, i.e. by solving subproblems. Hence, line 7 shows the optimal substructure property.

12. Consider the following code to find the nth fibonacci term using dynamic programming:

1. int fibo(int n) 2. int fibo_terms[100000] //arr to store the fibonacci numbers 3. fibo_terms[0] = 0 4. fibo_terms[1] = 1 5. 6. for i: 2 to n 7. fibo_terms[i] = fibo_terms[i - 1] + fibo_terms[i - 2] 8. 9. return fibo_terms[n]

Which technique is used by line 7 of the above code?

a) Greedy

b) Recursion

c) Memoization

d) None of the mentioned

View Answer

Explanation: Line 7 stores the current value that is calculated, so that the value can be used later directly without calculating it from scratch. This is memoization.

13. What is the time complexity of the ABOVE dynamic programming implementation used to compute the nth fibonacci term?

a) O(1)

b) O(n)

c) O(n^{2})

d) Exponential

View Answer

Explanation: To calculate the nth term, the for loop runs (n – 1) times and each time a for loop is run, it takes a constant time. Therefore, the time complexity is of the order of n.

14. What is the space complexity of the ABOVE dynamic programming implementation used to compute the nth fibonacci term?

a) O(1)

b) O(n)

c) O(n^{2})

d) Exponential

View Answer

Explanation: To calculate the nth term, we store all the terms from 0 to n – 1. So, it takes O(n) space.

15. What will be the output when the following code is executed?

#include<stdio. int fibo(int n) { int i; int fibo_terms[100]; fibo_terms[0]=0; fibo_terms[1]=1; for(i=2;i<=n;i++) fibo_terms[i] = fibo_terms[i-2] + fibo_terms[i-1]; return fibo_terms[n]; } int main() { int r = fibo(8); printf("%d",r); return 0; }

a) 34

b) 55

c) Compile error

d) 21

View Answer

Explanation: The program prints the 8th fibonacci term, which is 21.

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