Category: Data Structures & Algorithms Objective Questions

Data Structures & Algorithms Multiple Choice Questions on “Hamiltonian Path Problem”.

1. Which of the following algorithm can be used to solve the Hamiltonian path problem efficiently?
a) branch and bound
b) iterative improvement
c) divide and conquer
d) greedy algorithm
Answer: a
Clarification: The Hamiltonian path problem can be solved efficiently using branch and bound approach. It can also be solved using a backtracking approach.

2. The problem of finding a path in a graph that visits every vertex exactly once is called?
a) Hamiltonian path problem
b) Hamiltonian cycle problem
c) Subset sum problem
d) Turnpike reconstruction problem

Answer: a
Clarification: Hamiltonian path problem is a problem of finding a path in a graph that visits every node exactly once whereas Hamiltonian cycle problem is finding a cycle in a graph.

3. Hamiltonian path problem is _________
a) NP problem
b) N class problem
c) P class problem
d) NP complete problem

Answer: d
Clarification: Hamiltonian path problem is found to be NP complete. Hamiltonian cycle problem is also an NP- complete problem.

4. There is no existing relationship between a Hamiltonian path problem and Hamiltonian circuit problem.
a) true
b) false

Answer: b
Clarification: There is a relationship between Hamiltonian path problem and Hamiltonian circuit problem. The Hamiltonian path in graph G is equal to Hamiltonian cycle in graph H under certain conditions.

5. Which of the following problems is similar to that of a Hamiltonian path problem?
a) knapsack problem
b) closest pair problem
c) travelling salesman problem
d) assignment problem

Answer: c
Clarification: Hamiltonian path problem is similar to that of a travelling salesman problem since both the problem traverses all the nodes in a graph exactly once.

6. Who formulated the first ever algorithm for solving the Hamiltonian path problem?
a) Martello
b) Monte Carlo
c) Leonard
d) Bellman

Answer: a
Clarification: The first ever problem to solve the Hamiltonian path was the enumerative algorithm formulated by Martello.

7. In what time can the Hamiltonian path problem can be solved using dynamic programming?
a) O(N)
b) O(N log N)
c) O(N²)
d) O(N² 2^N)

Answer: d
Clarification: Using dynamic programming, the time taken to solve the Hamiltonian path problem is mathematically found to be O(N² 2^N).

8. In graphs, in which all vertices have an odd degree, the number of Hamiltonian cycles through any fixed edge is always even.
a) true
b) false

Answer: a
Clarification: According to a handshaking lemma, in graphs, in which all vertices have an odd degree, the number of Hamiltonian cycles through any fixed edge is always even.

9. Who invented the inclusion-exclusion principle to solve the Hamiltonian path problem?
a) Karp
b) Leonard Adleman
c) Andreas Bjorklund
d) Martello
Answer: c
Clarification: Andreas Bjorklund came up with the inclusion-exclusion principle to reduce the counting of number of Hamiltonian cycles.

10. For a graph of degree three, in what time can a Hamiltonian path be found?
a) O(0.251ⁿ)
b) O(0.401ⁿ)
c) O(0.167ⁿ)
d) O(0.151ⁿ)
Answer: a
Clarification: For a graph of maximum degree three, a Hamiltonian path can be found in time O(0.251ⁿ).

11. What is the time complexity for finding a Hamiltonian path for a graph having N vertices (using permutation)?
a) O(N!)
b) O(N! * N)
c) O(log N)
d) O(N)
Answer: b
Clarification: For a graph having N vertices traverse the permutations in N! iterations and it traverses the permutations to see if adjacent vertices are connected or not takes N iterations (i.e.) O(N! * N).

12. How many Hamiltonian paths does the following graph have?

a) 1
b) 2
c) 3
d) 4

Answer: a
Clarification: The above graph has only one Hamiltonian path that is from a-b-c-d-e.

13. How many Hamiltonian paths does the following graph have?

a) 1
b) 2
c) 0
d) 3

Answer: c
Clarification: The above graph has no Hamiltonian paths. That is, we cannot traverse the graph with meeting vertices exactly once.

& Algorithms.

Data Structures & Algorithms Multiple Choice Questions on “Linear Search Recursive”.

1. Is there any difference in the speed of execution between linear serach(recursive) vs linear search(lterative)?
a) Both execute at same speed
b) Linear search(recursive) is faster
c) Linear search(Iterative) is faster
d) Cant be said

Answer: c
Clarification: The Iterative algorithm is faster than the latter as recursive algorithm has overheads like calling function and registering stacks repeatedly.

2. Is the space consumed by the linear search(recursive) and linear search(iterative) same?
a) No, recursive algorithm consumes more space
b) No, recursive algorithm consumes less space
c) Yes
d) Nothing can be said

Answer: a
Clarification: The recursive algorithm consumes more space as it involves the usage the stack space(calls the function numerous times).

3. What is the worst case runtime of linear search(recursive) algorithm?
a) O(n)
b) O(logn)
c) O(n²)
d) O(nx)

Answer: a
Clarification: In the worst case scenario, there might be a need of calling the stack n times. Therfore O(n).

4. Linear search(recursive) algorithm used in _____________
a) When the size of the dataset is low
b) When the size of the dataset is large
c) When the dataset is unordered
d) Never used

Answer: a
Clarification: It is used when the size of the dataset is low as its runtime is O(n) which is more when compared to the binary search O(logn).

5. The array is as follows: 1,2,3,6,8,10. At what time the element 6 is found? (By using linear search(recursive) algorithm)
a) 4th call
b) 3rd call
c) 6th call
d) 5th call

Answer: a
Clarification: Provided that the search starts from the first element, the function calls itself till the element is found. In this case, the element is found in 4th call.

6. The array is as follows: 1,2,3,6,8,10. Given that the number 17 is to be searched. At which call it tells that there’s no such element? (By using linear search(recursive) algorithm)
a) 7th call
b) 9th call
c) 17th call
d) The function calls itself infinite number of times

Answer: a
Clarification: The function calls itself till the element is found. But at the 7th call it terminates as goes outside the array.

7. What is the best case runtime of linear search(recursive) algorithm on an ordered set of elements?
a) O(1)
b) O(n)
c) O(logn)
d) O(nx)

Answer: a
Clarification: The best case occurs when the given element to be found is at the first position. Therefore O(1) is the correct answer.

8. Which of the following code snippet performs linear search recursively?
a)

        for(i=0;i<n;i++)
	{
		if(a[i]==key)
		printf("element found");
	}

b)

        LinearSearch(int[] a, n,key)
	{
		if(n<1)
		return False
		if(a[n]==key)
		return True
		else
		LinearSearch(a,n-1,key)
	}

c)

        LinearSearch(int[] a, n,key)
	{
		if(n<1)
		return True
		if(a[n]==key)
		return False
		else
		LinearSearch(a,n-1,key)
	}

d)

        LinearSearch(int[] a, n,key)
	{
		if(n<1)
		return False
		if(a[n]==key)
		return True
		else
		LinearSearch(a,n+1,key)
	}

Answer: b
Clarification: Compare n with first element in arr[]. If element is found at first position, return it. Else recur for remaining array and n.

9. Can linear search recursive algorithm and binary search recursive algorithm be performed on an unordered list?
a) Binary search can’t be used
b) Linear search can’t be used
c) Both cannot be used
d) Both can be used
Answer: a
Clarification: As binary search requires comparison, it is required that the list be ordered. Whereas this doesn’t matter for linear search.

10. What is the recurrence relation for the linear search recursive algorithm?
a) T(n-2)+c
b) 2T(n-1)+c
c) T(n-1)+c
d) T(n+1)+c
Answer: c
Clarification: After each call in the recursive algorithm, the size of n is reduced by 1. Therefore the optimal solution is T(n-1)+c.

Data Structures & Algorithms Multiple Choice Questions on “Comb Sort”.

1. Comb sort is an improved version of _______
a) Selection sort
b) Bubble sort
c) Insertion sort
d) Merge sort

Answer: b
Clarification: Comb sort compares two elements at a variable gap from each other in each iteration unlike bubble sort where the gap remains 1. This reduces the average time complexity of comb sort.

2. The gap between two elements being compared shrinks by a factor of _______ after every iteration.
a) 1.1
b) 1.2
c) 1.3
d) 1.4

Answer: c
Clarification: It has been found experimentally that the gap between the two elements should shrink by a factor of 1.3 after each iteration for the most efficient sorting.

3. The initial gap between two elements being compared _______
a) is equal to number of elements in the array
b) is equal to 1.3
c) depends on the number of iterations
d) depends on the compiler being used

Answer: a
Clarification: Initial gap is taken to be equal to the number of elements in the array and shrinks by a factor of 1.3 in each iteration, initial gap is independent of the number of iterations and compiler being used.

4. What is the worst case time complexity of comb sort?
a) O(n²)
b) O(n log n)
c) O(n)
d) O(n²/2^a) (a=number of increment)

Answer: a
Clarification: Worst case complexity is observed when the input array is reverse sorted. This is same as the worst case complexity of bubble sort.

5. The gap value after 3 iterations in an array with 6 elements will be _______
a) 4
b) 3
c) 2
d) 1

Answer: c
Clarification: Initially the gap value will be 6. For the first iteration, it will be 6/1.3=4 then 4/1.3=3 for second iteration and 3/1.3=2 for the third iteration.

6. Auxiliary space used by comb sort is _______
a) O(1)
b) O(n)
c) O(log n)
d) O(n log n)

Answer: a
Clarification: Auxiliary space used by comb sort is O(1) as it does not use any extra space for manipulating the input.

7. The given array is arr={7,4,5,8,1,2}. The number of iterations required to sort the array using comb sort and bubble sort respectively will be _______
a) 7 and 8
b) 5 and 6
c) 5 and 5
d) 4 and 5

Answer: d
Clarification: Comb sort takes 1 iteration less as compared to bubble sort as it makes use of variable gap value whereas bubble sort uses a constant gap value of 1.

8. Comb sort is a stable sorting algorithm.
a) True
b) False
Answer: b
Clarification: Comb sort is not a stable sorting algorithm as it does not sort the repeated elements in the same order as they appear in the input.

9. What is the best case time complexity of comb sort and bubble sort respectively?
a) O(n²) and O(n log n)
b) O(n log n) and O(n)
c) O(n) and O(n²)
d) O(n²/2^a) (a=number of increment) and O(n²)

Answer: b
Clarification: Best case complexity for comb sort and bubble sort is O(n log n) and O(n) respectively. It occurs when the input array is already sorted.

10. What is the advantage of comb sort over merge sort?
a) Comb sort is an in place sorting algorithm
b) Comb sort is a stable sorting algorithm
c) Comb sort is more efficient
d) It has no advantage
Answer: a
Clarification: Comb sort does not require auxiliary space for manipulating input so it is an in place sorting algorithm but merge sort does require O(n) of auxiliary space which makes comb sort better in terms of space complexity.

Data Structures & Algorithms Multiple Choice Questions & Answers (MCQs) on “Recursive Insertion Sort”.

1. Which of the following is an advantage of recursive insertion sort over its iterative version?
a) it has better time complexity
b) it has better space complexity
c) it is easy to implement
d) it has no significant advantage

Answer: d
Clarification: Recursive insertion sort has no significant advantage over iterative insertion sort. It is just a different way to implement the same.

2. Insertion sort is an online sorting algorithm.
a) true
b) false

Answer: a
Clarification: Insertion sort does not require the entire input data in the beginning itself in order to sort the array. It rather creates a partial solution in every step, so future elements are not required to be considered. Hence it is an online sorting algorithm.

3. What will be the recurrence relation of the code of recursive insertion sort?
a) T(n) = 2T(n/2) + n
b) T(n) = 2T(n/2) + c
c) T(n) = T(n-1) + n
d) T(n) = T(n-1) + c
Answer: c
Clarification: The recurrence relation of the code of recursive insertion sort is T(n) = T(n-1) + n. It can be solved by the method of substitution and is found to be equal to n².

4. Which of the following sorting algorithm is stable?
a) Selection sort
b) Quick sort
c) Insertion sort
d) Heap sort
Answer: c
Clarification: Out of the given options insertion sort is the only algorithm which is stable. It is because the elements with identical values appear in the same order in the output array as they were in the input array.

5. Which of the following is a variant of insertion sort?
a) selection sort
b) shell sort
c) odd-even sort
d) stupid sort

Answer: b
Clarification: Shell sort is a variation of insertion sort. It has a better running time in practical applications.

6. Recursive insertion sort is a comparison based sort.
a) True
b) False

Answer: a
Clarification: In insertion sort, we need to compare elements in order to find the minimum element in each iteration. So we can say that it uses comparisons in order to sort the array. Thus it qualifies as a comparison based sort.

7. What is the average case time complexity of recursive insertion sort?
a) O(n)
b) O(n log n)
c) O(n²)
d) O(log n)
View Answer

Answer: c
Clarification: The overall recurrence relation of recursive insertion sort is given by T(n) = T(n-1) + n. It is found to be equal to O(n²).

8. What is the best case time complexity of recursive insertion sort?
a) O(n)
b) O(n log n)
c) O(n²)
d) O(log n)
View Answer

Answer: a
Clarification: The best case time complexity of recursive insertion sort is O(n). It occurs in the case when the input is already/almost sorted.

9. What is the worst case time complexity of recursive insertion sort?
a) O(n)
b) O(n log n)
c) O(n²)
d) O(log n)
View Answer

Answer: c
Clarification: The overall recurrence relation of recursive insertion sort is given by T(n) = T(n-1) + n. It is found to be equal to O(n²).

10. How many swaps will be required in the worst case to sort an array having n elements using binary insertion sort?
a) n
b) 1
c) n * log n
d) log n
View Answer

Answer: d
Clarification: In a normal insertion sort at most n comparisons are required to sort the array. But if we also implement the concept of a binary sort in insertion sort then we can sort by having log n comparisons only.

11. What will be the base case for the code of recursive insertion sort ?
a)

if(n 
b) 
if(n == 0)
return;
c) 
if(n 
d) 
If(n == 2)
return;
Answer: c
Clarification: The most appropriate condition for the base case of recursive insertion sort is when n is less than or equal 1 then return. It is because we know that an array with only 1 element is always sorted.
 
 
12. What is the auxiliary space complexity of recursive insertion sort?
a) O(n)
b) O(1)
c) O(n log n)
d) O(n2)

Answer: b
Clarification: The auxiliary space required by recursive insertion sort is O(1). So it qualifies as an in place sorting algorithm.
13. Which of the following is an adaptive sorting algorithm?
a) recursive insertion sort
b) merge sort
c) heap sort
d) selection sort

Answer: a
Clarification: Insertion sort is an adaptive algorithm. It is because the time complexity of the algorithm improves when the input array is almost sorted.
14. Which of the following sorting algorithm is in place?
a) recursive insertion sort
b) merge sort
c) radix sort
d) counting sort

Answer: a
Clarification: Out of the given options recursive insertion sort is the only algorithm which is in place. It is because the auxiliary space required by recursive bubble sort is O(1). 
15.Choose the correct function for recursive insertion sort?
a)







void RecInsertionSort(int arr[], int n) 
{ 
	if (n <= 1) 
		return; 
	RecInsertionSort( arr, n-1 ); 	
	int key = arr[n-1]; 
	int j = n-2; 	
	while (j >= 0 && arr[j] > key) 
	{ 
		arr[j+1] = arr[j]; 
		j--; 
	} 
	arr[j+1] = key; 
}







b)







void RecInsertionSort(int arr[], int n) 
{ 
 
	if (n < 1) 
		return; 
 
	RecInsertionSort( arr, n-1 ); 
 
	int key = arr[n-1]; 
	int j = n-2; 
 
	while (j >= 0 || arr[j] > key) 
	{ 
		arr[j+1] = arr[j]; 
		j--; 
	} 
	arr[j] = key; 
}







c)







void RecInsertionSort(int arr[], int n) 
{ 
 
	if (n <1) 
		return; 
	RecInsertionSort( arr, n-1 ); 
 
	int key = arr[n-1]; 
	int j = n-2; 
 
	while (j >= 0 && arr[j] > key) 
	{ 
		arr[j+1] = arr[j]; 
		j--; 
	} 
	arr[j] = key; 
}







d)







void RecInsertionSort(int arr[], int n) 
{ 
 
	if (n <= 1) 
		return; 
 
	RecInsertionSort( arr, n-1 ); 
 
	int key = arr[n-1]; 
	int j = n-1; 
 
	while (j >= 0 || arr[j] > key) 
	{ 
		arr[j+1] = arr[j]; 
		j--; 
	} 
	arr[j+1] = key; 
}







Answer: a
Clarification: The base case of recursive bubble sort should be when n equal or less than 1 then return. Also we need to insert the element being chosen as key at its correct position in the sorted array to its left. All this needs to be done in a recursive code.
 
 
 & Algorithms.
 and Answers.

Data Structures & Algorithms Multiple Choice Questions on “Chan’s Algorithm”.

1. Chan’s algorithm is used for computing _________
a) Closest distance between two points
b) Convex hull
c) Area of a polygon
d) Shortest path between two points

Answer: b
Clarification: Chan’s algorithm is an output-sensitive algorithm used to compute the convex hull set of n points in a 2D or 3D space. Closest pair algorithm is used to compute the closest distance between two points.

2. What is the running time of Chan’s algorithm?
a) O(log n)
b) O(n log n)
c) O(n log h)
d) O(log h)

Answer: c
Clarification: The running time of Chan’s algorithm is calculated to be O(n log h) where h is the number of vertices of the convex hull.

3. Who formulated Chan’s algorithm?
a) Timothy
b) Kirkpatrick
c) Frank Nielsen
d) Seidel
Answer: a
Clarification: Chan’s algorithm was formulated by Timothy Chan. Kirkpatrick and Seidel formulated the Kirkpatrick-Seidel algorithm. Frank Nielsen developed a paradigm relating to Chan’s algorithm.

4. The running time of Chan’s algorithm is obtained from combining two algorithms.
a) True
b) False
Answer: a
Clarification: The O(n log h) running time of Chan’s algorithm is obtained by combining the running time of Graham’s scan [O(n log n)] and Jarvis match [O(nh)].

5. Which of the following is called the “ultimate planar convex hull algorithm”?
a) Chan’s algorithm
b) Kirkpatrick-Seidel algorithm
c) Gift wrapping algorithm
d) Jarvis algorithm

Answer: b
Clarification: Kirkpatrick-Seidel algorithm is called as the ultimate planar convex hull algorithm. Its running time is the same as that of Chan’s algorithm (i.e.) O(n log h).

6. Which of the following algorithms is the simplest?
a) Chan’s algorithm
b) Kirkpatrick-Seidel algorithm
c) Gift wrapping algorithm
d) Jarvis algorithm

Answer: a
Clarification: Chan’s algorithm is very practical for moderate sized problems whereas Kirkpatrick-Seidel algorithm is not. Although, they both have the same running time. Gift wrapping algorithm is a non-output sensitive algorithm and has a longer running time.

7. What is the running time of Hershberger algorithm?
a) O(log n)
b) O(n log n)
c) O(n log h)
d) O(log h)

Answer: b
Clarification: Hershberger’s algorithm is an output sensitive algorithm whose running time was originally O(n log n). He used Chan’s algorithm to speed up to O(n log h) where h is the number of edges.

8. Which of the following statements is not a part of Chan’s algorithm?
a) eliminate points not in the hull
b) recompute convex hull from scratch
c) merge previously calculated convex hull
d) reuse convex hull from the previous iteration

Answer: b
Clarification: Chan’s algorithm implies that the convex hulls of larger points can be arrived at by merging previously calculated convex hulls. It makes the algorithm simpler instead of recomputing every time from scratch.

9. Which of the following factors account more to the cost of Chan’s algorithm?
a) computing a single convex hull
b) locating points that constitute a hull
c) computing convex hull in groups
d) merging convex hulls
Answer: c
Clarification: The majority of the cost of the algorithm lies in the pre-processing (i.e.) computing convex hull in groups. To reduce cost, we reuse convex hulls from previous iterations.

10. Chan’s algorithm can be used to compute the lower envelope of a trapezoid.
a) true
b) false
Answer: a
Clarification: An extension of Chan’s algorithm can be used for proving solutions to complex problems like computing the lower envelope L(S) where S is a set of ‘n’ line segments in a trapezoid.