Algorithms subject wise ugc net question analysis

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Q1➡ | NET June 2021
In the following table, the left column contains the names of standard graph algorithms and the right column contains the time complexities of the algorithms. Here, n and m are number of vertices and edges, respectively. Match each algorithm with its time complexity.


Choose the correct answer from the options given below: Options:-
i ➥ A – II, B – I, C – III, D – IV
ii ➥ A – II, B – IV, C – III, D – I
iii ➥ A – III, B – I, C – II, D – IV
iv ➥ A – III, B – IV, C – I, D – II

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Answer: III
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Q1➡ | NTA UGC NET November 2020

Consider the undirected graph below:
Consider the undirected graph below:  Using Prim’s algorithm to construct a minimum spanning tree starting with node a, which one of the following sequences of edges represents a possible order in which the edges would be added to construct the minimum spanning tree?
Using Prim’s algorithm to construct a minimum spanning tree starting with node a, which one of the following sequences of edges represents a possible order in which the edges would be added to construct the minimum spanning tree?
i ➥ (a,b), (a,h), (g,h), (f,g), (c,f), (c,i), (c,d), (d,e)
ii ➥ (a,b), (b,h), (g,h), (g,i), (c,i), (c,f), (c,d), (d,e)
iii ➥ (a,b), (b,c), (c,i), (c,f), (f,g), (g,h), (c,d), (d,e)
iv ➥ (a,b), (g,h), (g,f), (c,f), (c,i), (f,e), (b,c), (d,e)

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Answer: I, III
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Q2➡ | NTA UGC NET November 2020


Choose the correct answer from the options given below:
i ➥ A-I, B-III, C-IV, D-II
ii ➥ A-III, B-I, C-IV, D-II
iii ➥ A-III, B-I, C-II, D-IV
iv ➥ A-I, B-III, C-II, D-IV

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Answer: III
Explanation:
Topological sort of DAG takes (V+E) time.
Bellman ford single source shortest path takes O(VE) time
Floyd Warshall’s all pair shortest path algorithm takes O(V3) time.
Kruskal’s algorithm: Sorted edges will take O(ElogV) and unsorted edges will take O(ElogE)

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Q3➡ | NTA UGC NET November 2020
If algorithm A and another algorithm B take log2(n) and √n microseconds, respectively, to solve a problem, then the largest size n of a problem these algorithms can solve, respectively, in one second are________ and ________.
i ➥ 2106 and 106
ii ➥ 2106 and 1012
iii ➥ 2106 and 6 X 106
iv ➥ 2106 and 6 X 1012

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Answer: II
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Q4➡ | NTA UGC NET November 2020
The running time of an algorithm is O(g(n)) if and only if
i ➥ its worst-case running time is O(g(n)) and its best-case running time is (g(n))(O=big O)
ii ➥ its worst-case running time is Ω(g(n)) and its best-case running time is O(g(n))(O=big O)
iii ➥ O(g(n))=Ω(g(n))(O=big O)
iv ➥ o(g(n)) ∩ ω(g(n))is non-empty set, (o=small o)

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Answer: I
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Q5➡ | NTA UGC NET December 2020
Consider the following statements:
(a) The running time of dynamic programming algorithm is always θ(ρ) where ρ is number of sub-problems.
(b) When a recurrence relation has cyclic dependency, it is impossible to use that recurrence relation (unmodified) in a correct dynamic program.
(c) For a dynamic programming algorithm, computing all values in a bottom-up fashion is asymptotically faster than using recursion and memorization.
(d) If a problem X can be reduced to a known NP-hard problem, then X must be NP-hard.
Which of the statement(s) is (are) true?
i ➥ Only (b) and (a)
ii ➥ Only (b)
iii ➥ Only (b) and (c)
iv ➥ Only (b) and (d)

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Answer: II
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Q6➡ | NTA UGC NET December 2019
The time complexity to multiply two polynomials of degree n using Fast Fourier transform method is:
i ➥ θ(n lg n)
ii ➥ θ(n2)
iii ➥ θ(n)
iv ➥ θ(lg lg n )

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Answer: I
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Q 7➡ | NTA UGC NET December 2019

i ➥
ii ➥
iii ➥
iv ➥

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Answer: I
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Q8➡ | NTA UGC NET December 2019
Give asymptotic upper and lower bound for T(n) given below. Assume T(n) is constant for n≤2.
T(n)= 4T(√n)+lg^2 n
i ➥ T(n)= θ(lg*(lg^2 n)lg n)
ii ➥ T(n)= θ(lg^2n lg n)
iii ➥ T(n)= θ(lg^2 n (lg lg n))
iv ➥ T(n)= θ(lg (lg n)lg n)

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Answer: III
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Q9➡ | NTA UGC NET December 2019
When using Dijkstra’s algorithm to find shortest path in a graph, which of the following statement is not true?
i ➥ It can find shortest path within the same graph data structure
ii ➥ Every time a new node is visited, we choose the node with smallest known distance/cost (weight) to visit first
iii ➥ Shortest path always passes through least number of vertices
iv ➥ The graph needs to have a non-negative weight on every edge

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Answer: III
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Q10➡ | NTA UGC NET December 2019
The weight of minimum spanning tree in graph G, calculated using Kruskal’s algorithm is :

Graph-G
i ➥ 14
ii ➥ 15
iii ➥ 17
iv ➥ 18

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Answer: II
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Q11➡ | NTA UGC NET JUNE 2019
i ➥ (a)-(ii); (b)-(iv); (c)-(i); (d)-(iii)
ii ➥ (a)-(iii); (b)-(iv); (c)-(i); (d)-(ii)
iii ➥ (a)-(ii); (b)-(i); (c)-(iv); (d)-(iii)
iv ➥ (a)-(iii); (b)-(i) (c)-(iv); (d)-(ii)

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Answer: II
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Q12➡ | NTA UGC NET JUNE 2019
Consider the following steps:
S1 : Characterize the structure of an optimal solution
S2 : Computer the value of an optimal solution in bottom-up fashion
Which of the step(s) is/are common to both dynamic programming and greedy algorithms?
i ➥ Only S1
ii ➥ Only S2
iii ➥ both S1 and S2
iv ➥ neither S1 nor S2

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Answer: I
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Q13➡ | NTA UGC NET JUNE 2019
Which of the following is best running time to sort n integers in the range 0 to n2-1
i ➥ O(log n)
ii ➥ O(n)
iii ➥ O(n log n)
iv ➥ O(n2)

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Answer: II
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Q14➡ | NTA UGC NET JUNE 2019
Which of the following is an application of depth-first search?
i ➥ Only topological sort
ii ➥ Only strongly connected components
iii ➥ Both topological sort and strongly connected components
iv ➥ Neither topological sort not strongly connected components

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Answer: III
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Q15➡ | NTA UGC NET JUNE 2019
There are many sorting algorithms based on comparison. The running time of heap sort algorithm is O(nlogn). Let P, but unlike Q, heapsort sorts in place where (P,Q) is equal to
i ➥ Merge sort, Quick sort
ii ➥ Quick sort, Insertion sort
iii ➥ Insertion sort, Quick sort
iv ➥ insertion sort, Merge sort

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Answer: IV
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Q16➡ | NTA UGC NET JUNE 2019

Consider the complexity class CO-NP as the set of languages L such that L’ ε NP, and the following two statements:
S1 : P ⊆ CO-NP
S2 : If NP ≠ CO-NP, then P ≠ NP

Which of the following is/are correct?
i ➥ Only S1
ii ➥ Only S2
iii ➥ Both S1 and S2
iv ➥ Neither S1 nor S2

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Answer: III
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Q17➡ | NTA UGC NET December 2018

Use Kruskal’s algorithm to find the minimum spanning tree of the graph. The weight of this minimum spanning tree is:
i ➥ 13
ii ➥ 16
iii ➥ 17
iv ➥ 14

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Answer: II
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Q18➡ | NTA UGC NET December 2018
let r= a(a + b)*, s=aa*b and t=a*b be three regular expressions.
Consider the following:

(i) L(s) ⊆ L(r) and L(s) ⊆ L(t)
(ii). L(r) ⊆ L(s) and L(s) ⊆ L(t)
i ➥ Only (i) is correct
ii ➥ Both (i) and (ii) are correct
iii ➥ Only (ii) is correct
iv ➥ Neither (i) nor (ii) is correct

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Answer: I
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Q19➡ | NTA UGC NET December 2018

The second smallest of n elements can be found with _ comparisons in worst case.
i ➥ n + ceil(lg n) -2
ii ➥ n-1
iii ➥ lg n
iv ➥ 3n/1

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Answer: I
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Q20➡ | NTA UGC NET December 2018
Consider two sequences X and Y :
X = <0,1,2,1,3,0,1 >
Y = <1,3,2,0,1,0 >

The length of longest common sub-Sequence between X and Y is
i ➥ 4
ii ➥ 2
iii ➥ 3
iv ➥ 5

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Answer: I
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Q21➡ | NTA UGC NET December 2018
The solution of recurrence relation : T(n)=2T(sqrt(n))+lg(n) is
i ➥ O(lg (n) lg(n))
ii ➥ O(lg (n))
iii ➥ O(n lg (n))
iv ➥ O(lg (n) lg(lg(n)))

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Answer: IV
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Q22➡ | NTA UGC NET JUNE 2018
The solution of the recurrence relation T(m)=T(3m/4)+1 is :
i ➥ θ(lg m)
ii ➥ θ(m)
iii ➥ θ(mlg m)
iv ➥ θ(lglg m)

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Answer: I
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Q23➡ | NTA UGC NET JUNE 2018
Which of the following algorithms solves the single-source shortest paths ?
i ➥ Prim’s algorithm
ii ➥ Floyd – Warshall algorithm
iii ➥ Johnson’s algorithm
iv ➥ Dijkstra’s algorithm

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Answer: IV
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Q24➡ | NTA UGC NET JUNE 2018
A text is made up of the characters A, B, C, D, E each occurring with the probability 0.08, 0.40, 0.25, 0.15 and 0.12 respectively. The optimal coding technique will have the average length of :
i ➥ 2.4
ii ➥ 1.87
iii ➥ 3.0
iv ➥ 2.15

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Answer: IV
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Q25➡ | NTA UGC NET JUNE 2018


Code:
i ➥ (a)-(iv),(b)-(i), (c)-(iii), (d)-(ii)
ii ➥ (a)-(iv),(b)-(iii), (c)-(i), (d)-(ii)
iii ➥ (a)-(iii),(b)-(iv), (c)-(ii), (d)-(i)
iv ➥ (a)-(iv),(b)-(iii), (c)-(ii), (d)-(i)

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Answer: IV
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Q26➡ | NTA UGC NET JUNE 2018
The maximum number of comparisons needed to sort 9 items using radix sort is (assume each item is 5 digit octal number) :
i ➥ 45
ii ➥ 72
iii ➥ 360
iv ➥ 450

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Answer: III
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Q27➡ | NTA UGC NET November 2017
Consider an array representation of an n element binary heap where the elements are stored from index 1 to index n of the array. For the element stored at index i of the array (i<=n), the index of the parent is:
i ➥ floor ((i+1)/2)
ii ➥ ceiling ((i+1)/2)
iii ➥ floor (i/2)
iv ➥ ceiling (i/2)

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Answer: III
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Q28➡ | NTA UGC NET November 2017
The following numbers are inserted into an empty binary search tree in the given order: 10, 1, 3, 5, 15, 12, 16 What is the height of the binary search tree ?
i ➥ 3
ii ➥ 4
iii ➥ 5
iv ➥ 6

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Answer: I
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Q29➡ | NTA UGC NET November 2017
Let G be an un-directed connected graph with distinct edge weight. Let E max be the edge with maximum weight and E min the edge with minimum weight. Which of the following statements is false?
i ➥ Every minimum spanning tree of G must contain E min
ii ➥ If E max is in minimum spanning tree, then its removal must disconnect G.
iii ➥ No minimum spanning tree contains E max
iv ➥ G has a unique minimum spanning tree

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Answer: III
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Q30➡ | NTA UGC NET November 2017
A list of n strings, each of length n, is sorted into lexicographic order using merge – sort algorithm. The worst case running time of this computation is :
i ➥ O(n log n)
ii ➥ O(n2 log n)
iii ➥ O(n2 + log n)
iv ➥ O(n3 )

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Answer: II
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Q31➡ | NTA UGC NET November 2017
You are given a sequence of n elements to sort. The input sequence consists of n/k sub-sequences,each containing k elements. The elements in a given sub-sequence are all smaller than the elements in the succeeding sub-sequence and larger than the elements in the preceding sub-sequence. Thus, all that is needed to sort the whole sequence of length n is to sort the k elements in each of the n/k sub-sequences. The lower bound on the number of comparisons needed to solve this variant of the sorting problem is
i ➥ Ω(n)
ii ➥ Ω(n/k)
iii ➥ Ω(nlogk )
iv ➥ Ω(n/klogn/k)

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Answer: III
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Q32➡ | NTA UGC NET November 2017
T(n)= 8T (n/2)+Cn, if n>1 = b, if n=1 Consider the recurrence relation: Where b and c are constants. The order of the algorithm corresponding to above recurrence relation is:
i ➥ N
ii ➥ n2
iii ➥ n log n
iv ➥ n3

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Answer: IV
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Q33➡ | NTA UGC NET November 2017
Consider the following two sequences :
X = < B, C, D, C, A, B, C > and
Y = < C, A, D, B, C, B >

The length of longest common subsequence of X and Y is :
i ➥ 5
ii ➥ 3
iii ➥ 4
iv ➥ 2

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Answer: III
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Q34➡ | NTA UGC NET November 2017
A text is made up of the characters a, b, c, d, e each occurring with the probability 0.11, 0.40, 0.16, 0.09 and 0.24 respectively. The optimal Huffman coding technique will have the average length of:
i ➥ 2.40
ii ➥ 2.16
iii ➥ 2.26
iv ➥ 2.15

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Answer: II
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Q35➡ | NTA UGC NET November 2017
An undirected graph G (V, E) contains n (n > 2) nodes named v1 , v2 ,…,vn. Two nodes vi and vj are connected if and only if 0 <│i−j│≤2. Each edge (vi , vj ) is assigned a weight i+j. The cost of the minimum spanning tree of such a graph with 10 nodes is:
i ➥ 88
ii ➥ 91
iii ➥ 49
iv ➥ 21

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Answer: II
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Q36➡ | NTA UGC NET November 2017
If b is the branching factor and m is the maximum depth of the search tree, what is the space complexity of greedy search?
i ➥ O(b+m)
ii ➥ O(bm)
iii ➥ O(bm)
iv ➥ O(mm)

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Answer: III
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Q37➡ | NTA UGC NET January 2017
The asymptotic upper bound solution of the recurrence T(n)= 2T(n/2)+(n/lg) is relation given by
i ➥ O(n2)
ii ➥ O(n lg n)
iii ➥ O(n lg lg n)
iv ➥ O(lg lg n)

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Answer: III
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Q38➡ | UGC NET January 2017
Any decision tree that sorts n elements has height_______.
i ➥ Ω(lg n)
ii ➥ Ω(n)
iii ➥ Ω(n lg n)
iv ➥ Ω(n2)

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Answer: III
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Q39➡ | UGC NET January 2017
Red-black trees are one of many search tree schemes that are “balanced” in order to guarantee that basic dynamic-set operations take________time in the worst case.
i ➥ O(1)
ii ➥ O(lg n)
iii ➥ O(n)
iv ➥ O(n lg n)

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Answer: II
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Q40➡ | UGC NET January 2017
The minimum number of scalar multiplication required, for parenthesization of a matrix-chain product whose sequence of dimensions for four matrices is <5, 10, 3, 12, 5> is.
i ➥ 630
ii ➥ 580
iii ➥ 480
iv ➥ 405

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Answer: IV
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Q41➡ | UGC NET January 2017

Dijkstra’s algorithm is based on
i ➥ Divide and conquer paradigm
ii ➥ Dynamic programming
iii ➥ Greedy Approach
iv ➥ Backtracking paradigm

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Answer: III
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Q41➡ | UGC NET January 2017
i ➥
ii ➥ a-ii, b-i, c-iv, d-iii
iii ➥ a-ii, b-i, c-iii, d-iv
iv ➥ b-iii, b-ii, c-i, d-iv

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Answer: IV
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Q42➡ | UGC NET August 2016
Consider the problem of a chain < A1, A2, A3, A4 > of four matrices. Suppose that the dimensions of the matrices A1, A2, A3 and A4 are 30 × 35, 35 × 15, 15 × 5 and 5 × 10 respectively. The minimum number of scalar multiplications needed to compute the product A1 A2A3A4 is___
i ➥ 14875
ii ➥ 21000
iii ➥ 9375
iv ➥ 11875

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Answer: III
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Q43➡ | UGC NET August 2016
Consider a weighted complete graph G on the vertex set {ν1, ν2, …. νn} such that the weight of the edge (νi, νj) is 4 | i – j|. The weight of minimum cost spanning tree of G is :
i ➥ 4n2
ii ➥ N
iii ➥ 4n-4
iv ➥ 2n-2

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Answer: III
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Q44➡ | UGC NET August 2016
If there are n integers to sort, each integer has d digits, and each digit is in the set {1, 2, …,k}, radix sort can sort the numbers in:
i ➥ O (k (n + d))
ii ➥ O(d (n + k))
iii ➥ O((n + k) lg d)
iv ➥ O((n + d) lg k)

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Answer: II
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Q45➡ | UGC NET August 2016
i ➥ a-i, b-iii, c-iv, d-ii
ii ➥ a-i, b-iii, c-ii, d-iv
iii ➥ a-iii, b-i, c-iv, d-ii
iv ➥ a-iii, b-i, c-ii, d-iv

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Answer: III
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Q46➡ | UGC NET July 2016
The runtime for traversing all the nodes of a binary search tree with n nodes and printing them in an order is:
i ➥ O(lg n)
ii ➥ O(n lg n)
iii ➥ O(n)
iv ➥ O(n 2 )

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Answer: III
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Q47➡ | UGC NET July 2016

i ➥ (a)-(i), (b)-(ii), (c)-(iv), (d)-(iii)
ii ➥ (a)-(i), (b)-(iv), (c)-(ii), (d)-(iii)
iii ➥ (a)-(iii), (b)-(ii), (c)-(iv), (d)-(i)
iv ➥ (a)-(iii), (b)-(iv), (c)-(ii), (d)-(i)

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Answer: III
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Q48➡ | UGC NET December 2015
In general, in a recursive and non-recursive implementation of a problem (program):
i ➥ Both time and space complexities are better in recursive than in non-recursive program.
ii ➥ Both time and space complexities are better in non-recursive than in recursive program
iii ➥ Time complexity is better in recursive version but space complexity is better in non-recursive version of the program.
iv ➥ Space complexity is better in recursive version but time complexity is better in non-recursive version of the program.

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Answer: II
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