# Recent questions and answers in Algorithms

1
Let $A[1,\ldots,n]$ be an array storing a bit ($1$ or $0$) at each location, and $f(m)$ is a function whose time complexity is $\Theta(m)$. Consider the following program fragment written in a C like language: counter = 0; for (i=1; i<=n; i++) { if a[i] == 1) counter++; ... 0;} } The complexity of this program fragment is $\Omega(n^2)$ $\Omega (n\log n) \text{ and } O(n^2)$ $\Theta(n)$ $o(n)$
2
The number of elements that can be sorted in $\Theta(\log n)$ time using heap sort is $\Theta(1)$ $\Theta(\sqrt{\log} n)$ $\Theta(\frac{\log n}{\log \log n})$ $\Theta(\log n)$
3
Which one of the following is the tightest upper bound that represents the number of swaps required to sort $n$ numbers using selection sort? $O(\log n$) $O(n$) $O(n \log n$) $O(n^{2}$)
4
Let $n$ be a large integer. Which of the following statements is TRUE? $2^{\sqrt{2\log n}}< \frac{n}{\log n}< n^{1/3}$ $\frac{n}{\log n}< n^{1/3}< 2^{\sqrt{2\log n}}$ $2^\sqrt{{2\log n}}< n^{1/3}< \frac{n}{\log n}$ $n^{1/3}< 2^\sqrt{{2\log n}}<\frac{n}{\log n}$ $\frac{n}{\log n}< 2^\sqrt{{2\log n}}<n^{1/3}$
5
How to analyze this problem For (I=1;I<n;I++) { For (j=1;j<n;j=j+I) { Print("℅d℅d", I,j); } } Please suggest I'm detail how do we unroll I & j.
6
what is the time-complexity in kruskal algorithm for the overall step 2 where for each vertex Make-set function is called ? How come overall time for this step is O(v log v) ? We are performing this Operation for all the vertices in the Initial phase only so for ... Make-set operation only once right because after we come out of loop we have v sets of 1 vertex each . Please explain this clearly .
7
what is the time complexity of the $pow(m,n)$ ?
8
In a permutation $a_1 ... a_n$, of $n$ distinct integers, an inversion is a pair $(a_i, a_j)$ such that $i < j$ and $a_i > a_j$. What would be the worst case time complexity of the Insertion Sort algorithm, if the inputs are restricted to permutations of $1. . . n$ with at most $n$ inversions? $\Theta(n^2)$ $\Theta(n\log n)$ $\Theta(n^{1.5})$ $\Theta(n)$
9
An array $A$ contains $n$ integers. We wish to sort $A$ in ascending order. We are told that initially no element of $A$ is more than a distance $k$ away from its final position in the sorted list. Assume that $n$ and $k$ are large and $k$ is much smaller ... with constant $.n \log k$ comparisons but not with fewer comparisons. $A$ can be sorted with constant $.k^{2}n$ comparisons but not fewer.
10
Is it possible that a disconnected graph is an Euler graph ?
11
Suppose $P, Q, R, S, T$ are sorted sequences having lengths $20, 24, 30, 35, 50$ respectively. They are to be merged into a single sequence by merging together two sequences at a time. The number of comparisons that will be needed in the worst case by the optimal algorithm for doing this is ____.
12
Assume that a mergesort algorithm in the worst case takes $30$ seconds for an input of size $64$. Which of the following most closely approximates the maximum input size of a problem that can be solved in $6$ minutes? $256$ $512$ $1024$ $2018$
13
The worst case running times of Insertion sort , Merge sort and Quick sort, respectively are: $\Theta (n \log n)$, $\Theta (n \log n)$ and $\Theta(n^2)$ $\Theta (n^2)$, $\Theta (n^2)$ and $\Theta(n \log n)$ $\Theta (n^2)$, $\Theta (n \log n)$ and $\Theta (n \log n)$ $\Theta (n^2)$, $\Theta (n \log n)$ and $\Theta (n^2)$
14
Assume that the algorithms considered here sort the input sequences in ascending order. If the input is already in the ascending order, which of the following are TRUE? Quicksort runs in $\Theta (n^2)$ time Bubblesort runs in $\Theta (n^2)$ time Mergesort runs in $\Theta (n)$ time Insertion sort runs in $\Theta (n)$ time I and II only I and III only II and IV only I and IV only
15
Suppose there are $\lceil \log n \rceil$ sorted lists of $\lfloor n /\log n \rfloor$ elements each. The time complexity of producing a sorted list of all these elements is: (Hint:Use a heap data structure) $O(n \log \log n)$ $\Theta(n \log n)$ $\Omega(n \log n)$ $\Omega\left(n^{3/2}\right)$
16
Consider the following C program that attempts to locate an element $x$ in an array $Y[ \ ]$ using binary search. The program is erroneous. f (int Y[10] , int x) { int u, j, k; i= 0; j = 9; do { k = (i+ j) / 2; if( Y[k] < x) i = k;else j = k; } while (Y[k] != x) && (i < j)) ; if(Y[k] ... $x > 2$ $Y$ is $[2 \ 4 \ 6 \ 8 \ 10 \ 12 \ 14 \ 16 \ 18 \ 20]$ and $2 < x < 20$ and $x$ is even
17
For parameters $a$ and $b$, both of which are $\omega(1)$, $T(n) = T(n^{1/a})+1$, and $T(b)=1$. Then $T(n)$ is $\Theta (\log_a \log _b n)$ $\Theta (\log_{ab} n$) $\Theta (\log_{b} \log_{a} \: n$) $\Theta (\log_{2} \log_{2} n$)
1 vote
18
Consider a set of 156 elements to find minimum and maximum elements in the given set, the minimum number of comparisons required is___? You have given an array of 512 elements,minimum number of comparisons required to find out second largest element among all will be___? 230 & 517 229 & 516 231 & 518 232 & 519
19
A min heap having $1024$ distinct elements with keys ranging from $0$ to $1023$ is stored in array of $1024$ indices. The maximum difference between $(n/2)^{th}$ element present at maximum level and minimum level is ________. [Assume root is present at $level-1$] ? Please Tell me the Approach
1 vote
20
What will be the path from A-H if BFS is used in the following graph?
21
Let $x_n$ denote the number of binary strings of length $n$ that contain no consecutive 0s. The value of $x_5$ is $5$ $7$ $8$ $16$
22
Let $G$ be a weighted connected undirected graph with distinct positive edge weights. If every edge weight is increased by the same value, then which of the following statements is/are TRUE? $P$: Minimum spanning tree of $G$ does not change. $Q$: Shortest path between any pair of vertices does not change. $P$ only $Q$ only Neither $P$ nor $Q$ Both $P$ and $Q$
23
An undirected graph $G(V,E)$ contains $n \: (n>2)$ nodes named $v_1,v_2, \dots, v_n$. Two nodes $v_i, v_j$ are connected if and only if $0 < \mid i-j\mid \leq 2$. Each edge $(v_i,v_j)$ is assigned a weight $i+j$. A sample graph with $n=4$ is shown below. The length of the path from $v_5$ to $v_6$ in the MST of previous question with $n=10$ is $11$ $25$ $31$ $41$
24
Give an optimal algorithm in pseudo-code for sorting a sequence of $n$ numbers which has only $k$ distinct numbers ($k$ is not known a Priori). Give a brief analysis for the time-complexity of your algorithm.
25
What is the time complexity of job sequencing with deadline using greedy algorithm? O(n) O(log n) O(n log n) O(n2) Made Easy Full Syllabus Test-6 : Basic Level : Practice Test-14 Q 19 Please give reference for this answer to this algorithm.
26
An element in an array $X$ is called a leader if it is greater than all elements to the right of it in $X$. The best algorithm to find all leaders in an array solves it in linear time using a left to right pass of the array solves it in linear time using a right to left pass of the array solves it using divide and conquer in time $\Theta (n\log n)$ solves it in time $\Theta( n^2)$
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An array $A$ contains $n$ integers in locations $A[0], A[1], \dots A[n-1]$. It is required to shift the elements of the array cyclically to the left by $K$ places, where $1\leq K \leq n-1$. An incomplete algorithm for doing this in linear time, without using another array is given below. Complete ... A[j]:=____; j:=(j+K) mod n; if j<min then min:=j; end; A[(n+i-K)mod n]:=____; i:=______; end;
29
Given two arrays of numbers $a_{1},...,a_{n}$ and $b_{1},...,b_{n}$ where each number is $0$ or $1$, the fastest algorithm to find the largest span $(i, j)$ such that $a_{i}+a_{i+1}+\dots+a_{j}=b_{i}+b_{i+1}+\dots+b_{j}$ or report that there is ... $\Theta (n)$ time and space Takes $O(\sqrt n)$ time only if the sum of the $2n$ elements is an even number
30
what is the recurrence relation for merge sort?
1 vote
31
Merge sort uses : Divide-and-conquer Backtracking Heuristic approach Greedy approach
32
Let $T$ be a Depth First Tree of a undirected graph $G$. An array $P$ indexed by the vertices of $G$ is given. $P[V]$ is the parent of vertex $V$, in $T$. Parent of the root is the root itself. Give a method for finding and printing the ... be proportional to the length of the cycle. Describe the algorithm in a PASCAL $(C)$ - like language. Assume that the variables have been suitably declared.
33
Consider the following graph: Among the following sequences: abeghf abfehg abfhge afghbe Which are the depth-first traversals of the above graph? I, II and IV only I and IV only II, III and IV only I, III and IV only
34
Which of the following standard algorithms is not Dynamic Programming based? Bellman-Ford Algorithm for single source shortest path Floyd Warshall Algorithm for all pairs shortest paths $0-1$ Knapsack problem Prim’s Minimum Spanning Tree
35
Find the odd one out Merge Sort TVSP Problem Knapsack Problem OBST Problem
36
US portal Denominations are 1,10,21,34,70 , 100 and 350 . A) Make 140 Cents . Verify whether Greedy choice fails or not B) Make 182 Cents. Verify whether greedy choice fails or not. 1)Greedy fails in A 2) Greedy fails in B 3)Greedy does not fails in A 4)Greedy does not fails in B
37
Single source shortest path problems can be implemented by greedy algorithms using A. Singly linked list B. Min heap C. AVL tree D. All of the above
Consider the following code segment to find the $n^{th}$ Fibonacci number: Fib(n) { if(n==0) {return 0;} if(n==1) {return 1;} else { return(Fib(n-1) + Fib(n-2)); } } The time complexity of the above code and time complexity of the same problem solved using dynamic programming is______ $A)O(n^{2}),O(n)$ $B)O(2^{n}),O(n)$ $C)O(2^{n}),O(n^{2})$ $D)$None of the above
Consider the following steps: $S_1$: Characterize the structure of an optimal solution $S_2$: Compute the value of an optimal solution in bottom-up fashion Which of the following step(s) is/are common to both dynamic programming and greedy algorithms? Only $S_1$ Only $S_2$ Both $S_1$ and $S_2$ Neither $S_1$ nor $S_2$