Recent questions and answers in Algorithms

8 votes
4 answers
1
Consider the following function. Function F(n, m:integer):integer; begin If (n<=0 or (m<=0) then F:=1 else F:F(n-1, m) + F(n, m-1); end; Use the recurrence relation ... What is the value of $F(n, m)$? How many recursive calls are made to the function $F$, including the original call, when evaluating $F(n, m)$.
25 votes
5 answers
2
What does the following algorithm approximate? (Assume $m > 1, \epsilon >0$). x = m; y = 1; While (x-y > ϵ) { x = (x+y)/2; y = m/x; } print(x); $\log \, m$ $m^2$ $m^{\frac{1}{2}}$ $m^{\frac{1}{3}}$
2 votes
2 answers
3
Consider the following directed graph: Which of the following is/are correct about the graph? The graph does not have a topological order A depth-first traversal starting at vertex $S$ classifies three directed edges as back edges The graph does not have a strongly connected component For each pair of vertices $u$ and $v$, there is a directed path from $u$ to $v$
42 votes
7 answers
4
In the following $C$ program fragment, $j$, $k$, $n$ and TwoLog_n are integer variables, and $A$ is an array of integers. The variable $n$ is initialized to an integer $\geqslant 3$, and TwoLog_n is initialized to the value of $2^*\lceil \log_2(n) \rceil$ for (k = 3; k <= n; k++) A[k] ... $\left\{m \mid m \leq n, \text{m is prime} \right\}$ { }
65 votes
5 answers
5
In an adjacency list representation of an undirected simple graph $G=(V, E)$, each edge $(u, v)$ has two adjacency list entries: $[v]$ in the adjacency list of $u$, and $[u]$ in the adjacency list of $v$. These are called twins of each other. A twin pointer is a pointer from an ... list? $\Theta\left(n^{2}\right)$ $\Theta\left(n+m\right)$ $\Theta\left(m^{2}\right)$ $\Theta\left(n^{4}\right)$
41 votes
6 answers
6
Which of the following statement(s) is/are correct regarding Bellman-Ford shortest path algorithm? P: Always finds a negative weighted cycle, if one exists. Q: Finds whether any negative weighted cycle is reachable from the source. $P$ only $Q$ only Both $P$ and $Q$ Neither $P$ nor $Q$
22 votes
4 answers
7
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 $\Theta (n^2)$
42 votes
9 answers
8
You have an array of $n$ elements. Suppose you implement quicksort by always choosing the central element of the array as the pivot. Then the tightest upper bound for the worst case performance is $O(n^2)$ $O(n \log n)$ $\Theta(n\log n)$ $O(n^3)$
28 votes
4 answers
9
Which one of the following is the recurrence equation for the worst case time complexity of the quick sort algorithm for sorting $n$ ( $\geq$ 2) numbers? In the recurrence equations given in the options below, $c$ is a constant. $T(n) = 2 T (n/2) + cn$ $T(n) = T ( n - 1) + T(1) + cn$ $T(n) = 2T ( n - 1) + cn$ $T(n) = T (n/2) + cn$
17 votes
6 answers
10
An array of $25$ distinct elements is to be sorted using quicksort. Assume that the pivot element is chosen uniformly at random. The probability that the pivot element gets placed in the worst possible location in the first round of partitioning (rounded off to $2$ decimal places) is ________
52 votes
11 answers
11
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
42 votes
9 answers
12
Let $G_1=(V,E_1)$ and $G_2 =(V,E_2)$ be connected graphs on the same vertex set $V$ with more than two vertices. If $G_1 \cap G_2= (V,E_1\cap E_2)$ is not a connected graph, then the graph $G_1\cup G_2=(V,E_1\cup E_2)$ cannot have a cut vertex must have a cycle must have a cut-edge (bridge) has chromatic number strictly greater than those of $G_1$ and $G_2$
4 votes
3 answers
13
Huffman tree is constructed for the following data :$\{A,B,C,D,E\}$ with frequency $\{0.17,0.11,0.24,0.33\ \text{and} \ 0.15 \}$ respectively. $100\ 00\ 01101$ is decoded as $BACE$ $CADE$ $BAD$ $CADD$
5 votes
3 answers
14
What is the complexity of the following code? sum=0; for(i=1;i<=n;i*=2) for(j=1;j<=n;j++) sum++; Which of the following is not a valid string? $O(n^2)$ $O(n\log\ n)$ $O(n)$ $O(n\log\ n\log\ n)$
0 votes
3 answers
15
Match the following with respect to algorithm paradigms: ... iii, b-i, c-ii, d-iv a-ii, b-i, c-iv, d-iii a-ii, b-i, c-iii, d-iv a-iii, b-ii, c-i, d-iv
2 votes
5 answers
16
Merge sort uses : Divide-and-conquer Backtracking Heuristic approach Greedy approach
2 votes
3 answers
17
Given two sorted list of size '$m$' and '$n$' respectively. The number of comparisons needed in the worst case by the merge sort algorithm will be : $m^{*}n$ minimum of $m, n$ maximum of $m, n$ $m+n-1$
3 votes
3 answers
18
For constants $a \geq 1$ and $b>1$, consider the following recurrence defined on the non-negative integers: $T(n) = a \cdot T \left(\dfrac{n}{b} \right) + f(n)$ Which one of the following options is correct about the recurrence $T(n)$? If $f(n)$ is $n \log_2(n)$, then $T(n)$ ... $\Theta(n^{\log_b(a)})$ If $f(n)$ is $\Theta(n^{\log_b(a)})$, then $T(n)$ is $\Theta(n^{\log_b(a)})$
0 votes
2 answers
19
What is the worst-case number of arithmetic operations performed by recursive binary search on a sorted array of size $n$? $\Theta ( \sqrt{n})$ $\Theta (\log _2(n))$ $\Theta(n^2)$ $\Theta(n)$
28 votes
7 answers
20
We have a binary heap on $n$ elements and wish to insert $n$ more elements (not necessarily one after another) into this heap. The total time required for this is $\Theta(\log n)$ $\Theta(n)$ $\Theta(n\log n)$ $\Theta(n^2)$
4 votes
5 answers
21
Consider the following recurrence relation. $T\left ( n \right )=\left\{\begin{array} {lcl} T(n/2)+T(2n/5)+7n & \text{if} \; n>0\\1 & \text{if}\; n=0 \end{array}\right.$ Which one of the following options is correct? $T(n)=\Theta (n^{5/2})$ $T(n)=\Theta (n\log n)$ $T(n)=\Theta (n)$ $T(n)=\Theta ((\log n)^{5/2})$
19 votes
5 answers
22
Consider the program below: #include <stdio.h> int fun(int n, int *f_p) { int t, f; if (n <= 1) { *f_p = 1; return 1; } t = fun(n-1, f_p); f = t + *f_p; *f_p = t; return f; } int main() { int x = 15; printf("%d/n", fun(5, &x)); return 0; } The value printed is: $6$ $8$ $14$ $15$
1 vote
2 answers
23
Define $R_n$ to be the maximum amount earned by cutting a rod of length $n$ meters into one or more pieces of integer length and selling them. For $i>0$, let $p[i]$ denote the selling price of a rod whose length is $i$ meters. Consider the array of prices: ... $R_7$? $R_7=18$ $R_7=19$ $R_7$ is achieved by three different solutions $R_7$ cannot be achieved by a solution consisting of three pieces
1 vote
2 answers
24
Show how to sort $n$ integers in the range $0$ to $n^3-1$ in $O(n)$ time.
1 vote
3 answers
25
Let $A$ be an array of $31$ numbers consisting of a sequence of $0$’s followed by a sequence of $1$’s. The problem is to find the smallest index $i$ such that $A[i]$ is $1$ by probing the minimum number of locations in $A$. The worst case number of probes performed by an optimal algorithm is $2$ $4$ $3$ $5$
0 votes
2 answers
26
Let $G$ be a graph with $n$ vertices and $m$ edges.What is the tightest upper bound on the running time of Depth First Search of $G$, when $G$ is represented using adjacency matrix? $O(n)$ $O(m+n)$ $O(n^2)$ $O(mn)$
0 votes
3 answers
27
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
0 votes
3 answers
28
A hash function $f$ defined as $f(\text{key})=\text{key mod }7$, with linear probing, insert the keys $37,38,72,48,98,11,56$ into a table indexed from $11$ will be stored in the location $3$ $4$ $5$ $6$
0 votes
3 answers
29
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 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 $\theta (n^{2})$
0 votes
2 answers
30
Consider the undirected graph below: Using Prim's algorithm to construct a minimum spanning tree starting with node $a$ ... $(a,b), (g,h), (g,f), (c,f), (c,i), (f,e), (b,c), (d,e)$
0 votes
2 answers
31
Match $\text{list I}$ with $\text{List II}$ ... : $A-I, B-III, C-IV, D-II$ $A-III, B-I, C-IV, D-II$ $A-III, B-I, C-II, D-IV$ $A-I, B-III, C-II, D-IV$
31 votes
7 answers
32
$\displaystyle \sum_{1\leq k\leq n} O(n)$, where $O(n)$ stands for order $n$ is: $O(n)$ $O(n^2)$ $O(n^3)$ $O(3n^2)$ $O(1.5n^2)$
0 votes
3 answers
33
Consider the following C code segment: int Ls Prime(n) { int i,n; for(i=2;i<=sqrt(n);i++) if(n%i ==0) { printf( NOT Prime.\n ); return 0; } return 1; } Let $T(n)$ denote the number of times the for loop is executed by the program on input $n.$ ... $T(n) = O(\sqrt{n})$ and $T(n) = \Omega (1)$ $T(n) = O(n)$ and $T(n) = \Omega (\sqrt{n})$ None of these
1 vote
2 answers
34
The most efficient algorithm for finding the number of connected components in a $n$ undirected graph on $n$ vertices and $m$ edges has time complexity $\Theta (n)$ $\Theta (m)$ $\Theta (m+n)$ $\Theta (mn)$
2 votes
1 answer
35
Let $A$ be array of $n$ integers that is not assumed to be sorted. You are given a number $x$. The aim is to find out if there are indices $k,\: l$ and $m$ such that $A[k] + A[l] + A[m] = x$. Design an algorithm for this problem that works in time $O(n^2)$.
0 votes
3 answers
36
Consider the following undirected graph with edge weights as shown: The number of minimum-weight spanning trees of the graph is ___________.
2 votes
3 answers
37
Consider the following three functions. $f_1=10^n\; f_2=n^{\log n}\;f_3=n^{\sqrt {n}}$ Which one of the following options arranges the functions in the increasing order of asymptotic growth rate? $f_3, f_2, f_1$ $f_2, f_1, f_3$ $f_1, f_2,f_3$ $f_2, f_3, f_1$
4 votes
3 answers
38
Let $G$ be a connected undirected weighted graph. Consider the following two statements. $S_1$: There exists a minimum weight edge in $G$ which is present in every minimum spanning tree of $G$. $S_2$: If every edge in $G$ has distinct weight, then $G$ has a unique minimum spanning ... $S_1$ is true and $S_2$ is false $S_1$ is false and $S_2$ is true Both $S_1$ and $S_2$ are false
2 votes
2 answers
39
Consider the string $\textrm{abbccddeee}$. Each letter in the string must be assigned a binary code satisfying the following properties: For any two letters, the code assigned to one letter must not be a prefix of the code assigned to the other letter. For any two letters ... assignments which satisfy the above two properties, what is the minimum length of the encoded string? $21$ $23$ $25$ $30$
0 votes
1 answer
40
Consider a $\textit{dynamic}$ hashing approach for $4$-bit integer keys: There is a main hash table of size $4$. The $2$ least significant bits of a key is used to index into the main hash table. Initially, the main hash table entries are empty. Thereafter, when more keys are hashed into it, to resolve ... in decimal notation)? $5,9,4,13,10,7$ $9,5,10,6,7,1$ $10,9,6,7,5,13$ $9,5,13,6,10,14$
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