GATE CS Algorithms questions

In a directed acyclic graph with a source vertex $\textsf{s}$, the $\textit{quality-score}$ of a directed path is defined to be the product of the weights of the edges on the path. Further, for a vertex $v$ other than $\textsf{s}$, the quality-score of $v$ is defined to be the maximum among the quality-scores of all the paths from $\textsf{s}$ to …

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

Consider a double hashing scheme in which the primary hash function is $h_1(k)= k \text{ mod } 23$, and the secondary hash function is $h_2(k)=1+(k \text{ mod } 19)$. Assume that the table size is $23$. Then the address returned by probe $1$ in the probe sequence (assume that the probe sequence begins at probe $0$) for key value $k=90$ …

Let $G = (V, E)$ be a weighted undirected graph and let $T$ be a Minimum Spanning Tree (MST) of $G$ maintained using adjacency lists. Suppose a new weighed edge $(u, v) \in V \times V$ is added to $G$. The worst case time complexity of determining if $T$ is still an MST of the resultant graph is

Let $G = (V,E)$ be a directed, weighted graph with weight function $w: E \rightarrow \mathbb{R}$. For some function $f: V \rightarrow \mathbb{R}$, for each edge$(u,v)\in E$, define ${w}'(u,v)$ as $w(u,v)+f(u)-f(v)$. Which one of the options completes the following sentence so that it is TRUE? “The shortest paths in $G$ under $w$ are shortest …

Consider the following C functions. int tob (int b, int* arr) { int i; for (i = 0; b>0; i++) { if (b%2) arr [i] = 1; else arr[i] = 0; b = b/2; } return (i); } int pp(int a, int b) { int arr[20]; int i, tot = 1, ex, len; ex = a; len = tob(b, arr); for (i=0; i<len ; i++) { if (arr[i] ==1) tot = tot * ex; ex= ex*ex; } return (tot) ; } The value returned …

Consider a graph $G = (V,E)$, where $V = \{v_1,v_2, \dots ,v_{100}\}$, $E = \{(v_i,v_j) \mid 1\leq i < j \leq 100\}$, and weight of the edge $(v_i,v_j)$ is $\mid i – j \mid$. The weight of minimum spanning tree of $G$ is _________

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 ________

Consider a sequence of $14$ elements: $A=[-5, -10, 6, 3, -1, -2, 13, 4, -9, -1, 4, 12, -3, 0]$. The sequence sum $S(i,j) = \Sigma_{k=i}^j A[k]$. Determine the maximum of $S(i,j)$, where $0 \leq i \leq j <14$. (Divide and conquer approach may be used.) Answer: ___________

Consider the following C function. void convert (int n ) { if (n<0) printf{“%d”, n); else { convert(n/2); printf(“%d”, n%2); } } Which one of the following will happen when the function convert is called with any positive integer $n$ as argument?

There are $n$ unsorted arrays: $A_1, A_2, \dots, A_n$. Assume that $n$ is odd.Each of $A_1, A_2, \dots, A_n$ contains $n$ distinct elements. There are no common elements between any two arrays. The worst-case time complexity of computing the median of the medians of $A_1, A_2, \dots , A_n$ is

Consider the weights and values of items listed below. Note that there is only one unit of each item. …

Consider the following undirected graph $G$: Choose a value for $x$ that will maximize the number of minimum weight spanning trees (MWSTs) of $G$. The number of MWSTs of $G$ for this value of $x$ is ____.

Consider the following program written in pseudo-code. Assume that $x$ and $y$ are integers. Count (x, y) { if (y !=1 ) { if (x !=1) { print("*"); Count (x/2, y); } else { y=y-1; Count (1024, y); } } } The number of times that the $print$ statement is executed by the call $Count(1024, 1024)$ is _____

Let $G$ be a graph with $100!$ vertices, with each vertex labelled by a distinct permutation of the numbers $1, 2,\ldots, 100.$ There is an edge between vertices $u$ and $v$ if and only if the label of $u$ can be obtained by swapping two adjacent numbers in the label of $v$. Let $y$ denote the degree of a vertex in $G$, and $z$ denote the …

Let $G$ be a simple undirected graph. Let $T_D$ be a depth first search tree of $G$. Let $T_B$ be a breadth first search tree of $G$. Consider the following statements. No edge of $G$ is a cross edge with respect to $T_D$. (A cross edge in $G$ is between two nodes neither of which is an ancestor of the other in $T_D$). For every edge …

Assume that multiplying a matrix $G_1$ of dimension $p \times q$ with another matrix $G_2$ of dimension $q \times r$ requires $pqr$ scalar multiplications. Computing the product of $n$ matrices $G_1G_2G_3 \dots G_n$ can be done by parenthesizing in different ways. Define $G_i G_{i+1}$ as an explicitly computed pair for a given …

Consider the following table: …

Consider the following functions from positive integers to real numbers: $10$, $\sqrt{n}$, $n$, $\log_{2}n$, $\frac{100}{n}$. The CORRECT arrangement of the above functions in increasing order of asymptotic complexity is:

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 $A\left [i \right ]$ is $1$ by probing the minimum number of locations in $A$. The worst case number of probes performed by an optimal algorithm is ____________.