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31
ISI 2019 | PCB CS | Question: 4
In a binary tree $T$, for a node $v$, the $\text{LEFT-HEIGHT} (v)$ is the length of the longest path from $v$ to any leaf in the left subtree of $v$. If $v$ has no left child then $\text{LEFT-HEIGHT} (v)=0$ ... Design an efficient algorithm that, given a binary tree, enumerates all the nodes which are properly balanced.
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32
ISI 2019 | PCB CS | Question: 5
Consider a stack machine where the only available workspace is a stack whose elements are unsigned integers. We will denote the configuration of the stack by a sequence. For example $[a, b, c, d]$ represents a stack with $a$ being the ... $a b$.
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isi2019-pcb-cs
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33
ISI 2019 | PCB CS | Question: 6
Consider the alphabet $\Sigma=\{0,1,2, \ldots, 9, \#\}$, and the language of strings of the form $x \# y \# z$, where $x, y$ and $z$ are strings of digit such that when viewed as numbers, satisfy the equation $x+y=z$. For example, the string $123 \# 45 \# 168$ is in this language because $123+45=168$. Is this language regular? Justify your answer.
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isi2019-pcb-cs
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34
ISI 2019 | PCB CS | Question: 7
Recall that in go-back-$N$ protocol, the transmitting window size is $N$ and the receiver window size is $1.$ Consider a pipelined, reliable transport protocol that uses go-back-$N$ with cumulative acknowledgment. Assume that the timeouts trigger retransmissions (but note ... $1 \mathrm{~Gb} / \mathrm{s}$. The bottleneck link rate is $2 \mathrm{~Gb} / \mathrm{s}$.
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isi2019-pcb-cs
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35
ISI 2019 | PCB CS | Question: 8
Let us assume that a disk scheduling algorithm is applied on a storage disk to access several cylinders (numbered as $0,1, \ldots, n$ ... first the disk head accesses all the cylinders while moving toward cylinder $0$ and then the disk head moves toward the other end.
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isi2019-pcb-cs
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36
ISI 2019 | PCB CS | Question: 9
Consider a byte addressable memory with $16$ bit addresses and a $2$- way set associative $\mathrm{L} 1$ cache of size $8 \mathrm{ kB}$ (kilobyte). Each cache line is $4$ words long. A process sequentially accesses the following memory ... replacement policy is used, indicate whether the cache access will result in a hit or a miss for each of the above addresses.
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isi2019-pcb-cs
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37
ISI 2019 | PCB CS | Question: 10
Let $R$ be a relation with functional dependencies $\mathcal{F}$. For any subset of attributes $X \subseteq R$, the closure of $X$ is defined as the set $ X^{+}=\{A \in R \mid X \rightarrow A \text { holds with respect to } \mathcal{F}\} . $ For two non-empty ... each of the following statements: $\left(Y^{+} Z\right)^{+}=(Y Z)^{+}$ $(Y Z)^{+}=Y^{+} Z^{+}$
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isi2019-pcb-cs
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38
ISI 2019 | PCB CS | Question: 11
Consider an array of length $n$ consisting only of positive and negative integers. Design an algorithm to rearrange the array so that all the negative integers appear before all the positive integers, using $O(n)$ time and only constant amount of extra space.
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isi2019-pcb-cs
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39
ISI 2019 | PCB CS | Question: 12
You can climb up a staircase of $n$ stairs by taking steps of one or two stairs at a time. Formulate a recurrence relation for counting $a_{n},$ the number of distinct ways in which you can climb up the staircase. Mention the boundary conditions for your recurrence relation. Find a closed form expression for $a_{n}$ by solving your recurrence.
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Aug 9
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isi2019-pcb-cs
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40
ISI 2019 | PCB CS | Question: 13
An $n$-variable Boolean function $f:\{0,1\}^{n} \rightarrow\{0,1\}$ is called symmetric if its value depends only on the number of $1 \text{'s}$ in the input. Let $\sigma_{n}$ denote the number of such functions. Calculate the value of $\sigma_{4}$. Derive an expression for $\sigma_{n}$ in terms of $n$.
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Aug 9
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isi2019-pcb-cs
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41
ISI 2019 | PCB CS | Question: 14
Let the valid moves along a staircase be $U$ (one step up) and $D$ (one step down). For example, the string $s=U U D U$ represents the sequence of moves as two steps up, then one step down, and then again one step up. Suppose a person ... to the base of the staircase after the final step. Show that $L$ is not regular. Write a context free grammar for accepting $L$.
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Aug 9
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isi2019-pcb-cs
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42
ISI 2019 | PCB CS | Question: 15
Consider a max-heap of $n$ distinct integers, $n \geq 4$, stored in an array $\mathcal{A}[1 \ldots n]$. The second minimum of $\mathcal{A}$ is the integer that is less than all integers in $\mathcal{A}$ except the minimum of $\mathcal{A}$. Find all possible array indices of $\mathcal{A}$ in which the second minimum can occur. Justify your answer.
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isi2019-pcb-cs
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43
ISI 2019 | PCB CS | Question: 16
The following function computes an array $\textsf{SPF},$ where, for any integer $1<1<1000, \textsf{SPF[i]}$ is the smallest prime factor of $\textsf{i}.$ For example, $\textsf{SPF[6]}$ is $2,$ and $\textsf{SPF[11]}$ is $11.$ There are five missing parts in the following code, ... ){ /* Blank 4 */ if (SPF[j] == j) { SPF[j] = _ _ _ _ _; /* Blank 5 */ } } } } }
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isi2019-pcb-cs
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44
ISI 2019 | PCB CS | Question: 17
A context switch from a process $P_{old}$ to a process $P_{new }$ consists of the following steps: - Step I: saving the context of $P_{old};$ Step II: running the scheduling algorithm to pick $P_{new};$ ... each process requires exactly one $\mathrm{CPU}$ burst of $20 \mathrm{~ms}$ and no $\mathrm{I} / \mathrm{O}$ burst.
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Aug 9
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isi2019-pcb-cs
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45
ISI 2019 | PCB CS | Question: 18
Consider a $5$-stage instruction pipeline. The stages and the corresponding stage delays are given below. Instruction Stage delay Fetch instruction (FI) 3 ns Decode instruction (DI) 4 ns Fetch operand (FO) 7 ns Execute instruction (EI) 10 ns Write ... flow through the pipeline stages in this processor. Calculate the time (in ns) needed to execute the program.
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isi2019-pcb-cs
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46
ISI 2019 | PCB CS | Question: 19
The data link layer uses a fixed-size sliding window protocol, where the window size for the connection is equal to twice the bandwidth-delay product of the network path. Consider the following three scenarios, in each of which only the given parameter changes as specified ( ... the round trip time $R$ increases to $1.8 R;$ the window size $W$ decreases to $W / 3.$
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isi2019-pcb-cs
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47
ISI 2019 | PCB CS | Question: 20
Consider two $n \times 1$ vectors u and v, stored as tables $\mathrm{U(ind, val)}$ and $\mathrm{V(ind, val)}$ with the same schema. A row $\left(i, u_{i}\right)$ of table $\mathrm{U}$ specifies that the $i$-th element of vector u ... a relational algebra expression or an $\text{SQL}$ query to compute the sum u $+$ v of the two vectors u and v. Explain your solution.
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isi2019-pcb-cs
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48
ISI 2021 | PCB CS | Question: 2
Let $P$ be a set of $n$ real numbers. For any two real numbers $a$ and $b$ $(a<b),$ define $ R(a, b)=|\{x \in P \mid a \leq x \leq b\}| . $ Design a suitable data structure $\mathcal{D}$ to store $P$ ... your data structure $\mathcal{D}$. Justify that $O(\log n)$ time is sufficient for reporting $R(a, b)$ using your data structure $\mathcal{D}$.
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Aug 8
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isi2021-pcb-cs
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49
ISI 2021 | PCB CS | Question: 3
Let $G$ be a simple undirected graph having $n$ vertices with the property that the average of the degrees of the vertices in $G$ is at least $4.$ ... $\textsf{deg(x)}$ denotes the degree of the vertex $\textsf{x}$.
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isi2021-pcb-cs
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50
ISI 2021 | PCB CS | Question: 4
Let $A$ be a matrix of size row $\times$ col. $A$ has to be filled in a spiral clockwise fashion with successive integers from $1,2, \ldots$, row $\times$ col starting from the top left corner. For example, a $3 \times 4$ ... (int *)); for(i=0;i<row;i++){ A[i] = (int *)calloc(col,sizeof(int)); } spiralFill(A,row,col); }
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Programming
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isi2021-pcb-cs
programming
programming-in-c
functions
matrix
descriptive
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51
ISI 2021 | PCB CS | Question: 5
Given a set $S$ of $n$ integers and a constant $k$ (positive integer), design an algorithm for finding a subset of $S$ of maximum possible size such that the sum of each pair of integers in this subset is not divisible by $k$. Note that full credit will be given for a polynomial (in $n$ ) time algorithm.
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isi2021-pcb-cs
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52
ISI 2021 | PCB CS | Question: 6
A ternary variable can assume the values $0,1$ or $2,$ and can be coded with two binary bits as $00,01$ and $10$ respectively. A ternary full-adder has three ternary digits $X, Y$ and a carry-in $C_{in}$ as inputs, and produces the ternary ... and $C_{o}=(1)_{3}.$ Design a circuit for this ternary full adder using binary gates as well as binary half and full adders.
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isi2021-pcb-cs
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53
ISI 2021 | PCB CS | Question: 7.a
Consider the single precision (i.e., $32$-bit) floating point representation of numbers in the normalized form where $8$ bits are used for the exponent with the bias of $127.$ What is the binary representation of $-10.4$ in the above form? The steps followed to arrive at the ... between $2^{-18}$ and $2^{-17}$ (i.e., excluding $2^{-18}$ and $\left.2^{-17}\right)?$
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CO and Architecture
Aug 8
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Lakshman Patel RJIT
54
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isi2021-pcb-cs
descriptive
co-and-architecture
number-representation
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54
ISI 2021 | PCB CS | Question: 8
Consider the following language $L$ over the alphabet $\Sigma=\{a, b, c\}$. $ L=\left\{a^{i} b^{j} c^{k} \mid i, j, k \geq 0 \text {, and if } i=1 \text { then } j=k\right\} $ Show that $L$ is not regular. Show that $L$ is a context-free language.
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isi2021-pcb-cs
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ISI 2021 | PCB CS | Question: 9
In relational algebra, for any pair of relations $R_{1}$ and $R_{2}$, the standard division operation is denoted by $\div$ ...
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Databases
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Lakshman Patel RJIT
57
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isi2021-pcb-cs
descriptive
databases
relational-algebra
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56
ISI 2021 | PCB CS | Question: 10
Consider sending a message of $10,000$ bits from the source node $S$ to the destination node $D$ passing through the two routers $R 1$ and $R 2$ ... -end latency of the message when it is broken into $10$ packets each of size $1000$ bits, and then transmitted to the destination.
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isi2021-pcb-cs
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57
ISI 2021 | PCB Mathematics | Question: 1
Consider a standard balance with two pans where weights can only be placed on the left pan, and the object to be weighed on the right pan. Find the minimum number of weights required to weigh any object whose weight in grams could be any integer ranging from $1$ to $127$. Give precise argument in favor of your answer.
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isi2021-pcb-mathematics
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58
ISI 2021 | PCB Mathematics | Question: 2
Consider three real numbers $a \geq b \geq c>0$. If $\left(a^{x}-b^{x}-c^{x}\right)(x-2)>0$ for any rational number $x \neq 2$, show that $a, b$ and $c$ can be the lengths of the three sides of a triangle $A B C;$ $A B C$ is a right-angled triangle.
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isi2021-pcb-mathematics
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ISI 2021 | PCB Mathematics | Question: 3
Consider a two-player game between Alice and Bob, in which the players take turns to roll a fair six-faced die. Alice rolls the die first. Then Bob rolls the die and he wins if he gets the same outcome as Alice. Otherwise, Alice rolls the ... first three rolls (two by Alice and one by Bob) of the die. What is the probability that Alice will win the game?
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isi2021-pcb-mathematics
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60
ISI 2021 | PCB Mathematics | Question: 5
Let $G$ be a group generated by $a$ and $b$ such that $\operatorname{ord}(a)=n, \operatorname{ord}(b)= 2$ and $a b=b a^{-1}$, where $n$ is a positive integer, $b \notin\langle a\rangle$ and ord $(x)$ denotes the order of the element $x$. Prove ... $H$ be a cyclic subgroup of $\langle a\rangle$. Show that $H$ is a normal subgroup of $G$.
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