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ISI 2019 | PCB CS | Question: 1
Let $A$ be a sorted array containing $n$ distinct integers, such that, for all $1 \leq i<j \leq n$, we have $A[i]<A[j]$. Note that the integers stored in the array $A$ ... time of the algorithm should be asymptotically better than $O(n)$. Prove the correctness of your algorithm and state its asymptotic time complexity.

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ISI 2019 | PCB CS | Question: 2
Let $K_{n}$ denote the complete graph on $n$ vertices, with $n \geq 3$, and let $u, v, w$ be three distinct vertices of $K_{n}$. Determine the number of distinct paths from $u$ to $v$ that do not contain the vertex $w$.

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ISI 2019 | PCB CS | Question: 3
When we add a pair of two-bit binary numbers, say $a b$ and $c d$, we get a number of at most three bits, say $p q r$. Using standard operators of Boolean algebra, namely $\operatorname{AND}\;(\wedge), \operatorname{OR}\; (\vee)$ and $\operatorname{NOT}\;(\neg)$, derive the Boolean expressions of $p, q$ and $r$ in terms of $a, b, c$ and $d$.

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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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5

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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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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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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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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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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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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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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12

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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13

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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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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15

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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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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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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18

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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19

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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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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