GATE CSE 2006

## GATE CSE

Consider the following C-program fragment in which i, j and n are integer variables.
for (i = n, j = 0; i > 0; i /= 2

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In a binary max heap containing n numbers, the smallest element can be found
in time

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Which one of the following in place sorting algorithms needs the minimum
number of swaps?

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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 algorit

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A 3-ary max heap is like a binary max heap, but instead of 2 children, nodes have 3
children. A 3-ary heap can be repres

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Given two arrays of numbers a1,......,an and b1,......, bn where each number is 0 or 1,
the fastest algorithm to find th

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A 3-ary max heap is like a binary max heap, but instead of 2 children, nodes have 3
children. A 3-ary heap can be repres

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Consider the following recurrence:
$$T\left( n \right){\rm{ }} = {\rm{ 2T(}}\left\lceil {\sqrt n } \right\rceil {\rm{) +

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The median of n elements can be found in O(n) time. Which one of the following is correct about the complexity of quick

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Consider a weighted complete graph G on the vertex set {v1, v2, ..vn} such that the weight of the edge (vi, vj) is $$2|i

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To implement Dijkstra’s shortest path algorithm on unweighted graphs so that it runs in linear time, the data structure

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Consider the following graph:
Which one of the following cannot be the sequence of edges added, in that order, to a min

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Let S be an NP-complete problem and Q and R be two other problems not known to be in NP. Q is polynomial time reducible

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Let SHAM3 be the problem of finding a Hamiltonian cycle in a graph G = (V, E) with |V| divisible by 3 and DHAM3 be th

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Consider the following grammar.
$$\eqalign{
& S \to S*E \cr
& S \to E \cr
& E \to F + E \cr
&a

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Which one of the following grammars generates the following language?
$$L = \left( {{a^i}{b^j}|i \ne j} \right)$$

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In the correct grammar of the previous question, what is the length of the derivation (number of steps starring from S)

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Consider the following grammar:
$$\eqalign{
& S \to FR \cr
& R \to *S\,|\,\varepsilon \cr
& F \to

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Consider the following translation scheme.
$$\eqalign{
& S \to ER \cr
& R \to *E\left\{ {pr{\mathop{\rm in

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Consider the following C code segment.
for (i = 0; i < n; i++)
{
for (j=0; j < n; j++)
{

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Consider the diagram shown below where a number of LANs are connected by (transparent) bridges. In order to avoid packet

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Consider the diagram shown below where a number of LANs are connected by (transparent) bridges. In order to avoid packet

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Station A uses 32 byte packets to transmit messages to Station B using a sliding window protocol. The round trip delay b

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Station A needs to send a message consisting of 9 packets to Station B using a sliding window (window size 3) and go-bac

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For which one of the following reasons does Internet Protocol (IP) use the time-to-live (TTL) field in the IP datagram h

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Two computers C1 and C2 are configured as follows. C1 has IP address 203.197.2.53 and netmask 255.255.128.0. C2 has IP a

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Given two three bit number $${a_2}{a_1}{a_0}$$ and $${b_2}{b_1}{b_0}$$ and $$c,$$ the carry in the function that repres

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A $$CPU$$ has a cache with block size $$64$$ bytes. The main memory has $$k$$ banks, each bank being $$c$$ bytes wide. C

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Consider two cache organization: The first one is $$32$$ $$KB$$ $$2$$-way set associate with $$32$$-byte block size. The

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Consider two cache organization: The first one is $$32$$ $$KB$$ $$2$$-way set associate with $$32$$-byte block size. The

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A CPU has five stages pipeline and runs at $$1$$ $$GHz$$ frequency. Instruction fetch happens in the first stage of the

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A $$CPU$$ has $$24$$-bit instructions. A program starts at address $$300$$ (in decimal). Which one of the following is a

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Consider the following program segment. Here R1, R2 and R3 are the general purpose registers.
Assume that the content

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Consider the following program segment. Here R1, R2 and R3 are the general purpose registers.
Assume that the content

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Consider the following program segment. Here R1, R2 and R3 are the general purpose registers.
Assume that the content

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An implementation of a queue Q, using two stacks S1 and S2, is given below:
void insert(Q, X){
push(S1, X);
}
void de

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The following function computes the value of mCn correctly for all legal values m and n (m≥1,n≥0 and m>n)
int func(

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A scheme for storing binary trees in an array X is as follows. Indexing of X starts at 1 instead of 0. the root is store

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In a binary tree, the number of internal nodes of degree 1 is 5, and the number of internal nodes of degree 2 is 10. The

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Suppose that we have numbers between 1 and 100 in a binary search tree and want to search for the number 55. Which of th

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Which of the following sequences of array elements forms a heap?

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An array X of n distinct integers is interpreted as a complete binary tree. The index of the first element of the array

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An array X of n distinct integers is interpreted as a complete binary tree. The index of the first element of the array

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An array X of n distinct integers is interpreted as a complete binary tree. The index of the first element of the array

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Consider the depth-first-search of an undirected graph with 3 vertices P, Q, and R. Let discovery time d(u) represent th

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Let T be a depth first search tree in an undirected graph G. Vertices u and v are leaves of this tree T. The degrees of

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Let T be a depth-first search tree in an undirected graph G. Vertices u and v are
leaves of this tree T. The degrees of

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Consider the relation enrolled (student, course) in which (student, course ) is the primary key, and the relation Paid (

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The following functional dependencies are given :
$$\eqalign{
& AB \to CD,\,AF \to D,\,\,DE \to F, \cr
& C

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Consider the relation account (customer, balance) where customer is a primary key and there are no null values. We would

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Consider the relation "enrolled (student, course)" in which (student, course) is the primary key, and the relation "paid

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Consider a database with three relation instances shown below. The primary keys for the Drivers and Cars relation are Di

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Consider a database with three relation instances shown below. The primary keys for the Drivers and Cars relation are Di

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Consider the relations r1(P, Q, R) and r2(R, S, T) with primary keys P and R respectively. The relation r1 contains 2000

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Which of the following relational query languages have the same expressive power?
I) Relational algebra
II) Tuple relati

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Consider a Boolean function $$f(w, x, y, z).$$ Suppose that exactly one of its inputs is allowed to change at a time. If

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Consider the circuit above. Which one of the following options correctly represents $$f(x,y,z)?$$

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You are given a free running clock with a duty cycle of $$50$$% and a digital waveform $$f$$ which changes only at the n

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Consider the circuit in the diagram. The $$ \oplus $$ operator represents $$EX$$-$$OR.$$ The $$D$$ flip-flops are initia

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Which one of the first order predicate calculus statements given below correctly expresses the following English stateme

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Consider the following propositional statements:
$${\rm P}1:\,\,\left( {\left( {A \wedge B} \right) \to C} \right) \equ

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In a certain town, the probability that it will rain in the afternoon is known to be 0.6. Moreover, meteorological data

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A logical binary relation $$ \odot $$, is defined as follows:
Let ~ be the unary negation (NOT) operator, with higher

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When a coin is tossed, the probability of getting a Head is p, 0 < p < 1. Let N be the random variable denoting th

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Consider the following first order logic formula in which $$R$$ is a binary relation symbol.
$$\forall x\forall y\left(

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Let $$P, Q$$, and $$R$$ be sets. Let $$\Delta $$ denote the symmetric difference operator defined as $$P\Delta Q = \left

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Let E, F and G be finite sets.
Let $$X = \,\left( {E\, \cap \,F\,} \right)\, - \,\left( {F\, \cap \,G\,} \right)$$
and

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A relation $$R$$ is defined on ordered pairs of integers as follows:
$$\left( {x,y} \right)R\left( {u,v} \right)\,if\,x

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Let $$X,. Y, Z$$ be sets of sizes $$x, y$$ and $$z$$ respectively. Let $$W = X x Y$$ and $$E$$ be the set of all

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The set $$\left\{ {1,\,\,2,\,\,3,\,\,5,\,\,7,\,\,8,\,\,9} \right\}$$ under multiplication modulo 10 is not a group. Give

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For the set $$N$$ of natural numbers and a binary operation $$f:N \times N \to N$$, an element $$z \in N$$ is called an

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Given a set of elements N = {1, 2, ....., n} and two arbitrary subsets $$A\, \subseteq \,N\,$$ and $$B\, \subseteq \,N\,

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Let S = {1, 2, 3,....., m} , m > 3. Let $${X_1},\,....,\,{X_n}$$ be subsets of S each of size 3. Define a function f

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Consider the set S = {a, b, c, d}. Consider the following 4 partitions $$\,{\pi _1},\,{\pi _2},\,{\pi _3},\,{\pi _4}$$ o

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If all the edge weights of an undirected graph are positive, then any subject of edges that connects all the vertices an

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Consider a weighted complete graph $$G$$ on the vertex set $$\left\{ {{v_1},\,\,\,{v_2},....,\,\,\,{v_n}} \right\}$$ suc

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Consider the polynomial $$P\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3},$$ where $${a_i} \ne 0,\forall i

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For each elements in a set of size $$2n$$, an unbiased coin in tossed. The $$2n$$ coin tosses are independent. An elemen

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What is the cardinality of the set of integers $$X$$ defined below?
$$X = $$ {$$n\left| {1 \le n \le 123,\,\,\,\,\,n} \r

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The $${2^n}$$ vertices of a graph $$G$$ correspond to all subsets of a set of size $$n$$, for $$n \ge 6$$. Two vertices

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The $${2^n}$$ vertices of a graph $$G$$ correspond to all subsets of a set of size $$n$$, for $$n \ge 6$$. Two vertices

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Consider the undirected graph $$G$$ defined as follows. The vertices of $$G$$ are bit strings of length $$n$$. We have a

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$$F$$ is an $$n$$ $$x$$ $$n$$ real matrix. $$b$$ is an $$n$$ $$x$$ $$1$$ real vector. Suppose there are two $$n$$ $$x$$

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What are the eigen values of the matrix $$P$$ given below?
$$$P = \left( {\matrix{
a & 1 & 0 \cr
1 &

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Consider three $$CPU$$-intensive process, which require $$10,20$$ and $$30$$ time units and arrive at times $$0,2$$ and

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Consider three processes (process id $$0,1,2,$$ respectively) with compute time bursts $$2, 4,$$ and $$8$$ time units. A

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Consider three processes, all arriving at time zero, with total execution time of $$10,20,$$ and $$30$$ units, respectiv

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The atomic fetch-and-set x, y instruction unconditionally sets the memory location x to 1 and fetches the old value of x

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Barrier is a synchronization construct where a set of processes synchronizes globally i.e. each process in the set arriv

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Barrier is a synchronization construct where a set of processes synchronizes globally i.e. each process in the set arriv

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A Computer system supports $$32$$-bit virtual addresses as well as $$32$$-bit physical addresses. Since the virtual addr

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Consider the following snapshot of a system running n processes. Process i is
holding xi instances of a resource R, for

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Consider this C code to swap two integers and these five statements:
void swap(int *px, int *py)
{
*px = *px - *py

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If $$s$$ is a string over $${\left( {0 + 1} \right)^ * }$$ then let $${n_0}\left( s \right)$$ denote the number of $$0'$

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Consider the regular language $$L = {\left( {111 + 11111} \right)^ * }.$$ The minimum number of states in any $$DFA$$ ac

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Let $${L_1} = \left\{ {{0^{n + m}}{1^n}{0^m}\left| {n,m \ge 0} \right.} \right\},$$
$$\,\,\,{L_2} = \left\{ {{0^{n + m}

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Consider the following statements about the context-free grammar
$$G = \left\{ {S \to SS,\,S \to ab,\,S \to ba,\,S \to \

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For $$s \in {\left( {0 + 1} \right)^ * },$$ let $$d(s)$$ denote the decimal value of $$s(e. g.d(101)=5)$$
Let $$L = \lef

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Let $${L_1}$$ be a regular language, $${L_2}$$ be a deterministic context-free language and $${L_3}$$ a recursively enum

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