# computer science paper

Discussion in 'Computer Science and IT Students' started by GATE, May 19, 2006.

1. ### GATEGuest

Which of the following is an example of spooled device?

(a) A line printer used to print the output of a number of jobs.

(b) A terminal used to enter input data to a running program.

(c) A secondary storage device in a virtual memory sytem.

(d) A graphic display device.

* A ROM is sued to store the table for multiplication of two
8-bit unsigned integers. The size of ROM required is

(a) 256 ´ 16 (b) 64 K ´ 8 (c) 4 K ´ 16 (d) 64 K ´ 16

* A critical section is a program segment

(a) which should run in a certain specified amount of time

(c) where shared resources are accessed

(d) which must be enclosed by a pair of semaphore operations, P and V

* Which two of the following four regular expressions are
equivalent? (e is the empty string).

(i) (00) * (e + 0)

(ii) (00)*

(iii) 0*

(iv) 0(00)*

(a) (i) and (ii) (b) (ii) and (iii) (c) (i) and (iii) (d) (iii) and
(iv)

* Which of the following statements is false?

(a) The Halting problem of Turing machines is undecidable.

(b) Determining whether a context-free grammar is ambiguous is
undecidbale.

(c) Given two arbitrary context-free grammars G1 and G2
it is undecidable whether L(G1) = L(G2).
(d) Given two regular grammars G1 and G2 it is undecidable
whether L(G1) = L(G2).

* Let L Í S* where S = {a, b}. which of the following is true?

(a)
(b)
(c)
(d)

L = {x x has an equal number of a's and b' }s is regular

L = { na bn n ³ }1 is regular

L = {x x has more a's and b' }s is regular

L = { ma bn m ,³ n ³ }1 is regular

* Which of the following is false?

(a)

≈ ’
100n log = n log n
« ◊

log n = 0 (log log n)
∆ 100 ÷
(b)

(c)

If 0 < x < y then nx

= 0 (n )y

(d)

2n ¹ 0 (n )k

* Consider the following statements:

(i) First-in-first out types of computations are efficiently supported by
STACKS.

(ii) Implementing LISTS on linked lists is more efficient than
implementing LISTS
on an array for almost all the basic LIST operations.
(iii) Implementing QUEUES on a circular array is more efficient than
implementing QUEUES on a linear array with two indices.

(iv) Last-in-first-out type of computations are efficiently supported by
QUEUES.

(a) (ii) and (iii) are true (b) (i) and (ii) are true

(c) (iii) and (iv) are true (d) (ii) and (iv) are true

* Two dice are thrown simultaneously. The probability that at least one
of them will have 6 facing up is

(a) 1
36

(b) 1
3

(c) 25
36

(d) 11
36

* The formula used to compute an approximation for the second
derivative of a function f at a point X0 is.

(a) (f 0x

(c) (f 0x

+ h) + (f 0x
2

+ h) + 2 (f 0x
2h

- h)

) + (f 0x

- h)

(b) (f 0x

(d) (f 0x

+ h) - (f 0x
2h

+ h) - 2 (f 0x
2h

- h)

) + (f 0x

- h)

* Let Ax = b be a system of linear equations where A is an
m ´ n matrix and b is a m ´ 1 column vector and X is a n ´ 1
column vector of unknows. Which of the following is false?

(a) The system has a solution if and only if, both A and the augmented
matrix [A
b] have the same rank.
(b) If m < n and b is the zero vector, then the system has
infinitely many solutions.

(c) If m = n and b is non-zero vector, then the system has a unique
solution.
(d) The system will have only a trivial solution when m = n,
b is the zero vector and rank (A) = n.

*
An advantage of chained hash table (external hashing) over

(a) Worst case complexity of search operations is less?

(b) Space used is less (c) Deletion is easier

(d) None of the above

*
The pass numbers for each of the following activities
(i) object code generation
(ii) literals added to literal table
(iii) listing printed
(iv) address resolution of local symbols that occur in a
two pass assembler respectively are

(a) 1, 2, 1, 2 (b) 2, 1, 2, 1 (c) 2, 1, 1, 2 (d) 1, 2, 2, 2

(a) a batch operating system

BLOCKED

(b) an operating system with a preemptive scheduler

(c) an operating system with a non-preemptive scheduler

(d) a uni-programmed operating system.

* Let A and B be sets and let cA

and cB denote the complements of the sets A and
B. the set (a œ b) È(b-a) È(aÇb) is equal to.

(a) A È B (b) cA È cB

(c) A Ç B (d) cA Ç cB

*
Let X = {2,3,6,12,24}, Let £ be the partial order defined by
X £ Y if x divides y. Number of edge as in the Hasse diagram of
(X,£) is

(a) 3 (b) 4

(c) 9 (d) None of the above

* Suppose X and Y are sets and

X and

Y are their respective cardinalities. It is
given that there are exactly 97 functions from X to Y. from
this one can conclude that

(a)

X = 1, Y = 97

(b)

X = 97, Y = 1

(c)

X = 97, Y = 97

(d) None of the above

* Which of the following statements is false?

(a) The set of rational numbers is an abelian group under addition.

(b) The set of integers in an abelian group under addition.

(c) The set of rational numbers form an abelian group under
multiplication.
(d) The set of real numbers excluding zero in an abelian group under
multiplication.

* Number of machine cycles required for RET instruction in 8085
microprocessor is

(a) 1 (b) 2 (c) 3 (d) 5

1.14 In the balanced binary tree in Fig.1.14 given below, how many nodes
will become unbalanced when a node is inserted as a child of the node —g“?

a

b e
c d f
g

(a) 1 (b) 3 (c) 7 (d) 8

* Which of the following sequences denotes the post order
traversal sequence of the tree of question 1.14?

(a) f e g c d b a (b) g c b d a f e (c) g c d b f e a (d) f e d g c b
a

1.16 Relative mode of addressing is most relevant to writing

(a) coroutines (b) position œ independent code

(c) shareable code (d) interrupt handlers

*
Both‘s algorithm for integer multiplication gives worst
performance when the multiplier pattern is

(a) 101010 …..1010 (b) 100000 …..0001

(c) 111111 …..1111 (d) 011111 …..1110

2. Write in your answer book the correct or the most appropriate
answer to the following multiple choice questions by writing the
corresponding letter a, b, c or d against the sub-question number.

* Which of the following is false? Read Ù as AND, Ú as OR, ~ as NOT, ‰
as one way implication and « as two way implication.

(a)

(c)

(( x ® )y Ù x ) ® y

( x ® ( x Ú y ))

(b)

(d)

((- x ® )y Ù (- x Ù - )y ) ® x

(( x Ú )y « (- x ®- )y )

* Let R denotes the set of real numbers. Let f:R´R ‰ R´R be
a bijective function defined by f(x,y)=(x+y,x-y). the inverse function of
f is given by

(a)

-1f

≈ ’
( x, )y = 1 1,
∆ ÷
« x + y x - y ◊
(b)
-1f
( x, )y = ( x - ,y x + )y

(c)

-f 1 (

x y =, ) x

+ y x - y ’
,

1f -

x, y = 2»

x - y , 2

x + y ÿ⁄
∆ 2 2 ÷
(d) ( ) ( ) ( )
« ◊

* Let R be a non-empty relation on a collection of sets
defined by A R B if and only if A Ç B = f. Then, (pick the true
statement)

(a) R is reflexive and transitive

(b) R is symmetric and not transitive

(c) R is an equivalence relation

(d) R is not reflexive and not symmetric

* For the daisy chain scheme of connecting I/O devices, which
of the following statements is true?

(a) It gives non-uniform priority to various devices.

(b) It gives uniform priority to all devices.

(c) It is only useful for connecting slow devices to a processor device.

(d) It requires a separate interrupt pin on the processor for each
device.

* Consider the following floating-point number representation.

31 24 23 0

Exponent Mantissa

The exponent is in 2‘s complement representation and mantissa is
in the sign magnitude representation. The range of the
magnitude of the normalized numbers in this representation is

(a) 0 to 1 (b) 0.5 to 1

(c)

2-23

to 0.5

(d)

0.5 to (1- -2 23 )

* The probability that top and bottom cards of a randomly shuffled deck
are both aces in

(a) 4 ´ 4

(b) 4 ´ 3

(c) 4 ´ 3

(d) 4 ´ 4
52 52
52 52
52 51
52 51

* If L1 and L2 are context free languages and R a regular
set, one of the languages below is not necessarily a context free
language. Which one?

(a) L1, L2 (b) L1 Ç L2 (c) L1 Ç R (d) L1 È L2

*
Which one of the following is false?
(a) The set of all bijective functions on a finite set forms a
group under function composition.
(b) The set {1,2, ……., p œ1} forms a group under multiplication
mod p where p is a prime number.
(c) The set of all strings over a finite alphabet forms a group under
concatenation.

(d) A subset s ¹ f of G is a subgroup of the group <G, *>
if and only if for any pair of elements a, b Î s, a* b-1 Î s.

* Newton-Raphson iteration formula for finding 3

c , where c > 0 is,

(a)

xn +1

3x c
= 2 n + 3
3 2

(b)

32x c
= n - 3
xn +1 2
nx 3xn

2 3 3
=x xn + c
(d) x
2x c
= n -
(c)
n +1 2x3 n
n +1 23x
n

»
* The matrices q -

q ÿ » ÿ
and
cos sin a 0
… Ÿ … Ÿ commute under multiplication
sinq
cos q 0 b
⁄ ⁄

(a) if a = b or q = np, is an integer (b) always

(c) never (d) if a cos q ¹ b sin q

* The minimum number of interchanges needed to convert the array

89, 19, 40, 17, 12, 10, 2, 5, 7, 11, 6, 9, 70

(a) 0 (b) 1 (c) 2 (d) 3

* A binary search tree is generated by inserting in order the following
integers:

50, 15, 62, 5, 20, 58, 91, 3, 8, 37, 60, 24
The number of nodes in the left subtree and right subtree of the root
respectively is

(a) (4, 7) (b) (7, 4) (c) (8, 3) (d) (3, 8)

* Define a context free languages L £ {0,1} init (L) = {u/uv
Î L for some n in
{{0,1}} (in other words, init(L) is the set of prefixes of L)

Let L {w/w is nonempty and has on equal number of 0‘s and 1‘s

Then init (L) is

(a) the set of all binary strings with unequal number of 0‘s and 1‘s

(b) the set of all binary strings including the null string
(c) the set of all binary strings with exactly one more 0‘s
than the number of 1‘s or one more 1 than the number of 0‘s.

(d) None of the above

2.10. The grammar whose productions are
‰ if id then <stmt>
‰ if id then <stmt> else <stmt>
‰ id:=id

is ambiguous because

(a) the sentence

if a then if b then c:=d
(b) the left most and right most derivations of the sentence if a then if
b then c:=d

give rise top different parse trees

(c) the sentence
if a then if b then c:=d else c:=f has more than two parse trees

(d) the sentence
if a then if then c:=d else c:=f has two parse trees

* The correct matching for the following pairs is

(B) Location counter (2) Garbage collection

(C) Reference counts (3) Subroutine call

(a) A œ 3 B œ 4 C œ 1 D - 2 (b) A œ 4 B œ 3 C œ 1 D - 2

(c) A œ 4 B œ 3 C œ 2 D - 1 (d) A œ 3 B œ 4 C œ 2 D - 1

2.12. The recurrence relation

T(1) = 2

≈ ’
T(n) = 3T n + n
∆ ÷
4« ◊

Has the solution T(n) equal to

(a) O(n) (b) O (log n)

≈ 3 ’
(c) O n∆ 4 ÷
÷∆ (d) None of the above
◊«
* The average number of key comparisons done on a successful
sequential search in list of length n is

(a) log n (b)

n - 1
2

(c)

n (d)
2

n + 1
2

* Quick-sort is run on two inputs shown below to sort in ascending
order

(i) 1,2,3 ….n

(ii) n, n œ 1, n œ 2, …. 2, 1

Let C1 and C2 be the number of comparisons made for the
inputs (i) and (ii)
respectively. Then,

(a) C1 < C2 (b) C1 > C2 (c) C1 = C2

(d) we cannot say anything for arbitrary n.

* Which of the following macros can put a macro assembler into an
infinite loop?

(i) MACRO M1, X
IF EQ, X: if X=0 then …. M1 X + 1

ENDC

IF NE, X: if X ¹ O then ……

WORD X: address (X) is stored here
ENDC ENDM
(ii) MACRO M2,X IF EQ, X
M2 X ENDC
IF NE, X WORD X + 1
ENDC ENDM

(a) (ii) only (b) (i) only

(c) both (i) and (ii) (d) None of the above

* A solution to the Dining Philosophers Problem which avoids deadlock
is

(a) ensure that all philosophers pick up the left fork before the right
fork

(b) ensure that all philosophers pick up the right fork before the left
fork
(c) ensure that one particular philosopher picks up the left
fork before the right fork, and that all other philosophers pick up the
right fork before the left fork

(d) None of the above

* A 1000 Kbyte memory is managed using variable partitions but
to compaction. It currently has two partitions of sizes 200 Kbytes
and 260 Kbytes respectively. The smallest allocation request in Kbytes that
could be denied is for

(a) 151 (b) 181 (c) 231 (d) 541

* Consider the following state table in Fig.2.23 for a
sequential machine. The number of states in the minimized machine will
be
input
x
Present 0 1
state
A D0 B1

B A0 C1

C A0 B1

D A1 C1

(a) 4 (b) 3 (c) 2 (d) 1

* A micro program control unit is required to generate a
total of 25 control signals. Assume that during any
microinstruction, at most two control signals are active. Minimum
number of bits required in the control word to generate the
required control signals will be

(a) 2 (b) 2.5 (c) 10 (d) 12

* Four jobs to be executed on a single processor system
arrive at time 0+ in the order A, B, C, D. their burst CPU
time requirements are 4, 1, 8, 1 time units respectively. The
completion time of A under round robin scheduling with time slice
of one time unit is

(a) 10 (b) 4 (c) 8 (d) 9

* Consider the circuit in Fig.2.21 which has a four fit
binary number 3b 2b 1b 0b as
input and a five bit binary number, 4d 3d 2d 1d 0d as output.
”f‘
b3 b2 b1 b0
(a) Binary of Hex conversion

(b) Binary to BCD conversion

(c) Binary to grey code conversion

C out
C in
”f‘
4dd3d2d1 d0

* What is the equivalent Boolean expression in
product-of-sums form for the
Karnaugh map given in Fig.2.24.

(a) BD + BD
BA
00 01 11 20
CD

(b) (B + C + )D (B + C + )D

(c) (B + )D (B + )D

(d) (B + )D (B + )D
00 1 1

10 1 1

11 1 1

01 1 1

2.22. Consider the circuit in Fig.2.22 f implements

0C

1C Multiplex f
2C
3C

A B

(a) A BC + ABC + ABC

(c) A Å B Å C

(b) A + B + C

(d) AB+BC+CA

3. Let f be a function defined by

x2

(f x ) = ax2 + bx + c
x + d

for x £ 1 for 1 x £ 2 for x > 2
Find the values for the constants a; b; c and d so that f
is continuous and differentiable everywhere on the real line.

* Let Q = (q1, q2) (a,b), (a, b, Z) d, qi, Z, f) be a
pushdown automaton accepting by empty stack for the language
which is the set of all nonempty even palindromes over
the set {a,b}. Below is an incomplete specification of
the transition d. complete the specification. The top of stack is
assumed to be at the right end of the string representing stack
contents.

(1)

d ( 1q , ,a )Z

= ({ 1q , Za)}

(2)

d ( 1q , ,b

Z ) = ({ 1q , Zb)}

(3)

(4)

(5)

(6)

d ( 1q , ,a a) = {>> ,> }>

d ( 1q , ,b b) = {>> ,> }>

d ( 2q , ,a a) = ({ 2q , e )}

d ( 2q , ,b b) = ({ 2q , e )}

(7)

d ( 2q , e , )Z

= ({ 2q , e )}

* A demand paged virtual memory system uses 16 bit virtual
256 bytes, and has 1 Kbyte of main memory. LRU page
replacement is implemented using list, whose current status (page
numbers in decimal) is

17 1 63

LRU page

00FF, 010D, 10FF 11B0

indicate,

(i) the new status of the list

(ii) page faults, if any, and

(iii) page replacements, if any.

* The Fibonacci sequence { 1f 2, f 3, f > n }f

is defined by the following recurrence:

nf + 2 = nf +1 + n ,f n ³ 1; 2f = 1 1: f = 1 :

Prove by induction that every third element of the sequence is even.

» ÿ » ÿ
10. Let A = 1a 1 1a 2 and B = 1b 1 1b 2
… Ÿ … Ÿ be two matrices such that
2a 1 2a 2 ⁄ 2b 1 2b 2 ⁄

»1 0ÿ
AB = I. Let C = A … Ÿ and CD = 1. Express the elements of D
in terms of the

elements of B.
1 1⁄

* A logic network has two data inputs A and B, and two control
inputs C0 and C1. It implements the function F according to the following
table.

C1 C2 F

0 0 A + B

0 1 A + B

1 0 A Å B

Implement the circuit using one 4 to AB Multiplexor, one 2-input
Exclusive OR
gate, one 2-input AND gate, one 2-input OR gate and one Inverter.

* A logic network has two data inputs A and B, and two control
inputs C0 and C1. It implements the function F according to the following
table.

C1 C2 F

0 0 A + B

0 1 A + B

1 0 A Å B

Implement the circuit using one 4 to AB Multiplexor, one 2-input
Exclusive OR
gate, one 2-input AND gate, one 2-input OR gate and one Inverter.

* Let F be the collection of all functions f:{1,2,3} ‰ {1,2,3}.
If f and g Î F, define an equivalence relation ~ by f ~ g if and only
if f(3) = g.

(a) Find the number of equivalence classes defined by ~

(b) Find the number of elements in each equivalence class.

* Let G be a context-free grammar where G = ({S, A, B, C},
{a, b, d}, P,S) with the productions in P given below.
S ‰ ABAC A ‰ aA|e
B ‰ bB ‰ e
C ‰ d

(e denoted the null string). Transform the grammar G to an
equivalent context- free grammar G that has no e productions and
no unit productions. ( A unit production is of the form x ‰ y, and y
are non terminals).

* A two dimensional array A[1- n] [1 œ n] of integers is partially sorted
if

" ,i je 1»

- n - 1ÿ⁄

A i» ÿ⁄ j» ÿ⁄ < A i» ÿ⁄ j»

+ 1ÿ⁄

and

A i» ÿ⁄ j» ÿ⁄ <

Fill in the blanks:

i» + 1ÿ⁄ j» ⁄ ÿ⁄

(a) The smallest item in the array is at A [j] where i =

and j = …………
(b) The smallest item is deleted. Complete the following O(n) procedure
to insert item x (which is guaranteed to be smaller than any
item in the last row or column) still keeping A partially sorted.
procedure insert (x integer) var j ¢ j:integer; begin

(1) i: =1; j:=A [j]:=x;

(2) while (x > or x >) do

(3) if A[ i + 1] [j] Ç ]j] then begin

(4) A [j]:= A[i +1] [j]; i: =i + 1

(5) end

(6) else begin

(7)

(8) end

(9) A [j]:=

end

* Let G be the directed, weighted graph shown below in Fig.4

B
14
6 12
22
9f
A C
2 12

6f

D E
5f

51
f7 1f
F

We are interested in the shortest paths from A.
(a) Output the sequence of vertices identified by the
Dijkstra‘s algorithm for single source shortest path when the algorithm
is started at node A.

(b) Write down sequence of vertices in the shortest path from A to E.

(c) What is the cost of the shortest path from A to E?

* A file system with a one-level directory structure is
implemented on a disk with disk block size of 4 K bytes. The disk is
used as follows:
Disk-block 0: File Allocation Table, consisting of one 8-bit entry
per date block, representing the data block address of the next
date block in the file
Disk block 1: Directory, with one 32 bit entry per file: Disk block 2:
Data block 1;

Disk block 3: Data block 2; etc.

(a) What is the maximum possible number of files?

(b) What is the maximum possible file size in blocks?

* Consider the following program that attempts to locate an
element x in an array a [ ] using binary search. Assume N >
1. The program is erroneous. Under what conditions does the program
fail?

var i.j.k:integer; x:integer.
a:=array; [1 … N] of integer. begin i:1;=n;

repeat ki+j) div 2;
if a [k] < x then i:=k else j:=k
until (a⁄ =x) or (iÇ=j);

if (a [k] = x) then

writeln (”x is not in the array‘)

else

writeln (”x‘ is not in the array‘)

end;

* A computer system has a three level memory hierarchy, with
access time and hit ratios as shown below:

Level 1 (Cache memory) Level 2 (main memory) Level 3

Access time = 50 nsec/byte Access time = 200 nsec/byte Access time = 5
µsec/byte

Size Hit ratio Size Hit ratio Size Hit ratio

8 M byte 0.80 4M byte 0.98

16 M byte 0.90 16 M byte 0.99

260 Mbyte 1.0

64 M byte 0.95 64 M byte 0.995

(a) What should be the minimum sizes of level 1 and 2 memories
to achieve an average access time of less than 100 nsec?

(b) What is the average access time achieved using the chosen
sizes of level 1
and level 2 memories?

* Consider the syntax-directed translation schema (SETS) shown below: E
‰ E + E {print —+“}
E ‰ E * E {print —.“}
E ‰ id {print id.name} E ‰ (E)
An LR œ parser executes the actions associated with the
productions immediately after a reduction by the corresponding
production. Draw the parse tree and write the translation for the
sentence.

(a + b)* (c + d), using SDTS given above.

* The concurrent programming constructs fork and join are as below:

fork <label> which creates a new process executing from the specified label

join <variable> which decrements the specified synchronization
variable (by 1)
and terminates the process if the new value is not 0.
Show the precedence graph for S1, S2, S3, S4 and S5 of the
concurrent program below.

N = 2
M = 2 fork L3 fork L4

S1
L1 : join N S3
L2: join M S5

L3:S2

goto L1

L4:S4

goto L2

next:

* A hard disk is connected to a 50 MHz processor
through a DMA controller. Assume that the initial set-up of a
DMA transfer takes 1000 lock cycles for the processor, and assume
that the handling of the interrupt at DMA completion requires
500 clock cycles for the processor. The hard disk has a transfer
rate of
2000 Kbytes/sec and average block size transferred is 4 K bytes. What
fraction of the processor time is consumed by the disk, if the
disk is actively transferring
100% of the time?

Level 1 (Cache memory) Level 1 (Cache memory)

Access time = 50 nsec/byte Access time = 200 nsec/byte

Size Hit ratio Size Hit ratio

8 Kbytes 0.80 4 Kbytes 0.98

16 Kbytes 0.90 16 Kbytes 0.99

64 Kbytes 0.95 64 Kbytes 0.995

Size Hit ratio

250 M bytes 1.0

* A computer system uses the Banker‘s Algorithm to deal
with deadlocks. Its current state is shown in the tables below,
where P0, P1, P2 are processes, and R0, R1, R2 are resoures types.

Maximum Need Current Allocation Available

R0 R1 R2 R0 R1 R2 R0 R1 R2

P0 4 1 2 P0 1 0 2 2 2 0

P1 1 5 1 P1 0 3 1

P2 1 2 3 P2 1 0 2

(a) Show that the system can be in this state.

(b) What will system do on a request by process P0 for one
unit of resource type
R1?

* A binary search tree is used to locate the number 43. Which
of the following probe sequences are possible and which are not? Explain.

(a) 61 52 14 17 40 43

(b) 2 3 50 40 60 43

(c) 10 65 31 48 37 43

(d) 81 61 52 14 41 43

(e) 17 77 27 66 18 43

* A library relational database system uses the following schema
USERS (User #, User Name, Home Town) BOOKS (Books # Book Title, Author
Name) ISSUED (Book #, User #, Date)
Explain in one English sentence, what each of the following
relational algebra queries is designed to determine

(a) s User #=6 (11 User #, Book Title ((USERS ISSUED) BOOKS))

(b) s Author Name (BOOKS (s Home Town) = Delhi (USERS ISSUED)))

* Consider the following program in pseudo-Pascal syntax. What is
printed by the program if parameter a is procedure test 1 is passed as

(i) call-by-reference parameter?

(ii) call-by-value-result parameter?

program Example (input, output)

var b: integer;

procedure test 2;

begin b: = 10 end

procedure test 1 (a:integer);

begin a:5;

writeln (”point 1: ”a,b);

test 2;

wrote;m(”point: ”a,b);

end;
begin (*Example*) b:=3; test ] (b); writeln (”point 3: ”b) end

* Insert the characters of the string K R P C S N Y T J M
into a hash table of size 10.

Use the hash funtion

H(x) = (ord (x) œ ord (”a‘) + 1) mod 10

And linear probing to resolve collisions.

(a) Which insertions cause collisions?

(b) Display the final hash table?

* A complete, undirected, weighted graph G is given on the vertex
{0, 1 …., n œ 1}
for any fixed ”n‘. Draw the minimum spanning tree of G if

(a) the weight of the edge (u, v) is |u œ v|

(b) the weight of the edge (u, v) is u + v