Printed Pages-3

ECSO72

Following Paper ID and Rolr No. to be filled in your Answer Book

B.Tech.

(sEM. Vrr) THEORY EXAMTNATTON 20n _12 COMPUTATIONAL COMPLEXITY Time

: 3 Hours

Total Marks

:

lA0

Note :-Attempt all questions.

L

Attempt any two parts of the following

(a)

(10x2:20)

:

What are the different models of computation ? Describe these models in comparative manner.

(b)

Prove or disprove the following conjectures

:

(D f(n): o (g(n)) implies o (f(n)) (ir) f(n) + g(n):0 (min(f(n), g(n))) (iiD f(n): o (g(n)) implies s(n) =

2f

@)

-

O(2er,l;

where f(n) and g(n) are asymptotically positive functions.

(c)

Are the following sets closed under the following operation. Prove your answer

(i)

FIN (the set of finite languages) under the function odds L, defined on languages as follows : odds (L)

(ii)

:

:

{w

:

!

x e L(w: odds(x)))}.

INF(the set of infinite languages) under the function odds L.

ECS0T2|KLH-26644

I

[Turn Over

2.

Attempt any two parts of the following

(a)

:

(10x2=20)

What do you mean by complexity classes ? Discuss the relationship among the complexity classes.

(b) Show that if there is a reduction from P, to P, then (i) If P, is undecidable then so is P.. (ir) If P, is non-recursive enumerable then so is P,. (c) State and prove Rice theorem. :

3.

Attempt any two parts of the following

:

(10x2=20)

(a) Explain the general steps in establishing NP-completeness proof of

(b)

a given problem.

Explain the Gddel's incompleteness theorem with the help

of examples.

(c) 4.

Write the randomized version of Quick sort algorithm.

Attempt any two parts ofthe

following:

(a) State the circuit satisfiability

(10x2=20)

problem. Prove the circuit

satisfiability problem belongs to the class NP.

(b)

Consider the problem of multiplication of an nxn matrix

A

:

(aij) by an n-vector

Y: 0i) is given Yi =

\n

f

x = (x,). The resulting n-vector

bY the equation

au x,

;

:

for i=7,2,

"",n.

j=r

Write an al gorithm to perform matrix-vector multipl i cation by computing all the entries

(c)

ofy in parallel.

Write short note on Interactive proofs.

ECS072tK7H-26644

5.

Attempt any two parts of the following

(a) Explain the completeness

:

(10x2=20)

and soundness properties

of

probabilistically checkable proof system.

(b) Explain

the following class of problems

(i)

BPP

(ir)

RP

:

(iii) coRP

'\/

(iv\

(c)

ZPP.

Write short note on Quantum computing.

BCffi72tKtH-t6644

3

2462s

l it

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