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Chapter 0: Q1E (page 1)

Question: Answer each part TRUE or FALSE.

a.2n=O(n)b.n2=O(n)路Ac.n2=O(nlog2n)路Ad.nlogn=O(n2)e.3n=2O(n)f.22n=O(22n)

Short Answer

Expert verified

(a) 2n=O(n)is True.

(b)n2=O(n)isFalse.

(c)n2=O(nlog2n)isFalse.

(d)nlogn=O(n2)isTrue.

(e)3n=O(2n)isFalse.

(f)22n=O(22n) is True.

Step by step solution

01

To check whether 2n = O(n)  True or false

a)2n=O(n)isTrue

Suppose.c=2鈥�2n猢�cn=2n鈥�for鈥�all鈥�n>=1

Thus Big-O holds.

02

To check whether  O(n)n2=O(n) True or false

b)n2=O(n)isFalse

n2=cn鈥�n2<=cn鈥�for鈥�all鈥�n>=n0doesn鈥檛 hold.

03

To check whether  n2=O(nlog2n) True or false

c)n2=O(nlog2n) isFalse

There doesn鈥檛 exist positive constants n0 and c such that n2猢�cn鈥�log2n鈥�for鈥�all鈥�n>=n0.

04

To Find whether  nlogn=O(n2) True or false

d)nlogn=O(n2)isTrue

log鈥�n=O(n) there exist positive constants c and n0 such that log n猢�cn鈥�for鈥�all鈥�n>=n0.

05

To Find whether  n = 2O(n) True or false

e) 3n=2O(n)isFalse

there doesn鈥檛 exist constants c and n0 such that 3n猢�c2n鈥�for鈥�all鈥�n>=n0.

06

To Find whether 22n=O(22n)  True or false

f) 22n=O(22n)isTrue

We know any function fn鈥�is鈥�O(f(n)).

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Most popular questions from this chapter

Write formal descriptions of the following sets.

  1. The set containing the numbers1,10, and100
  2. The set containing all integers that are greater than5
  3. The set containing all natural numbers that are less than5
  4. The set containing the string aba
  5. The set containing the empty string
  6. The set containing nothing at all

For each of the following languages, give two strings that are members and two strings that are not members鈥攁 total of four strings for each part. Assume the 危=a,balpha-alphabet in all parts.

a.a*b*b.aba*bc.a*鈭�b*d.aaa*e.危*a危*b危*a危*f.aba鈭�babg.(蔚鈭�a)bh.(a鈭�ba鈭�bb)危*

In both parts, provide an analysis of the time complexity of your algorithm.

a. Show thatEQDFA鈭�P.

b. Say that a languageAisstar-closedifA=A*. Give a polynomial time algorithm

to test whether a DFArecognizes a star-closed language. (Note that EQNFAis not

known to be in P.)

Let危={a,b} . For each k猢�1, let Ckbe the language consisting of all strings that contain an a exactly K places from the right-hand end.

ThusCk=危*补危k-1 . Describe an NFA with k+1states that recognizes Ckin terms of both a state diagram and a formal description.

Give an example in the spirit of the recursion theorem of a program in a real programming language (or a reasonable approximation thereof) that prints itself out.
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