Faculty of Computer Science and Engineering Department of Computer Science Part 1 potx

Reorder the following efficiencies from the smallest to the largest: a.. Determine the big-O notation for the following: a... Prove that for any positive functions f and g, fn + gn and m

DATA STRUCTURES & ALGORITHMS

Tutorial 1 Questions COMPUTATIONAL COMPLEXITY

Required Questions

Question 1

Reorder the following efficiencies from the smallest to the largest:

a 2n3 + n5

b 2000

c 4n+1

d n4

e (n-1)!

f nlog2(n)

g 2klogk(n) (k is a predefined constant)

Question 2

Determine the big-O notation for the following:

a 2n6

b 100n + 10.5n2/3 + 5.2n5/7

c 9log2(n)

d 5n10 + 2logn

e n! + k! (k is a predefined constant)

f 100log2(n) + 5(n+2n)

Question 3

Calculate the run-time efficiency of the following program segment:

1 i = n-1

2 loop (i >= n/2)

1 j = 1

2 loop (j < n/2)

1 print(i, j)

2 j = j + 2

3 end loop

4 i = i - 2

3 end loop

Question 4

Calculate the run-time efficiency of the following program segment:

1 i = n3

2 loop (i >= n/2)

1 j = 1

2 loop (j < n/2)

1 print(i, j)

2 j = j + 2

3 end loop

4 i = i / 2

3 end loop

b<g<f<d<a<c<e

O(n^6) O(n) O(log2(n))

O(n!) O(n) O(n^10)

O(n^2)

n/4*log2(n^3)

nlog2(n)

Question 5

If the algorithm doIt has an efficiency factor of 2n, calculate the run time efficiency of

the following program segment:

1 i = 1

2 loop (i <= n)

1 j = 1

2 loop (j < n)

1 k = 1

2 loop (k <= n)

1 doIt(…)

2 k = k *2

3 end loop

4 j = j + 1

3 end loop

4 i = i + 1

3 end loop

Question 6

Given that the efficiency of an algorithm is 2nlog2(n 4), if a step in this algorithm takes 1

nanosecond (10−9), how long does it take the algorithm to process an input of size 1024?

Question 7

Write a recurrence equation for the running time T(n) of g(n), and solve that recurrence

Algorithm g (val n <integer>)

Return integer value of g corresponding to n

1 if (n = 1)

1 return 1

2 else

1 return g(n – 1)+ 1

End g

n^3*log2(n)

40*2^1024*10^-9

U

n = U +1

U = 1 O(n) n-1 1

Advanced Questions

Question 8

Prove that for any positive functions f and g, f(n) + g(n) and max(f(n), g(n)) are asymptotically

equivalent (i.e they yield the same big-O notations)

Question 9

Estimating the time complexity of the following program segment:

1 i = 0

2 loop (i < N)

1 j = i

2 loop (j < N)

1 k = 0

2 loop (k < M)

1 x = 0

2 loop (x < K)

1 print(i, j, k)

2 x = x + 3

3 end loop

4 k = k + 1

3 end loop

4 k = 0

5 loop (k < 2*K)

1 print(k)

2 k = k + 1

6 end loop

7 j = j + 1

3 end loop

4 i = i + 1

3 end loop

Question 10

Calculate the run-time efficiency of the following program segment:

1 i = n

2 k = n/3

2 loop (i >= k)

1 j = n – 2*k

2 loop (j < i)

1 print(i, j)

2 j = j + 2

3 end loop

4 i = i - 1

3 end loop

Question 11

Write a recurrence equation for the running time T(n) of f(n), and solve that recurrence

Algorithm f (val n <integer>)

Return integer value of f corresponding to n

1 if (n <= 1)

1 return 1

2 else

1 return f(n – 1) + f(n – 2)

End

Question 12

Solve recurrence f(n) = 2f(√n) + log2n (Hint : change variables, with m = 2 n)

End