Faculty of Computer Science and Engineering Department of Computer Science Released on 24/08/2012 20:06:39 1/4 DATA STRUCTURES & ALGORITHMS Tutorial 1 Questions COMPUTATIONAL COMPLEXITY Required Questions Question 1. Reorder the following efficiencies from the smallest to the largest: a. 2n 3 + n 5 b. 2000 c. 4 n+1 d. n 4 e. (n-1)! f. nlog 2 (n) g. 2klog k (n) (k is a predefined constant) Question 2. Determine the big-O notation for the following: a. 2n 6 b. 100n + 10.5n 2/3 + 5.2n 5/7 c. 9log 2 (n) d. 5n 10 + 2logn e. n! + k! (k is a predefined constant) f. 100log 2 (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 = n 3 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) Faculty of Computer Science and Engineering Department of Computer Science Released on 24/08/2012 20:06:39 2/4 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 2 n log 2 (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>) Pre n must be greater than 0 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 Faculty of Computer Science and Engineering Department of Computer Science Released on 24/08/2012 20:06:39 3/4 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>) Pre n must be greater than 0 Return integer value of f corresponding to n 1 if (n <= 1) 1 return 1 2 else Faculty of Computer Science and Engineering Department of Computer Science Released on 24/08/2012 20:06:39 4/4 1 return f(n – 1) + f(n – 2) End Question 12. Solve recurrence f(n) = 2f(√n) + log 2 n. (Hint : change variables, with m = 2 n ) End . 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 ^10 24 *10 ^-9 U n = U +1 U = 1 O(n) n -1 1 Faculty of Computer Science and Engineering. Faculty of Computer Science and Engineering Department of Computer Science Released on 24/08/2 012 20:06:39 1/ 4 DATA STRUCTURES & ALGORITHMS Tutorial 1 Questions COMPUTATIONAL. 2 else Faculty of Computer Science and Engineering Department of Computer Science Released on 24/08/2 012 20:06:39 4/4 1 return f(n – 1) + f(n – 2) End Question 12 . Solve recurrence