Lab 2 Fall 2009

This lab is to review Chapter 0 of the text book. Please go to your lab, do the problems to get your full credit for the lab.
  1. Using L'Hospital's rule, prove that the function log(n) grows slower than n0.5 (in fact log(n) grows slower than even n x for 0 < x &le 1). (optional:Use your proof to show log(log(n)) grows slower than (log(n))2).
  2. Arrange the following functions in increasing order of their growth rate.
    (3/2) n;n; n!; 106 ; 7; 3 n; 3 sqrt(log(n)); n n; n log(n) ; n 3 n; n 4/3 ; 3 3 n ; n 5 ; log(log(n)); 0.5 n log(n+5);
  3. Write a funcion MERGE3 that takes three sorted lists and merges the, into one list. (Write a pseudo code, state the worst case running time).
  4. Write a new merge sort procedure that uses MERGE3 that takes 3 sorted lists and merges them. ( Implement this algorithm and test it for Home Work 2 and not for this lab .)
  5. Show that 18n 3 logn + 100 n 2 +500 = O(n3log(n));
  6. Can big Oh in the above problem be replaced with theta?
  7. Optional: Show that log(n!) = theta(n log (n)). (Hint: we have shown in class n! is O(n n ) - please reconstruct the argument. Show n! is greater than (n/2) n/2 - combining these two you will get the theta bound).