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COMP 731 - Data Struc & Algorithms - Lecture 9 - Tuesday, July 11, 2000
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Today: 

o Quicksort analysis 

* Analyze quicksort under the assumption that one gets an 
  n/10 - 9n/10 split every time:

       T(n) = T(n/10) + T(9n/10) + n 

  Use a recursion tree to find that 

       T(n) \le n log_{10/9} n 
            \in O(n\lg n)


* Now use the method from the Motwani/Raghavan book to analyze the
  expected-case behavior of Quicksort, getting the
  recurrence relation 
                                 2
      T(n) =      sum        --------
              1<= i<j <= n     j-i+1

  Do an example (i=5) in order for the student to see that

               n    n-i+1
           =  sum  -------   
              i=2     i

                n     1
          <= n sum  ----
               i=2    i

   Work out upper and lower bounds for H_n, the n-th Harmonic number, using
   an integral to get

           \in O(n lg n)