CMPSCI 311: Theory of Algorithms

David Mix Barrington

Fall, 2003

Discussion Notes #3

from Wednesday 24 Sept 2003

Applying the Master Theorem (Levitin Appendix B)

Here we will analyze some recursive algorithms and get recurrences that we may solve using the Master Theorem. In each case explain and ve your recurrence to get a Θ solution, that is, a function f(n) that is Θ(T(n)) where T(n) is the actual worst-case running time on inputs of length n.

Remember that because of the Smoothness Theorem we can ignore issues caused by divide-and-conquer procedures not dividing the input into equal pieces.

Please answer the questions during the discussion period.

• Question 1: The Stoogesort algorithm sorts an array of Comparables. Assume for the discussion that it sorts correctly and analyze its running time.

  
       void stoogesort (Comparable [] a, int i, int j) 
       {// sorts items i through j of array a
           int size = j-i+1;            // number of items to sort
           int twothirds = 2*size/3;
           if (size <= 3) {
                   trivialsort(a,i,j);  // assume this is Theta(1)
                   return;}
           stoogesort (a,i,i+twothirds-1);
           stoogesort (a,j-twothirds+1,j);
           stoogesort (a,i,i+twothirds-1);}
  

• Question 2: The Stoogesearch algorithm finds the target element in a sorted array or returns -1 if it is not there. I've probably messed up the code, but the idea is that it searches either the first 2/3 or last 2/3 of the given range.

       int stoogesearch (Comparable target, Comparable [] a, 
                         int low, int high)
       {// returns index of target if it is in a[low..high]
           int size = high-low+1;
           if (size == 1) if (a[low].equals(target)) return low;
                          else return -1;
           else {int twothirds = 2*size/3;
                 if (a[high-twothirds].compareTo(target) < 0) 
                    stoogesearch(target, a, high-twothirds+1, high);
                 else
                    stoogesearch(target, a, low, low+twothirds-1);}}
  

• Question 3: Here is a recursive algorithm for the closest pair problem, given as pseudocode. Analyze its running time.

       Given a set S of n points
           If n <= 3 check all pairs and return closest, else
           Divide S into two sets A and B of n/2 points each
           Check each pair (a,b) with a in A and b in B 
               and find the closest
           Find the closest pair within A recursively
           Find the closest pair within B recursively
           Return the closest of the three pairs you've found
  

Last modified 24 September 2003