CS6363: Computer Algorithms
Spring 2008
Assignment 3

Due: Monday, February 11 (in class)

1. Consider the two-assembly-line scheduling problem in Chapter 15.
   Suppose we are given an integer k that specifies the maximum number
   of times a job may switch lines.  Design an efficient dynamic
   programming algorithm that finds the fastest way through the factory
   among schedules that change lines at most k times.  Show the recursive
   formulation, the resulting dynamic program and its running time analysis.

2. In solving a problem, we got the following recurrence:

   f(i,j) = max[0, max{1+f(i,k)+f(k,j)}, over i<k<j, whenever g(i,j,k) is true]

   Assume that $f(i,i) = f(i,i+1) = 0$, and that there is a Boolean
   function g(i,j,k) (which returns true/false).
   The following recursive algorithm computes it:

   f(i,j)
   {
     if (i=j or i+1=j) then
       return 0
     else {
       v := 0
       for k := i+1 to j-1 do {
         if (g(i,j,k)) {
           x := 1 + f(i,k) + f(k,j)
           if (x > v) then v := x
         }
       }
       return v
     }
   }

   Write a dynamic programming algorithm that computes f(i,j),
   for all 1 <= i <= j <= n.  What is its running time?

3. A sequence is called "alternating" if it alternates between
   odd and even integers (i.e., for any two successive elements,
   one must be odd and the other even).  Give an O(n^2)-time algorithm
   to find the longest monotonically increasing alternating subsequence
   of a sequence of n numbers.

4. Problem 15-2 (p. 364)  [1st ed: Prob 16-2, p. 325]  (Printing neatly).