Greedy Algorithms

• A greedy algorithm generally works by
  • Making a local "best" choice which turns out to be the "best" choice globally.
  • This choice, however, might not be the obvious choice.
• A read of 4.1 is probably good.
• They solve the scheduling problem.
  • Given a set of tasks $T = \{ T_1, T_2, ... T_n\}, T_i = \{s_i, f_i\}, s_i \le f_i$, find the maximum sized subset S where not two tasks overlap.
• They consider four different greedy selection methods.
  1. Select the next available task that starts earliest.
     • This is easy to find a counter example, assume each block of xs is a task.

     •     1111111111111111111111111111111111111111 
              22222  33333 44444 55555 66666 
                    

     • Selecting task 1
  2. Select the task with the smallest interval

     •               111 
         2222222222222 333333333333333333 
                

     • Again this would force you to select task 1
  3. Select the task with the fewest conflicts.

     • 
                 1111111  22222222  33333333
                 4444444            55555555
                 6666666            77777777
                 8888888            99999999
             aaaaaaa  bbbbbbb  cccccccc   ddddddd
                     

     • Here task 2 has the fewest conflicts at 2
  4. Select the next available task that finishes first.
     • Try this with each of the previous cases.
• The Algorithm

  
  Schedule(T)
   Sort T by finish times
   currentEnd = T[0].f
   Add T[0] to S
   For i = 1 to n-1 do
       If T[i].s > currentEnd
         Add T[i] to S
         currentEnd = T[i].f
       EndIF
   EndFor
   Return S