Discussion 3 - Rogaway

1. HW - explain what is asked 

2. A lower bound: finding the min and the max:
      homework: showed how to do in 3n/2-2 comparisons
      claim:  for every n, you can't do it in less than 3n/2-2 comparisons
              by any comparison-based algorithm!!

  WLOG, assume keys are distinct.  
  Any algorithm that identify MAX and MIN:    
    - every element other than MAX must have lost some comparison
    - every element other than MIN must have won some comparions
  So:  when count each win as one unit of information, 
                  each loss as one unit of information.
  So:  Any comparion-based algorithm must provide (n-1) + (n-1) = 2n-2 
       units of information.
  Idea: Adversary tries to make lots of the comparisions give one unit of 
  information, only a few of the comparison give two units of information.

  Every element has a status: one of 
     N  - the element has neither won nor lost any comparison
     W  - the element has won but never lost
     L  - the element has lost but never won
     WL - the element has both won and lost

  compare     return    new status   inf units
  ----------------------------------------------
  N : N  	>	W  L     	2 
  W : N         >       W  L		1
  WL: N		>       WL L		1
  L : N		<	L  W		1

  W : W		>  	W  WL		1

  L : L		>	WL L		1
  W : L		>	W  L		0
  WL: L		>	WL L		0

  W : WL	>	W  WL		0
  WL: WL   consistent   WL WL		0

 Assign values as you go.   Adjust these values
 when necessary.  You will never get stuck:

  example:   n=6.   7 Comparisons necessary.
  call elements: a b c d e f g

         a   b   c   d   e   f   g 

 a:b     N   N   N   N   N   N   N
         2   1
         W   L   
 c:d             4   3
                 W   L
 d:e                     5
                         W
 f:g                         W   L    
                             7   6

 a:c     8       
                 WL

 b:c        
 d:g                             0 
                     WL
 c:d  >


   Always consistent with some ordering.

   n even:  at most n/2 2-information calls;
   Ask q questions.

   2 (n/2) + (q-n/2) >= 2n-2  
     n + q-(n/2) >= 2n-2 
          q >= 3n/2 - 3/2

   n odd:  at most n-1/2 2-information calls;
    2 ((n-1)/2) + (q-((n-1)/2)) >= 2n-2
    n - 1 + q - n/2 + 1/2 >=  2n-2 
            q             >= 3n/2 - 3/2

   



3.  Do dynamic programming problem from last year's midterms