Model Solution of Problem G

1. (CLR exercise 11.3-4)

m = 1000, A = (sqrt(5) - 1) / 2, h(k) = floor(m (k A mod 1))

      k          h(k)
-----------|-------------
     61          700
     62          318
     63          936
     64          554
     65          172
   
 
2. (CLR exercise 11.4-4)

Max probes on unsuccessful search is 1 / (1 - alpha)  [ Thm. 11.6 ]
Max probes on successful search is (1 / alpha)* ln( 1 / (1- alpha))
[ Thm. 11.8 ]

For alpha = 3/4, the upper bound is :
                 4 probes on unsuccessful search
                 1.85 probes on successful search

For alpha = 7/8, the upper bound is :
                 8 probes on unsuccessful search
                 2.38 probes on successful search
  

3. (CLR exercise 11.4-5) 
Hint: In Maple, the expression "solve( LHS(x) = RHS(x), x);" works where
LHS and RHS are arbitrary expressions involving variable x. It is likely
to present a messy formula for the solution(s). Evalf() can be used to get
floating point answer(s). For instance, at the following prompt you can
enter "evalf(%)", to get an approximation of the previous answer (which is
what the "%" refers to). Exit maple with "quit". 

  max unsuccessful probes = 2 * max successful probes

From Maple:

Solve  ( (1/ (1-a)) = 2 * ((1/a) * ln (1/ (1-a))), a);
              
              2 LambertW(-1, -1/2 exp (-1/2)) + 1
          1/2 ------------------------------------
                LambertW(-1, -1/2 exp (-1/2))

evalf (%);
                  0.7153318630

So alpha =  0.7153318630