COS 554 SPRING 2009 HW #6  DUE TUESDAY 3/17/09
p. 229: 11.2-2, 10 points
p. 229: 11.2-5, 10 points
p. 236: 11.3-3, 15 points
p. 236: 11.3-4, 10 points
p. 244: 11.4-1, 20 point
p.1105: C.2-2,  10 points
See Problem 1 below, 25 points
********** Problem Notes *********,
Solve problems 11.2-2, 11.3-3, 11.3-4, 11.4-1 and Problem 1 below by writing 
programs. These programs should use the data in the problem and produce 
the expected results. They should be easy to read and should include 
comments. You can add comments to Logo programs using ";". Unless explicit 
language requirements are stated, you may use either of the five languages 
C, Pascal, LISP, Python or Logo in this course. For many short problems, Python will 
be the fastest language for coding.

As noted in the syllabus, I expect programs to come with printed source 
code and printed sample output. The source code should be comprehensible 
and contain comments.

For problems C.2-2 and C.2-10 you must start with a sample space and then 
put a probability distribution on it. If I don't see a reasonable sample 
space for the problem, you will get 0 points for the solution regardless 
of how good it is.

PROBLEM 1.
Write a program that will do the following:

(a) Find a 12 digit number, p, that you can verify is prime.

(b) Pick a subset S, consisting of 100 integers at random from 0 to p-1 
(explain how you are sure that you are picking using the uniform 
distribution).

(c) Let m be 200. Pick 50 random functions of the type described in 
Theorem 11.5 on page 235 of the book and look at the average number of 
collisions for each of these hash functions. In other words, if h:S -> 
{0,...,m-1} is a hash function, let Col(h) = the number of pairs (x,y), 
with x and y in S, x <>y, and h(x) = h(y). I expect you count (x,y) and 
(y,x) as separate pairs, even though they really are the same pair. Once 
you have the values for each of the 50 functions, you can then average 
these values.

(d) Explain how well the result in (c) above compares with the theoretical 
results.