The following paper does a variety of interesting probabilistic analyses: P. Flajolet et.al., “Birthday paradox, coupon collectors, caching algorithms and self-organizing search”, Discrete Applied Mathematics 39 (1992), 207-229. ( http://algo.inria.fr/flajolet/Publications/FlGaTh92 ) It includes the following formula, which provides the expected number of coupons needed under a general discrete probability distribution P for m coupons:  (14b) where For m=3 and , the paper simplifies (14b) to: . Your task is write a C program to 1) evaluate (14b) for the special case that k of the probabilities are the value p and the other m – k probabilities are the value q and 2) implement a simple random simulation of generating coupons for this situation. Your program must compile and execute on omega.uta.edu. (Formula (1) in https://web.cs.wpi.edu/~hofri/CCP says essentially the same thing as the formula above.) 

CSE 5311-001 Lab Assignment 1

Due March 3, 2021

Goals:

1. Understanding of coupon collecting

.

2. Understanding of enumeration.
3. Understanding of random simulation to verify a probability result.

Requirements:

1. The following paper does a variety of interesting probabilistic analyses:

P. Flajolet et.al., “Birthday paradox, coupon collectors, caching algorithms and self-organizing search”, Discrete Applied

Mathematics 39 (1992), 207-229. ( http://algo.inria.fr/flajolet/Publications/FlGaTh92 )

It includes the following formula, which provides the expected number of coupons needed under a general discrete

probability distribution P for m coupons:

(14b) where

For m=3 and , the paper simplifies (14b) to:

.

Your task is write a C program to 1) evaluate (14b) for the special case that k of the probabilities are the value p and the

other m – k probabilities are the value q and 2) implement a simple random simulation of generating coupons for this
situation. Your program must compile and execute on omega.uta.edu.

(Formula (1) in https://web.cs.wpi.edu/~hofri/CCP says essentially the same thing as the formula above.)

2. Submit your C code on Canvas before 3:45 p.m. on March 3. Be sure to include comments regarding how to compile and

execute your code.

Getting Started:

1. m will not exceed 50. m is at least 2.

2. The input is very simple:

a. The first input line is m and k. k must be larger than 0 and smaller than m.
b. The second line will have the value of the probability p for each of the first k coupons. p must be larger than 0 and

such that k • p is smaller than 1. Each of the other m – k coupons will have a probability of q = (1 – k • p)/(m – k)
c. The third input line will be the number of times to execute a random simulation.
d. The fourth input line is a seed for the random number generator (e.g. srandom()). (The only significance of the

seed is in the reproducibility of the experiments.)

3. Your powerset approach should use Q(m2) time instead of the Q(2m) time needed by the original version of expression

(14b). Elementary combinatorics ( https://en.wikipedia.org/wiki/Pascal%27s_triangle ) will be
useful. You should output how many times innermost loop(s) execute for the theoretical part. The time for computing
binomial coefficients should not be included.

€

m−1−q
−1( )

q=0

m−1

∑

1
1− PJJ =q

∑
€

PJ = Pi
i∈J
∑

€

p1,p2,p3( ) = a,b,c( )

€

1−
1
1− a

−
1
1−b

−
1
1−c

+
1

1− a−b
+

1
1−b−c

+
1

1−c − a