CS5140 Assignment 1: Hash Functions and PAC Algorithms Solution

Turn in (a pdf) through Canvas by 2:45pm:
Overview
In this assignment you will experiment with random variation over discrete events.
It will be very helpful to use the analytical results and the experimental results to help verify the other is correct. If they do not align, you are probably doing something wrong (this is a very powerful and important thing to do whenever working with real data).
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1 Birthday Paradox (35 points)
Consider a domain of size n = 5000.
A: (5 points) Generate random numbers in the domain [n] until two have the same value. How many random trials did this take? We will use k to represent this value.
B: (10 points) Repeat the experiment m = 300 times, and record for each time how many random trials this took. Plot this data as a cumulative density plot where the x-axis records the number of trials required k, and the y-axis records the fraction of experiments that succeeded (a collision) after k trials. The plot should show a curve that starts at a y value of 0, and increases as k increases, and eventually reaches a y value of 1.
C: (10 points) Empirically estimate the expected number of k random trials in order to have a collision. That is, add up all values k, and divide by m.
D: (10 points) Describe how you implemented this experiment and how long it took for m = 300 trials.
Show a plot of the run time as you gradually increase the parameters n and m. (For at least 3 fixed values of m between 300 and 10,000, plot the time as a function of n.) You should be able to reach values of n = 1,000,000 and m = 10,000.
2 Coupon Collectors (35 points)
Consider a domain [n] of size n = 300.
A: (5 points) Generate random numbers in the domain [n] until every value i ∈ [n] has had one random number equal to i. How many random trials did this take? We will use k to represent this value.
B: (10 points) Repeat step A for m = 400 times, and for each repetition record the value k of how many random trials we required to collect all values i ∈ [n]. Make a cumulative density plot as in 1.B.
C: (10 points) Use the above results to calculate the empirical expected value of k.

D: (10 points) Describe how you implemented this experiment and how long it took for n = 300 and m = 400 trials.
Show a plot of the run time as you gradually increase the parameters n and m. (For at least 3 fixed values of m between 400 and 5,000, plot the time as a function of n.) You should be able to reach n = 20,000 and m = 5,000.
3 Comparing Experiments to Analysis (30 points)
[Show your work, including describing which formula you used.] How does this compare to your results from 1.C?
[Show your work, including describing which formula you used.] How does this compare to your results from 2.C?
4 BONUS : PAC Bounds (2 points)
Consider a domain size n and let k be the number of random trials run, where each trial obtains each value i ∈ [n] with probability 1/n. Let fi denote the number of trials that have value i. Note that for each i ∈ [n] we have E[fi] = k/n. Let µ = maxi∈[n] fi/k.
Consider some parameter ε ∈ (0,1). As a function of parameter ε, how large does k need to be for Pr[|µ−1/n| ≥ ε] ≤ 0.05? That is, how large does k need to be for all counts to be within (ε·100)% of the average with probability 0.05? (Fine print: you don’t need to calculate this exactly, but describe a bound as a function of ε for the value k which satisfies PAC property. Chapter 2.3 in the M4D book should help.)
How does this change if we want Pr[|µ−1/n| ≥ ε] ≤ 0.005 (that is, only 0.005 probability of exceeding ε error)?
[Make sure to show your work.]