Foundations of Data Science

Exercise 7.8 DELETE? THIS IS IN CURRENT DEFINITION
Show that for a 2-universal hash family Prob (h(x) = z) = 1 for all x ∈ {1, 2, . . . , m}
M+1
and z ∈ {0, 1, 2, . . . , M}.
Exercise 7.9 Let p be a prime. A set of hash functions
H = {h| {0, 1, . . . , p − 1} → {0, 1, . . . , p − 1}}
is 3-universal if for all u,v,w,x,y, and z in {0, 1, . . . , p − 1} , u, v, and w distinct
Prob (h(x) = u) = 1 p , and
Prob (h (x) = u, h (y) = v, h (z) = w) = 1 p 3 .
(a) Is the set {h ab (x) = ax + b mod p | 0 ≤ a, b < p} of hash functions 3-universal?
(b) Give a 3-universal set of hash functions.
Exercise 7.10 Give an example of a set of hash functions that is not 2-universal.
Exercise 7.11 Select a value for k and create a set
H = { x|x = (x 1 , x 2 , . . . , x k ), x i ∈ {0, 1, . . . , k − 1} }
where the set of vectors H is two way independent and |H| < k k .
Analysis of distinct element counting algorithm
Counting the Number of Occurrences of a Given Element.
Exercise 7.12
(a) What is the variance of the method in Section 7.2.2 of counting the number of occurrences
of a 1 with log log n memory?
(b) Can the algorithm be iterated to use only log log log n memory? What happens to the
variance?
Exercise 7.13 Consider a coin that comes down heads with probability p. Prove that the
expected number of flips before a head occurs is 1/p.
Exercise 7.14 Randomly generate a string x 1 x 2 · · · x n of 10 6 0’s and 1’s with probability
1/ 2 of x i being a 1. Count the number of ones in the string and also estimate the number
of ones by the approximate counting algorithm. Repeat the process for p=1/4, 1/8, and
1/16. How close is the approximation?
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