 
Summary: Com S 633: Randomness in Computation
Lecture 5 Scribe: Rakesh Setty
1 Unique Elements
In the last class, we saw that the expected number of unique elements a set S can have
with respect to a randomly chosen hashing function belonging to H is N=2, where H is a
2Universal hash family U to T , jSj = N , and jT j = 2N .
1.1 Notations
U  Universal Set
S  an arbitrary subset of U
N  size of the set S i.e. jSj
T  hash table to store the elements of S
H  2Universal family of hash functions from U to T
X  A random variable dened as the number of unique elements in S with respect to h 2 H,
when h is chosen uniformly at random from H.
Y  N X
In the last class, we saw that when T 2N ,
E[X] N=2 (1.1)
Claim:
Pr[X N=3] 1=4 (1.2)
when jT j 2N .
