Foundations of Data Science

We now give an example of a 2-universal family of hash functions. Let M be a prime
greater than m. For each pair of integers a and b in the range [0, M − 1], define a hash
function
h ab (x) = ax + b (mod M)
To store the hash function h ab , store the two integers a and b. This requires only O(log M)
space. To see that the family is 2-universal note that h(x) = w and h(y) = z if and only
if ( ) ( ) ( )
x 1 a w
= (mod M)
y 1 b z
( ) x 1
If x ≠ y, the matrix is invertible modulo M.
y 1
28 Thus
( a
=
b)
( −1 ( )
x 1 w
y 1)
z
and for each ( w
z)
there is a unique
( a
b)
. Hence
and H is 2-universal.
(mod M)
Prob ( h(x) = w and h(y) = z ) = 1
M 2
Analysis of distinct element counting algorithm
Let b 1 , b 2 , . . . , b d be the distinct values that appear in the input. Then the set S =
{h(b 1 ), h(b 2 ), . . . , h(b d )} is a set of d random and pairwise independent values from the
set {0, 1, 2, . . . , M − 1}. We now show that M is a good estimate for d, the number of
min
distinct elements in the input, where min = min(S).
Lemma 7.1 With probability at least 2 − d , we have d ≤ M 3 M 6 min
smallest element of S.
≤ 6d, where min is the
Proof: First, we show that Prob ( M
> 6d) < 1 + d . This part does not require pairwise
min 6 M
independence.
( ) (
M
Prob
min > 6d = Prob min < M ) (
= Prob ∃k, h (b k ) < M )
6d
6d
d∑
(
≤ Prob h(b i ) < M ) (
⌈
M
≤ d
⌉
) (
6d 1
≤ d
6d M 6d + 1 )
≤ 1 M 6 + d M .
i=1
28 The primality of M ensures that inverses of elements exist in ZM ∗
then x and y are not equal mod M.
and M > m ensures that if x ≠ y,
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