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Computer Science 340
Reasoning about Computation
Precept 4
Problem 1
Consider the family of hash functions discussed in class
h
a,b
(
x
) = (
ax
+
b
mod
p
)(
mod
n
) where
a, b
∈ {
0
, . . . p

1
}
and
a
6
= 0. Consider a hash function
h
a,b
drawn uni
formly and at random from this family. For any
x
1
, x
2
∈ {
0
, . . . , p

1
}
, x
1
6
=
x
2
, show
that
Pr[
h
a,b
(
x
1
) =
h
a,b
(
x
2
)]
≤
1
/n.
Problem 2
Consider a variant of the family of hash functions discussed in class
h
a
(
x
) = (
ax
mod
p
)(
mod
n
) where
a
∈ {
1
, . . . p

1
}
. Consider a hash function
h
a
drawn uniformly and at random
from this family. For any
x
1
, x
2
∈ {
0
, . . . , p

1
}
, x
1
6
=
x
2
, show that
Pr[
h
a
(
x
1
) =
h
a
(
x
2
)]
≤
2
/n.
Problem 3
Consider the algorithm outlined in class to estimate the number of distinct queries in
a query log using a random hash function
h
:
U
→
[0
,
1]. Here
U
is the universe of all
queries. Assume that we can use a truly random hash function. Recall that the estimator
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This homework help was uploaded on 01/29/2008 for the course COS 340 taught by Professor Charikarandchazelle during the Fall '07 term at Princeton.
 Fall '07
 CharikarandChazelle

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