CS 4540, Advanced Algorithms
Homework 2
Fri, Sept 10, 2010
Due Fri, Sept 17, 2010
Problem 1
Motwani and Raghavan, Problem 4.1, page 97. Note:
The purpose of this problem is to familiarize
you with the use of Chernoff bounds.
You may use any of the foffowing forms of Theorems 1
through 5 given at the Class Notes of 090810 (see Class Notes on the class web site).
Problem 2
Let 0
≤
y
1
≤
y
2
≤
. . .
≤
y
2
N
≤
1 be 2
N
real numbers, with
y
2
N
+1

j
= 1

y
j
, for 1
≤
j
≤
N
.
Let
X
1
, X
2
, . . . , X
n
be independent random variables such that, for 1
≤
i
≤
n
and for 1
≤
j
≤
N
,
Pr [
X
i
=
y
j
] =
p
ij
and Pr [
X
i
=
y
2
N
+1

j
] = Pr [
X
i
=
y
j
].
Let
p
i
=
E
[
X
i
] =
∑
2
N
j
=1
y
j
p
ij
, where
0
<p
i
<
1. Finally, let
X
=
∑
n
i
=1
X
n
and let
μ
=
E
[
X
].
(a) Prove that, for any
δ >
0, Pr [
X >
(1 +
δ
)
μ
]
<
h
e
δ
(1+
δ
)
1+
δ
i
μ
.
(b) Prove that, for any 1
> δ >
0, Pr [
X <
(1

δ
)
μ
]
<
h
e

δ
(1

δ
)
1

δ
i
μ
.
Note:
The purpose of this problem is to familiarize you with the proof of the basic form of Chernoff
bounds. You may use as guideline the proof of (13.42) in page 749 of Kleinberg and Tardos, and/or
the proofs of Theorems 4.1 and 4.2 in pages 68 and 70 (respectively) of Motwani and Raghavan.
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 Algorithms, Democracy, Motwani, class web site

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