This preview shows pages 1–2. Sign up to view the full content.
6.896 Sublinear Time Algorithms
February 15, 2007
Lecture 4
Lecturer: Ronitt Rubinfeld
Scribe: Huy Nguyen
1
Testing uniformity of a monotone distribution over a totally
ordered set
Defnition 1
Let
D
=1
,
2
, .., n
=[
n
]
. D is monotone
if
p
1
≥
p
2
≥···≥
p
n
.
We want to test whether a monotone distribution over a totally ordered set is uniform or far from
uniform. Since all totally ordered sets are equivalent, for simplicity, we assume that the domain is [n].
The key idea of the algorithm is to take samples and check if the number of sample in the left half is
close to the number of sample in the right half.
Lemma 2
If
p
(
{
1
···
k
}
)
≤
(1 +
±
)
p
(
{
k
+1
2
k
}
)
then

p
−
U
[
n
]
≤
±
ProoF
δ
i
←
p
i
−
1
n

j
←
largest i s.t.
p
i
≥
1
2
k
=
1
n
We only consider the case when
j
≤
k
. The case when
j>k
is similar.
We de±ne
A
1
,A
2
3
as follow:
A
1
=
X
i
≤
j
δ
i
A
2
=
X
j<i
≤
k
δ
i
A
3
=
X
n
≥
i>k
δ
i
A
1
=
A
2
+
A
3
because the sum of distribution p and the uniform distribution are both 1.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '04
 RonittRubinfeld
 Algorithms

Click to edit the document details