6.896 Sublinear Time Algorithms
February 8, 2007
Lecture 2
Lecturer: Ronitt Rubinfeld
Scribe: Adriana Lopez
In this lecture, we will prove a theorem from last lecture that states that there exists a
closeness
tester
in
L
2
. We will also give an algorithm for
L
1
distance testing.
Recall the following definitions and facts from last lecture:
Definition 1
p
x
= Pr[
p
outputs
x
]
Definition 2
(collision probability of
p
and
q
)
CP
(
p, q
) = Pr[
sample from
p
equals sample from
q
] =
x
p
x
q
x
Definition 3
(selfcollision probability of
p
)
SCP
(
p
) = Pr[
2 samples from
p
are equal
] =
x
p
2
x
=
p
2
2
1
L
2
Distance Testing
Theorem 4
For all
, p, q
, there exists an
O
(
−
4
)
sample tester
T
such that
•
If
p
−
q
2
<
2
then
Pr[
T accepts
]
≥
2
3
.
•
If
p
−
q
2
>
then
Pr[
T rejects
]
≥
2
3
.
We will prove that tester
T
from last lecture (reproduced below) satisfies the conditions stated in
Theorem 4.
T
(
p, q,
):
m
←
O
(
−
4
)
S
p
←
m
samples from
p
S
q
←
m
samples from
q
r
p
←
number of selfcollisions in
S
p
r
q
←
number of selfcollisions in
S
q
Q
p
←
m
(new) samples from
p
Q
q
←
m
(new) samples from
q
r
pq
←
number of pairs
i, j
s.t.
i
th sample in
Q
p
is equal to
j
th sample from
Q
q
r
←
2
m
m
−
1
(
r
p
+
r
q
)
s
←
2
r
pq
if
r
−
s >
m
2
2
2
then
reject
else
accept
Also recall the following fact from last lecture:
E
[
r
−
s
] =
m
2
p
−
q
2
2
Lemma 5
Let
b
= max
x
p
x
, q
x
. Then
Var[
r
−
s
]
≤
O
(
m
3
b
2
+
m
2
b
)
.
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, Equals sign, M3 Lee, M4 Sherman, World War II American infantry weapons, ij kl

Click to edit the document details