6.896 Sublinear Time Algorithms
February 13, 2007
Lecture 3
Lecturer: Ronitt Rubinfeld
Scribe: Jeremy Hurwitz
In this lecture, we will prove the correctness of the equality tester for
L
1
.
L
1
DistTest
(
p, q,
)
m
←
O
(max(
−
2
,
4)
n
2
3
log
n
)
S
p
←
m
samples from
p
S
q
←
m
samples from
q
S
←
S
p
∪
S
q
discard any element that appears less than (1
−
63
)
mn
−
2
3
times in
S
if
S
=
∅
ˆ
p
i
←
number of times
i
appears in
S
p
ˆ
q
i
←
number of times
i
appears in
S
q
fail if
i
∈
S

ˆ
p
i
−
ˆ
q
i

>
m
8
define
p
as output of following process
sample
x
∈
p
if
x /
∈
S
output
x
else
output
x
∈
U
D
define
q
similarly
else
p
←
p
,
q
←
q
run
T
(
p , q ,
2
√
n
) with
O
(
n
2
3
4
) samples
T
is a tester described in the last lecture which passes
p, q
for which the
L
2
distance is at most
/
2
and fails
p, q
for which the
L
2
distance is at least
. We assume that the error probability is at most 1/6.
1
Useful Claims, Observations, and a Lemma
We begin by proving a series of claims. We will then analyze the behavior of the algorithm.
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 Fall '04
 RonittRubinfeld
 Algorithms, qi , total failure rate

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