6.896 Sublinear Time Algorithms
November, 16, 2004
Lecture 18
Lecturer: Eli BenSasson
Scribe: John Danaher
1
The Blum Luby Rubinfeld (BLR) Linearity Test
Throughout our talk today,
G, H
are groups (not necessarily Abelian), written additively. A function
ϕ
:
G
→
H
is called a homomorphism if
∀
a, b
∈
G, ϕ
(
a
+
b
)=
ϕ
(
a
)+
ϕ
(
b
). In this lecture we will give a
test for closeness to a homomorphism. First we de±ne ”closeness”.
Defnition 1 (Distance)
The (relative) Hamming distance between two functions
f,g
:
G
→
H
,i
s
δ
(
f,g
)=Pr
x
∈
G
[
f
(
x
)
6
=
g
(
x
)]
. The distance of
f
from a set of functions
F
,is
δ
(
f,F
) = min
f
0
∈
F
δ
(
f,f
0
)
.
Finally, the distance of the set
F
is
δ
(
F
) = min
f
6
=
f
0
∈
F
δ
(
f,f
0
)
.(
I
f

F
≤
1
we de±ne
δ
(
F
)=1
.)
We now express our testing problem formally:
Defnition 2 (Homomorphism Tester)
Given groups
G, H
, de±ne their set of homomorphisms to be
Hom(
G, H
)=
{
ϕ
:
G
→
H
:
∀
a, b
∈
G, ϕ
(
a
+
b
)=
ϕ
(
a
)+
ϕ
(
b
)
}
A
(
q, ²
0
,c
)
tester for
Hom(
G, H
)
is a randomized polynomial time algorithm
T
f
with oracle access to a
function
f
:
G
→
H
, with the following properties.
Operation
T
f
tosses some coins, makes
q
queries to
f
and outputs
accept/reject
.
Completeness
If
f
∈
Hom(
G, H
)
then
Pr[
T
f
accepts
]=1
.
Soundness
Pr[
T
f
rejects
]
>
min(
c
·
δ
(
f,
Hom(
G, H
))
,²
0
)
.
We would like a tester with the smallest possible query complexity
q
, and with the largest possible
²
0
,c
(both parameters are at most 1). Our main Theorem, originally proved by Blum, Luby, Rubinfeld
[2], shows the existence of a good tester for any pair of groups
G, H
. The proof of the Theorem we
present today is due to Coppersmith.
Theorem 3 (BLR Linearity Testing)
For every pair of groups
G, H
, there exists a
(3
,
2
/
9
,
1
/
2)

tester for
Hom(
G, H