6.896 Sublinear Time Algorithms
April 3, 2007
Lecture 15
Lecturer: Kevin Matulef
Scribe: Huy Le Nguyen
1
Defnitions and ProoF Plan
Lemma 1
(Szemeredi’s Regularity Lemma). For any
±
,thereex
is
ts
C
±
such that any graph
G
=(
V,E
)
can be partitioned into sets
V
0
,V
1
,...,V
k
k
≤
C
±
such that
1.

V
0
≤
±V
2.

V
1

=

V
2

=
...
=

V
k

3. All but at most
±
(
k
2
)
pairs
(
V
i
j
)
are
±
regular
Defnition 2
The density between disjoint vertex sets A, B is
d
(
A, B
)=
e
(
A,B
)

A
·
B

.
We also de±ne
d
(
A, A
2
e
(
A,A
)

A

2
.
Defnition 3
AandBare
±
regular if
∀
A
0
⊆
A,

A
0

>±

A

,B
0
⊆
B,

B
0


B

, we have

d
(
A
0
0
)
−
d
(
A, B
)

<±
.
One application of Szemeredi’s Regularity Lemma that we mentioned last time was testing triangle
freeness of graph. The algorithm is very simple, just taking random triples and testing if they form
triangles. We used an edge removal argument to prove the correctness of the algorithm. After partitioning
the graph into components by Szemeredi’s Regularity Lemma, we cleaned it up by removing internal
edges of the components and edges between non
±
regular pairs. Afterwards, since all pairs of components
are
±
regular, we can restrict ourselves to testing
±
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 Fall '04
 RonittRubinfeld
 Algorithms, Graph Theory, Vertex, regularity lemma, Endre Szemerédi, pV pW

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