6.896 Sublinear Time Algorithms
March 20, 2007
Lecture 13
Lecturer: Ronitt Rubinfeld
Scribe: Huy Ngoc Nguyen
1
Bipartiteness testing on dense graph
Definition 1
G
is ”
−
far
” from bipartiteness if must remove at least
n
2
edges to make it bipartite.
Definition 2
(equivalent to definition 1)
∀
partitions
(
V
1
, V
2
)
of
V
there are at least
n
2
edges
(
w, v
)
s.t either
u, v
∈
V
1
or
u, v
∈
V
2
(in this case
(
u, v
)
is a
violating pair
).
Algorithm (1):
:
TestBipartite(
A
,
G
)
(
A
 adjacency matrix for
G
)
choose sample
S
uniformly s.t

S

= Θ(
1
2
log
1
)
1
query
A
(
u, v
)
∀
u, v
∈
S
.
2
FAIL if subgraph not bipartite else PASS.
3
Note: total number of queries :
O
(
1
ε
4
log
2 1
ε
)
Theorem 1
The algorithm is a property tester for bipartiteness. More precisely,
(1) if
G
is bipartite, TestBipartite PASSES.
(2) if
G
is
ε
−
far
from bipartiteness,
Pr
[
TestBipartite FAILS
]
≥
3
4
.
Proof
Consider the following algorithm,
Algorithm (2)
:
Choose
U
uniformly, s.t

U

=
O
(
1
ε
log
1
ε
)
1
Choose a bunch of pairs (
u
i
, v
i
)
→
W
(

W

=
m
=
O
(
1
ε
2
log
1
ε
)
)
2
Query (
u, v
)
∀
u, v
∈
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 Fall '04
 RonittRubinfeld
 Algorithms, Topology, Graph Theory, high degree, Partition of unity, bipartite, bipartiteness

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