Lecture Notes for Randomized Algorithms
Last Updated by Eric Vigoda on February 2, 2006
8.1
Maximal Independent Sets
For a graph
( u, v )
∈
E
G either
= ( V, E ), an independent set is a set u
6∈
S and/or v
6∈
S
S
. The independent set
⊂
S
V which contains no edges of is a maximal independent set
G , i.e., for all if for all v
∈
V , either v
∈
S or N ( v )
∩
S =
∅ where N ( v ) denotes the neighbors of v .
It’s easy to find a maximal independent set. For example, the following algorithm works:
1.
I = ∅ , V
0
= V .
2. While ( V
0
=
∅
) do
(a) Choose any v
∈
V
0
.
(b) Set I = I ∪ v .
(c) Set V
0
= V
0
\
( v
∪
N ( v )).
3. Output I .
Our focus is finding an independent set using a parallel algorithm. The idea is that in every round we find a set S which is an independent set. Then we add from the current graph V
0
. If S
∪
N ( S
S to our current independent set
) is a constant fraction of

V
0

I , and we remove
, then we will only need
S
∪
N ( S )
O (log

V

) rounds. We will instead ensure that by removing S
∪
N ( S ) from the graph, we remove a constant fraction of the edges.
To choose S in parallel, each vertex v independently adds themselves to S with a well chosen probability p ( v ). We want to avoid adding adjacent vertices to S . Hence, we will prefer to add low degree vertices. But, if for some edge ( u, v ), both endpoints were added to S , then we keep the higher degree vertex.
Here’s the algorithm:
The Algorithm
Problem : Given a graph find a maximal independent set.
1.
I =
∅
, G
0
= G .
2. While ( G
0 is not the empty graph) do
(a) Choose a random set of vertices S
∈
G
0 by selecting each vertex independently with probability 1 / (2 d ( v )).
v
81
82 Lecture 8: Luby’s Alg. for Maximal Independent Sets using Pairwise Independence
(b) For every edge ( u, v )
∈
E ( G
0
) if both endpoints are in S then remove the vertex of lower degree from new set S
0
.
S (Break ties arbitrarily). Call this
(c) I
V
0
= I
S
S
0
.
G
0
= G
0
\
( S
0
S is the previous vertex set.
N ( S
0
)), i.e., G
0 is the induced subgraph on V
0
\
( S
0
S
N ( S
0
)) where
3. output I
Fig 1: The algorithm
Correctness : We see that at each stage the set S
0 that is added is an independent set. Moreover since we remove, at each stage, from G
0
S
0
S
N ( S
0
) the set I remains an independent set. Also note that all the vertices removed at a particular stage are either vertices in I or neighbours of some vertex in I . So the algorithm always outputs a maximal independent set. We also note that it can be easily parallelized on a CREW PRAM.
8.2
Expected Running Time
In this section we bound the expected running time of the algorithm and in the next section we derandomize it. Let G j
= ( V j
, E j
) denote the graph after stage j .
Main Lemma : For some c < 1,
E (

E j
 
E j
−
1
) < c

E j
−
1

.
Hence, in expectation, only O (log m ) rounds will be required, where m =

E
0

.
We say vertex v is BAD if more than 2 / 3 of the neighbors of v are of higher degree than is BAD if both of its endpoints are bad; otherwise the edge is GOOD.
v . We say an edge
The key claims are that at least half the edges are GOOD, and each GOOD edge is deleted with a constant probability. The main lemma then follows immediately.
Lemma 8.1
At least half the edges are GOOD.
Proof: f ( e
1
)
∩ f
Denote the set of bad edges by
( e
2
) =
∅
. This proves
E
B
. We will define

E
B
 ≤ 
E

/ 2, and we’re done.
f : E
B
→
E
2 so that for all e
1
= e
2
∈
E
B
,
The function f is defined as follows. For each ( u, v ) as in the algorithm. Now, suppose ( u, v )
∈
E
B
∈
E , direct it to the higher degree vertex. Break ties
, and is directed towards v . Since v is BAD, it has at least twice as many edges out as in. Hence we can pair two edges out of v with every edge into v . This gives our mapping.
Lemma 8.2
If v is GOOD then Pr ( N ( v )
∩
S =
∅
)
≥
2 α , where α :=
1
2
(1
− e
−
1 / 6
) .
Proof: Define L ( v ) :=
{ w
∈
N ( v )
 d ( w )
≤ d ( v )
}
.
Lecture 8: Luby’s Alg. for Maximal Independent Sets using Pairwise Independence
By definition,

L ( v )
 ≥ d ( v )
3 if v is a GOOD vertex.
Pr ( N ( v )
∩
S =
∅
) =
=
≥
=
≥
≥
≥
1
−
Pr ( N ( v )
∩
S =
∅
)
Y
1
−
Pr ( w
6∈
S ) w
∈
N ( v )
Y
1 − w
∈
L ( v )
Pr ( w 6∈ S )
1
−
Y
1
1
−
2 d ( w ) w ∈ L ( v )
1
−
Y
1
1
−
2 d ( v ) w
∈
L ( v )
1
− exp(
−
L ( v )

/ 2 d ( v ))
1
− exp(
−
1 / 6) , using full independence
Note, the above lemma is using full independence in its proof.
Lemma 8.3
Pr ( S
0
 w
∈
S )
≤
1 / 2 .
Proof: Let H ( w ) = N ( w )
\
L ( w ) =
{ z
∈
N ( w ) : d ( z ) > d ( w )
}
.
Pr ( S
0
 w
∈
S ) =
≤
≤
=
=
=
≤
≤
Pr ( H ( w )
∩
S =
∅ 
X
Pr ( z
∈
S
 w
∈
S w
∈
S )
) z ∈ H ( w )
X z ∈ H ( w )
Pr ( z
∈
S, w
∈
S )
Pr ( w
∈
S )
X z ∈ H ( w )
Pr ( z
∈
S ) Pr ( w
∈
S )
Pr ( w
∈
S )
X
Pr ( z
∈
S ) z
∈
H ( w )
X
1 z
∈
H ( w )
2 d ( z )
X
1 z ∈ H ( w )
2 d ( v )
1
2
.
using pairwise independence
Lemma 8.4
If v is GOOD then Pr ( v
∈
N ( S
0
) )
≥
α
83
84 Lecture 8: Luby’s Alg. for Maximal Independent Sets using Pairwise Independence
Proof: Let V
G denote the GOOD vertices. We have
Pr ( v
∈
N ( S
0
)
 v
∈
V
G
) =
=
≥
≥
=
Pr (
(1 / 2)(2 α )
α
N ( v )
∩
S
0
=
∅  v
∈
V
G
)
Pr ( N ( v )
∩
S
0
=
∅ 
Pr ( w
∈
S
0
 w
∈
N (
N v )
( v )
∩
S
∩
S, v
=
∅
, v
∈
V
G
∈
V
G
) Pr ( N
) Pr (
( v )
∩
N
S
( v )
∩
=
∅ 
S v
=
∅ 
∈
V
G v
)
∈
V
G
)
Corollary 8.5
If v is GOOD then the probability that v gets deleted is at least α .
Corollary 8.6
If an edge e is GOOD then the probability that it gets deleted is at least α .
Proof:
Pr ( e = ( u, v )
∈
E j
−
1
\
E j
)
≥
Pr ( v gets deleted ) .
We now restate the main lemma :
Main Lemma :
E (

E j
 
E j − 1
)
≤ 
E j − 1

(1
−
α/ 2) .
Proof:
E (

E j
 
E j
−
1
) =
≤
≤
X
1
−
Pr ( e gets deleted ) e ∈ E j
−
1

E j
−
1
 −
α

GOOD edges


E j − 1

(1
−
α/ 2) .
The constant α is approximately 0 .
07676.
Thus,
E (

E j

)
≤ 
E
0

(1
−
α
) j
2
≤ m exp(
− jα/ 2) < 1 , for j >
2
α log m . Therefore, the expected number of rounds required is
≤
4 m = O (log m ).
8.3
Derandomizing MIS
The only step where we use full independence is in Lemma 8.2 for lower bounding the probability that a
GOOD vertex gets picked.The argument we used was essentially the following :
Lemma 8.7
Let independent then
X i
, 1
≤ i
≤ n be
{
0 , 1
} random variables and p i
:= Pr (
Pr n
X
X i
> 0
!
≥
1
− n
Y
(1
− p i
)
1 1
X i
= 1 ) . If the X i are fully
Lecture 8: Luby’s Alg. for Maximal Independent Sets using Pairwise Independence 85
Here is the corresponding bound if the variables are pairwise independent
Lemma 8.8
Let independent then
X i
, 1
≤ i
≤ n be
{
0 , 1
} random variables and p i
:= Pr (
Pr n
X
X i
> 0
!
≥
1
1
2 min {
1
2
,
X p i
} i
X i
= 1 ) . If the X i are pairwise
Proof: Suppose
P i p i
≤ some algebra at the end)
1 / 2 .
Then we have the following, (the condition
P i p i
≤
1 / 2 will only come into
Pr
X
X i
> 0 i
!
≥
=
≥
=
≥
X
Pr ( X i
= 1 ) −
1
2
X
Pr ( X i i i = j
= 1 , X j
= 1 )
X p i
−
1
2
X p i p j i i = j
X p i
−
1
2 i
X p i
!
2 i
X p i
1
−
1
2 i
X p i
!
i
1
2
X p i i when
X i p i
≤ 1 .
If
1 /
P
2 i
≤ p i
P
> i ∈ S
1 / p i
2, then we restrict our index of summation to a set and the same basic argument works. Note, if
≤
1
P i p i
>
S
⊆
[ n ] such that 1 / 2
1, there always must exist a subset
≤
S
P
⊆ i ∈ S
[ n p i
≤
1,
] where
Using the construction described earlier we can now derandomize to get an algorithm that runs in O ( mn/α ) time (this is asymptotically as good as the sequential algorithm). The advantage of this method is that it can be easily parallelized to give an N C
2 algorithm (using O ( m ) processors).
8.4
History and Open Questions
The k
− wise independence derandomization approach was developed in [CG], [ABI] and [L85]. The maximal independence problem was first shown to be in NC in [KW].They showed that MIS in is N C
4
. Subsequently, improvements on this were found by [ABI] and [L85]. The version described in these notes is the latter.
The question of whether MIS is in N C
1 is still open.
References
[1] N Alon, L. Babai, A. Itai, “A fast and simple randomized parallel algorithm for the Maximal Independent
Set Problem”, Journal of Algorithms , Vol. 7, 1986, pp. 567583.
[2] B. Chor, O. Goldreich, “On the power of twopoint sampling”, pp.96106.
Journal of Complexity , Vol. 5, 1989,
86 Lecture 8: Luby’s Alg. for Maximal Independent Sets using Pairwise Independence
[3] R.M. Karp, A. Wigderson, A fast parallel algorithm for the maximal independent set problem , Proc.
16th ACM Symposium on Theory of Computing, 1984, pp. 266272.
[4] M. Luby, A simple parallel algorithm for the maximal independent set problem , Proc, 17th ACM Symposium on Theory of Computing, 1985, pp. 110.