CME 305: Discrete Mathematics and Algorithms
Instructor: Professor Amin Saberi ([email protected])
February 25, 2010
Lecture 11: Randomized Algorithms II
1
Randomized Algorithm for Finding a Perfect Matching
In this lecture, we extend the randomized algorithm from last lecture that determined the existence of a
perfect matching in a graph
G
(
V, E
), to actually finding a perfect matching in
G
. The key computational
step will be a matrix inversion.
First we establish a key (and surprising) property of subsets of random numbers:
Definition:
A
set system
(
S, F
) consists of a finite set
S
of elements,
S
=
{
x
1
, x
2
, . . . , x
n
}
and a family
F
of subsets of
S
,
F
=
{
S
1
, . . . , S
k
}
,
∅ ⊂
S
j
⊆
S
.
For each element of
S
, assign weight
w
i
to
x
i
, where
w
i
is chosen uniformly at random and independently
from
{
1
, . . . ,
2
n
}
. Denote the weight of set
S
j
to be
∑
x
i
∈
S
j
w
i
.
Lemma 1 (Isolating lemma)
The probability that there is a unique minimum weight set is at least
1
/
2
.
Proof:
Fix the weight of all elements except
x
i
.
Given
F
, define the
threshold
for element
x
i
to be the
number
α
i
such that if
w
i
> α
i
then
x
i
is in no set with minimum weight, and if
w
i
≤
α
i
then
x
i
is in some
set with minimum weight.
Clearly, if
w
i
< α
i
, then
x
i
is in
every
set with minimum weight. The only ambiguity occurs when
w
i
=
α
i
.
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 Winter '09
 Graph Theory, Bipartite graph, minimum weight

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