0.1
Introduction
Here are some of the typical uses of randomness in parallel computation.
The first use is for coordinating processors; usually random methods are faster
than deterministic ones, or in someways inferior.
For example, if two processors
contend for the same resource, then we can flip a coin to decide who gets it.
A more general application is what is called “symmetry breaking” – there are
some
P
processors wanting one of some
k
resources. The problem is how to coordi
nate who gets what, and so that as many processors as possible are given a resource.
One idea would be to each processor make a request to a random resource: this
would distribute the processors reasonably evenly amongst the resources, and then
use coin tossing to remove further contention. In this case, deterministic methods
might require more computation, or might not distribute the load uniformly.
A related use is in packet routing algorithms, where we use random priorities
for messages. This effectively puts messages into batches, with each batch getting
roughly the same number on the average. Notice that this is a strong statement: by
randomly assigning messages into batches we are likely to get a good partitioning
on all links in the system. Whereas it is very easy for deterministic methods to get
a perfectly good partitioning on a single link, but doing it simultaneously on all
links is quite hard.
1
Probability
Probability space
is defined as the set of all the possible outcomes of an experiment.
This is also known as
sample space
.
Let us denote the sample space by
P
.
Any
subset of the probability space is known as an
event
.
Example 1:
If we perform the experiment of flipping two coins then the probability
space is
C
2
=
{
tt, th, ht, hh
}
. The event that there is atleast one tail is given by
{
tt, th, ht
}
.
Example 2:
If the experiment performed is drawing a card from the deck, then
the probability space will be the entire deck (
D
).
A
probability distribution
is a function
P
from the set of events to
<
. This function
should satisfy the following axioms of probability.
Axioms of Probability
1. For any event
A
,
P
(
A
)
≥
0
2.
P
(
P
) = 1,
3. For any infinite sequence of disjoint events
A
1
, A
2
, . . .
,
P
∞
[
i
=1
A
i
!
=
∞
X
i
=1
P
(
A
i
)
1.1
Basic Properties of Probability
1.
P
(
∅
) = 0.
1
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2. For any finite sequence of disjoint events
A
1
, A
2
, . . . A
n
P
n
[
i
=1
A
i
!
=
n
X
i
=1
P
(
A
i
)
3. For any event
A
,
P
(
A
c
) = 1

P
(
A
).
4. For any event
A
, 0
≤
P
(
A
)
≤
1.
5. If
A
⊂
B
, then
P
(
A
)
≤
P
(
B
).
1.2
Random Variables
A
random variable
X
is a function from the probability space to the set of real
numbers. Formally,
X
:
P → <
.
Example 1:
Let
H
2
be the no. of heads in 2 coin flips, hence in our sample space
C
2
, we have,
H
2
(
tt
) = 0
, H
2
(
ht
) = 1
, H
2
(
th
) = 1
, H
2
(
hh
) = 2.
Example 2:
Let
C
be 1 if the card drawn is a club, 0 otherwise. Let
A
be 1 if the
card drawn is an ace, 0 otherwise.
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 Summer '09
 PROF.RANADE
 Algorithms, Probability theory

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