<s>
In	O
mathematics	O
and	O
computer	B-General_Concept
science	I-General_Concept
,	O
the	O
probabilistic	O
method	O
is	O
used	O
to	O
prove	O
the	O
existence	O
of	O
mathematical	O
objects	O
with	O
desired	O
combinatorial	O
properties	O
.	O
</s>
<s>
The	O
method	B-Algorithm
of	I-Algorithm
conditional	I-Algorithm
probabilities	I-Algorithm
,	O
converts	O
such	O
a	O
proof	O
,	O
in	O
a	O
"	O
very	O
precise	O
sense	O
"	O
,	O
into	O
an	O
efficient	O
deterministic	B-General_Concept
algorithm	I-General_Concept
,	O
one	O
that	O
is	O
guaranteed	O
to	O
compute	O
an	O
object	O
with	O
the	O
desired	O
properties	O
.	O
</s>
<s>
The	O
method	O
is	O
particularly	O
relevant	O
in	O
the	O
context	O
of	O
randomized	B-Algorithm
rounding	I-Algorithm
(	O
which	O
uses	O
the	O
probabilistic	O
method	O
to	O
design	O
approximation	B-Algorithm
algorithms	I-Algorithm
)	O
.	O
</s>
<s>
When	O
applying	O
the	O
method	B-Algorithm
of	I-Algorithm
conditional	I-Algorithm
probabilities	I-Algorithm
,	O
the	O
technical	O
term	O
pessimistic	B-Algorithm
estimator	I-Algorithm
refers	O
to	O
a	O
quantity	O
used	O
in	O
place	O
of	O
the	O
true	O
conditional	O
probability	O
(	O
or	O
conditional	O
expectation	O
)	O
underlying	O
the	O
proof	O
.	O
</s>
<s>
We	O
first	O
show	O
the	O
existence	O
of	O
a	O
provably	O
good	O
approximate	O
solution	O
using	O
the	O
probabilistic	O
method	O
...	O
[	O
We	O
then ]	O
show	O
that	O
the	O
probabilistic	O
existence	O
proof	O
can	O
be	O
converted	O
,	O
in	O
a	O
very	O
precise	O
sense	O
,	O
into	O
a	O
deterministic	O
approximation	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
(	O
Raghavan	O
is	O
discussing	O
the	O
method	O
in	O
the	O
context	O
of	O
randomized	B-Algorithm
rounding	I-Algorithm
,	O
but	O
it	O
works	O
with	O
the	O
probabilistic	O
method	O
in	O
general	O
.	O
)	O
</s>
<s>
To	O
apply	O
the	O
method	B-Algorithm
of	I-Algorithm
conditional	I-Algorithm
probabilities	I-Algorithm
,	O
one	O
focuses	O
on	O
the	O
conditional	O
probability	O
of	O
failure	O
,	O
given	O
the	O
choices	O
so	O
far	O
as	O
the	O
experiment	O
proceeds	O
step	O
by	O
step	O
.	O
</s>
<s>
The	O
method	B-Algorithm
of	I-Algorithm
conditional	I-Algorithm
probabilities	I-Algorithm
replaces	O
the	O
random	O
root-to-leaf	O
walk	O
in	O
the	O
random	O
experiment	O
by	O
a	O
deterministic	O
root-to-leaf	O
walk	O
,	O
where	O
each	O
step	O
is	O
chosen	O
to	O
inductively	O
maintain	O
the	O
following	O
invariant	O
:	O
</s>
<s>
Using	O
a	O
pessimistic	B-Algorithm
estimator	I-Algorithm
:	O
In	O
some	O
cases	O
,	O
as	O
a	O
proxy	O
for	O
the	O
exact	O
conditional	O
expectation	O
of	O
the	O
quantity	O
Q	O
,	O
one	O
uses	O
an	O
appropriately	O
tight	O
bound	O
called	O
a	O
pessimistic	B-Algorithm
estimator	I-Algorithm
.	O
</s>
<s>
The	O
pessimistic	B-Algorithm
estimator	I-Algorithm
is	O
a	O
function	O
of	O
the	O
current	O
state	O
.	O
</s>
<s>
Typically	O
,	O
a	O
good	O
pessimistic	B-Algorithm
estimator	I-Algorithm
can	O
be	O
computed	O
by	O
precisely	O
deconstructing	O
the	O
logic	O
of	O
the	O
original	O
proof	O
.	O
</s>
<s>
This	O
example	O
demonstrates	O
the	O
method	B-Algorithm
of	I-Algorithm
conditional	I-Algorithm
probabilities	I-Algorithm
using	O
a	O
conditional	O
expectation	O
.	O
</s>
<s>
To	O
apply	O
the	O
method	B-Algorithm
of	I-Algorithm
conditional	I-Algorithm
probabilities	I-Algorithm
,	O
first	O
model	O
the	O
random	O
experiment	O
as	O
a	O
sequence	O
of	O
small	O
random	O
steps	O
.	O
</s>
<s>
Because	O
of	O
its	O
derivation	O
,	O
this	O
deterministic	B-General_Concept
algorithm	I-General_Concept
is	O
guaranteed	O
to	O
cut	O
at	O
least	O
half	O
the	O
edges	O
of	O
the	O
given	O
graph	O
.	O
</s>
<s>
The	O
next	O
example	O
demonstrates	O
the	O
use	O
of	O
pessimistic	B-Algorithm
estimators	I-Algorithm
.	O
</s>
<s>
Let	O
Q(t )	O
denote	O
the	O
above	O
quantity	O
,	O
which	O
is	O
called	O
a	O
pessimistic	B-Algorithm
estimator	I-Algorithm
for	O
the	O
conditional	O
expectation	O
.	O
</s>
<s>
The	O
proof	O
showed	O
that	O
the	O
pessimistic	B-Algorithm
estimator	I-Algorithm
is	O
initially	O
at	O
least	O
|V|	O
/	O
( D+1	O
)	O
.	O
</s>
<s>
The	O
algorithm	O
will	O
make	O
each	O
choice	O
to	O
keep	O
the	O
pessimistic	B-Algorithm
estimator	I-Algorithm
from	O
decreasing	O
,	O
that	O
is	O
,	O
so	O
that	O
Q( t+1	O
)	O
≥	O
Q(t )	O
for	O
each	O
t	O
.	O
Since	O
the	O
pessimistic	B-Algorithm
estimator	I-Algorithm
is	O
a	O
lower	O
bound	O
on	O
the	O
conditional	O
expectation	O
,	O
this	O
will	O
ensure	O
that	O
the	O
conditional	O
expectation	O
stays	O
above	O
|V|	O
/	O
( D+1	O
)	O
,	O
which	O
in	O
turn	O
will	O
ensure	O
that	O
the	O
conditional	O
probability	O
of	O
failure	O
stays	O
below	O
1	O
.	O
</s>
<s>
If	O
u	O
already	O
has	O
a	O
neighbor	O
in	O
S	O
,	O
then	O
u	O
is	O
not	O
added	O
to	O
S	O
and	O
(	O
by	O
inspection	O
of	O
Q(t )	O
)	O
,	O
the	O
pessimistic	B-Algorithm
estimator	I-Algorithm
is	O
unchanged	O
.	O
</s>
<s>
By	O
calculation	O
,	O
if	O
u	O
is	O
chosen	O
randomly	O
from	O
the	O
remaining	O
vertices	O
,	O
the	O
expected	O
increase	O
in	O
the	O
pessimistic	B-Algorithm
estimator	I-Algorithm
is	O
non-negative	O
.	O
</s>
<s>
Conditioned	O
on	O
choosing	O
a	O
vertex	O
in	O
R(t )	O
,	O
the	O
probability	O
that	O
a	O
given	O
term	O
1/	O
(d(w )	O
+1	O
)	O
is	O
dropped	O
from	O
the	O
sum	O
in	O
the	O
pessimistic	B-Algorithm
estimator	I-Algorithm
is	O
at	O
most	O
(d(w )	O
+1	O
)	O
/	O
|	O
R(t )	O
|	O
,	O
so	O
the	O
expected	O
decrease	O
in	O
each	O
term	O
in	O
the	O
sum	O
is	O
at	O
most	O
1/	O
|	O
R(t )	O
|	O
.	O
</s>
<s>
Thus	O
,	O
there	O
must	O
exist	O
some	O
choice	O
of	O
u	O
that	O
keeps	O
the	O
pessimistic	B-Algorithm
estimator	I-Algorithm
from	O
decreasing	O
.	O
</s>
<s>
The	O
algorithm	O
below	O
chooses	O
each	O
vertex	O
u	O
to	O
maximize	O
the	O
resulting	O
pessimistic	B-Algorithm
estimator	I-Algorithm
.	O
</s>
<s>
By	O
the	O
previous	O
considerations	O
,	O
this	O
keeps	O
the	O
pessimistic	B-Algorithm
estimator	I-Algorithm
from	O
decreasing	O
and	O
guarantees	O
a	O
successful	O
outcome	O
.	O
</s>
<s>
For	O
the	O
method	B-Algorithm
of	I-Algorithm
conditional	I-Algorithm
probabilities	I-Algorithm
to	O
work	O
,	O
it	O
suffices	O
if	O
the	O
algorithm	O
keeps	O
the	O
pessimistic	B-Algorithm
estimator	I-Algorithm
from	O
decreasing	O
(	O
or	O
increasing	O
,	O
as	O
appropriate	O
)	O
.	O
</s>
<s>
The	O
algorithm	O
does	O
not	O
necessarily	O
have	O
to	O
maximize	O
(	O
or	O
minimize	O
)	O
the	O
pessimistic	B-Algorithm
estimator	I-Algorithm
.	O
</s>
<s>
Each	O
algorithm	O
is	O
analyzed	O
with	O
the	O
same	O
pessimistic	B-Algorithm
estimator	I-Algorithm
as	O
before	O
.	O
</s>
