Mcnteearlo
AlgOrithIIlS
for
Enumeration
and
Reliability
Problems
Richard M. Karpt
University
oJ
California at Berkeley
Michael Lubyt
01
Toronto
1. Introduction
We
present a simple but very general
MonteCarlo technique for the approximate
solution of enumeration and reliability prob
lems. Several applications are given, includ
ing:
1.
Estimating the number of triangulated
plane maps with a given number of ver
tices;
2.
Estimating the cardinality of a union of
sets;
3.
Estimating the number of input combi
nations for which
a
boolean function,
presented in disjunctive normal form,
assume the value
true;
4.
Estimating the failure probability of a
system with faulty components.
1.1 Randomized Approximation Algorithms
and Approximation Schemes
Let
f
be a function
~rom
some domain
D
into the positive reals.
shall be con
cerned with randomized algorithms which
accept as input any
wED
and produce as
output a positive real number
1
(w)
which is
an estimate of
f (w).
Since the algorithm
involves randomization,
1
is a random
variable,
rather
than a constant, for each
:fixed
w.
Such a randomized algorithm is called
an
(l:,o)
approximation algorithm for
J
if,
for every input
wED,
Pr
[I
l(
w
)
t
{w
}
I
>
l;
1
<
0
J{w) .
tResearch supported
by
NSF Grant MCS8105217
02725428/83/0000/0056$01.00
©
1983 IEEE
56
In a similar spirit, we can discuss random
ized approximation methods in which
~
0,
as'well as
w,
are
part of the input. Such a
randomized algorithm is called a random
ized approximation scheme for
f
if,
every
input
triple
(l:,O,w),
where
E
>
0
0
<
0
<
1,
the algorithm produces
as output a real number
l£,o(w)
such that
['le'~jl~f(W)
I
>
e
I
<
l5
In cases where
the domain
D
is a
set of
strings,
a
randomized
approximation
scheme is called
fully
polynomial if its exe
cution time is bounded
a polynomial in
;.
~
and the length of
We derive ran
domized
schemes
the
problems mentioned above.
In particular,
we
give
a
polynomial
scheme
estimating the
number of input combina
tions
that make
a
disjunctive normal form
boolean formula
true.
Thus we have
a
scheme for
a
IP
complete problem.
2. Counting Equivalence Classes
The general principles underlying all our
results
can be described abstractly as fol
lows.
S
be
a
finite
set on which an
equivalence relation
'"'J
is defined.
We wish
to
estimate the number of equivalence classes
into which
I'V
partitions
S.
The number of
equivalence classes will be denoted
1
S /I'J I.
We assume
that
1
s
I
J
the cardinality of
S,
is
known.
[x]
denote the equivalence class
containing
% .
two
MonteCarlo
methods for
estimating'S
/f'V ,.
Each of these methods
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t
trials.
The estimator of
IS/row
I
is
X +X
+
...
+X
1
2
t
,where
Xi
is the result of
t
the
it",
trial.
The random variables
~
are
independent and identically distributed, and
E
[~]
=
I
S
/~
I·
Each of
the methods requires
a
pro
cedure
for
choosing
elements
at random
from
S.
Method 1
assumes that, given
%, we
can determine the number of elements in
[x
].
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 Spring '08
 Staff
 Algorithms, Probability theory, Randomness, Equivalence relation, vertices, Plane Triangulations

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