<s>
In	O
computer	B-General_Concept
science	I-General_Concept
and	O
graph	O
theory	O
,	O
Karger	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
a	O
randomized	B-General_Concept
algorithm	I-General_Concept
to	O
compute	O
a	O
minimum	O
cut	B-Algorithm
of	O
a	O
connected	O
graph	O
.	O
</s>
<s>
All	O
other	O
edges	O
connecting	O
either	O
or	O
are	O
"	O
reattached	O
"	O
to	O
the	O
merged	O
node	O
,	O
effectively	O
producing	O
a	O
multigraph	B-Language
.	O
</s>
<s>
Karger	O
's	O
basic	O
algorithm	O
iteratively	O
contracts	O
randomly	O
chosen	O
edges	O
until	O
only	O
two	O
nodes	O
remain	O
;	O
those	O
nodes	O
represent	O
a	O
cut	B-Algorithm
in	O
the	O
original	O
graph	O
.	O
</s>
<s>
By	O
iterating	O
this	O
basic	O
algorithm	O
a	O
sufficient	O
number	O
of	O
times	O
,	O
a	O
minimum	O
cut	B-Algorithm
can	O
be	O
found	O
with	O
high	O
probability	O
.	O
</s>
<s>
A	O
cut	B-Algorithm
in	O
an	O
undirected	O
graph	O
is	O
a	O
partition	O
of	O
the	O
vertices	O
into	O
two	O
non-empty	O
,	O
disjoint	O
sets	O
.	O
</s>
<s>
The	O
cutset	B-Algorithm
of	O
a	O
cut	B-Algorithm
consists	O
of	O
the	O
edges	O
between	O
the	O
two	O
parts	O
.	O
</s>
<s>
The	O
size	O
(	O
or	O
weight	O
)	O
of	O
a	O
cut	B-Algorithm
in	O
an	O
unweighted	O
graph	O
is	O
the	O
cardinality	O
of	O
the	O
cutset	B-Algorithm
,	O
i.e.	O
,	O
the	O
number	O
of	O
edges	O
between	O
the	O
two	O
parts	O
,	O
</s>
<s>
Among	O
the	O
remaining	O
choices	O
,	O
swapping	O
the	O
roles	O
of	O
and	O
does	O
not	O
change	O
the	O
cut	B-Algorithm
,	O
so	O
each	O
cut	B-Algorithm
is	O
counted	O
twice	O
;	O
therefore	O
,	O
there	O
are	O
distinct	O
cuts	O
.	O
</s>
<s>
The	O
minimum	O
cut	B-Algorithm
problem	O
is	O
to	O
find	O
a	O
cut	B-Algorithm
of	O
smallest	O
size	O
among	O
these	O
cuts	O
.	O
</s>
<s>
A	O
cut	B-Algorithm
is	O
sometimes	O
called	O
a	O
“	O
global	O
cut	B-Algorithm
”	O
to	O
distinguish	O
it	O
from	O
an	O
“	O
-	O
cut	B-Algorithm
”	O
for	O
a	O
given	O
pair	O
of	O
vertices	O
,	O
which	O
has	O
the	O
additional	O
requirement	O
that	O
and	O
.	O
</s>
<s>
Every	O
global	O
cut	B-Algorithm
is	O
an	O
-	O
cut	B-Algorithm
for	O
some	O
.	O
</s>
<s>
Thus	O
,	O
the	O
minimum	O
cut	B-Algorithm
problem	O
can	O
be	O
solved	O
in	O
polynomial	O
time	O
by	O
iterating	O
over	O
all	O
choices	O
of	O
and	O
solving	O
the	O
resulting	O
minimum	O
-	O
cut	B-Algorithm
problem	O
using	O
the	O
max-flow	B-Algorithm
min-cut	I-Algorithm
theorem	I-Algorithm
and	O
a	O
polynomial	O
time	O
algorithm	O
for	O
maximum	B-Algorithm
flow	I-Algorithm
,	O
such	O
as	O
the	O
push-relabel	B-Algorithm
algorithm	I-Algorithm
,	O
though	O
this	O
approach	O
is	O
not	O
optimal	O
.	O
</s>
<s>
Better	O
deterministic	O
algorithms	O
for	O
the	O
global	O
minimum	O
cut	B-Algorithm
problem	O
include	O
the	O
Stoer	B-Algorithm
–	I-Algorithm
Wagner	I-Algorithm
algorithm	I-Algorithm
,	O
which	O
has	O
a	O
running	O
time	O
of	O
.	O
</s>
<s>
The	O
fundamental	O
operation	O
of	O
Karger	B-Algorithm
’s	I-Algorithm
algorithm	I-Algorithm
is	O
a	O
form	O
of	O
edge	O
contraction	O
.	O
</s>
<s>
The	O
contraction	O
algorithm	O
repeatedly	O
contracts	O
random	O
edges	O
in	O
the	O
graph	O
,	O
until	O
only	O
two	O
nodes	O
remain	O
,	O
at	O
which	O
point	O
there	O
is	O
only	O
a	O
single	O
cut	B-Algorithm
.	O
</s>
<s>
The	O
key	O
idea	O
of	O
the	O
algorithm	O
is	O
that	O
it	O
is	O
far	O
more	O
likely	O
for	O
non	O
min-cut	O
edges	O
than	O
min-cut	O
edges	O
to	O
be	O
randomly	O
selected	O
and	O
lost	O
to	O
contraction	O
,	O
since	O
min-cut	O
edges	O
are	O
usually	O
vastly	O
outnumbered	O
by	O
non	O
min-cut	O
edges	O
.	O
</s>
<s>
Subsequently	O
,	O
it	O
is	O
plausible	O
that	O
the	O
min-cut	O
edges	O
will	O
survive	O
all	O
the	O
edge	O
contraction	O
,	O
and	O
the	O
algorithm	O
will	O
correctly	O
identify	O
the	O
min-cut	O
edge	O
.	O
</s>
<s>
When	O
the	O
graph	O
is	O
represented	O
using	O
adjacency	B-Data_Structure
lists	I-Data_Structure
or	O
an	O
adjacency	B-Algorithm
matrix	I-Algorithm
,	O
a	O
single	O
edge	O
contraction	O
operation	O
can	O
be	O
implemented	O
with	O
a	O
linear	O
number	O
of	O
updates	O
to	O
the	O
data	O
structure	O
,	O
for	O
a	O
total	O
running	O
time	O
of	O
.	O
</s>
<s>
Alternatively	O
,	O
the	O
procedure	O
can	O
be	O
viewed	O
as	O
an	O
execution	O
of	O
Kruskal	B-Algorithm
’s	I-Algorithm
algorithm	I-Algorithm
for	O
constructing	O
the	O
minimum	O
spanning	O
tree	O
in	O
a	O
graph	O
where	O
the	O
edges	O
have	O
weights	O
according	O
to	O
a	O
random	O
permutation	O
.	O
</s>
<s>
Removing	O
the	O
heaviest	O
edge	O
of	O
this	O
tree	O
results	O
in	O
two	O
components	O
that	O
describe	O
a	O
cut	B-Algorithm
.	O
</s>
<s>
In	O
this	O
way	O
,	O
the	O
contraction	O
procedure	O
can	O
be	O
implemented	O
like	O
Kruskal	B-Algorithm
’s	I-Algorithm
algorithm	I-Algorithm
in	O
time	O
.	O
</s>
<s>
In	O
a	O
graph	O
with	O
vertices	O
,	O
the	O
contraction	O
algorithm	O
returns	O
a	O
minimum	O
cut	B-Algorithm
with	O
polynomially	O
small	O
probability	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
success	O
probability	O
for	O
this	O
algorithm	O
is	O
much	O
better	O
than	O
the	O
probability	O
for	O
picking	O
a	O
cut	B-Algorithm
at	O
random	O
,	O
which	O
is	O
at	O
most	O
.	O
</s>
<s>
To	O
further	O
establish	O
the	O
lower	O
bound	O
on	O
the	O
success	O
probability	O
,	O
let	O
denote	O
the	O
edges	O
of	O
a	O
specific	O
minimum	O
cut	B-Algorithm
of	O
size	O
.	O
</s>
<s>
The	O
contraction	O
algorithm	O
returns	O
if	O
none	O
of	O
the	O
random	O
edges	O
deleted	O
by	O
the	O
algorithm	O
belongs	O
to	O
the	O
cutset	B-Algorithm
.	O
</s>
<s>
The	O
minimum	O
degree	O
of	O
is	O
at	O
least	O
(	O
otherwise	O
a	O
minimum	O
degree	O
vertex	O
would	O
induce	O
a	O
smaller	O
cut	B-Algorithm
where	O
one	O
of	O
the	O
two	O
partitions	O
contains	O
only	O
the	O
minimum	O
degree	O
vertex	O
)	O
,	O
so	O
.	O
</s>
<s>
An	O
extension	O
of	O
Karger	B-Algorithm
’s	I-Algorithm
algorithm	I-Algorithm
due	O
to	O
David	O
Karger	O
and	O
Clifford	O
Stein	O
achieves	O
an	O
order	O
of	O
magnitude	O
improvement	O
.	O
</s>
<s>
To	O
determine	O
a	O
min-cut	O
,	O
one	O
has	O
to	O
touch	O
every	O
edge	O
in	O
the	O
graph	O
at	O
least	O
once	O
,	O
which	O
is	O
time	O
in	O
a	O
dense	O
graph	O
.	O
</s>
<s>
The	O
Stein	O
's	O
min-cut	O
algorithm	O
takes	O
the	O
running	O
time	O
of	O
,	O
which	O
is	O
very	O
close	O
to	O
that	O
.	O
</s>
