<s>
Borůvka	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
for	O
finding	O
a	O
minimum	O
spanning	O
tree	O
in	O
a	O
graph	B-Application
,	O
</s>
<s>
or	O
a	O
minimum	O
spanning	O
forest	O
in	O
the	O
case	O
of	O
a	O
graph	B-Application
that	O
is	O
not	O
connected	O
.	O
</s>
<s>
This	O
algorithm	O
is	O
frequently	O
called	O
Sollin	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
especially	O
in	O
the	O
parallel	B-Operating_System
computing	I-Operating_System
literature	O
.	O
</s>
<s>
The	O
algorithm	O
begins	O
by	O
finding	O
the	O
minimum-weight	O
edge	O
incident	O
to	O
each	O
vertex	O
of	O
the	O
graph	B-Application
,	O
and	O
adding	O
all	O
of	O
those	O
edges	O
to	O
the	O
forest	O
.	O
</s>
<s>
Each	O
repetition	O
of	O
this	O
process	O
reduces	O
the	O
number	O
of	O
trees	O
,	O
within	O
each	O
connected	O
component	O
of	O
the	O
graph	B-Application
,	O
to	O
at	O
most	O
half	O
of	O
this	O
former	O
value	O
,	O
</s>
<s>
The	O
following	O
pseudocode	O
illustrates	O
a	O
basic	O
implementation	O
of	O
Borůvka	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
A	O
tie-breaking	O
rule	O
is	O
necessary	O
to	O
ensure	O
that	O
the	O
created	O
graph	B-Application
is	O
indeed	O
a	O
forest	O
,	O
that	O
is	O
,	O
it	O
does	O
not	O
contain	O
cycles	O
.	O
</s>
<s>
For	O
example	O
,	O
consider	O
a	O
triangle	O
graph	B-Application
with	O
nodes	O
 { a , b , c } 	O
and	O
all	O
edges	O
of	O
weight	O
1	O
.	O
</s>
<s>
input	O
:	O
A	O
weighted	O
undirected	O
graph	B-Application
G	O
=	O
(	O
V	O
,	O
E	O
)	O
.	O
</s>
<s>
Borůvka	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
can	O
be	O
shown	O
to	O
take	O
iterations	O
of	O
the	O
outer	O
loop	O
until	O
it	O
terminates	O
,	O
and	O
therefore	O
to	O
run	O
in	O
time	O
,	O
where	O
is	O
the	O
number	O
of	O
edges	O
,	O
and	O
is	O
the	O
number	O
of	O
vertices	O
in	O
(	O
assuming	O
)	O
.	O
</s>
<s>
In	O
planar	O
graphs	O
,	O
and	O
more	O
generally	O
in	O
families	O
of	O
graphs	O
closed	O
under	O
graph	B-Application
minor	O
operations	O
,	O
it	O
can	O
be	O
made	O
to	O
run	O
in	O
linear	O
time	O
,	O
by	O
removing	O
all	O
but	O
the	O
cheapest	O
edge	O
between	O
each	O
pair	O
of	O
components	O
after	O
each	O
stage	O
of	O
the	O
algorithm	O
.	O
</s>
<s>
Image	O
components	O
Description200px	O
{A}{B}{C}{D}{E}{F}{G}This	O
is	O
our	O
original	O
weighted	O
graph	B-Application
.	O
</s>
<s>
Other	O
algorithms	O
for	O
this	O
problem	O
include	O
Prim	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
and	O
Kruskal	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Fast	O
parallel	O
algorithms	O
can	O
be	O
obtained	O
by	O
combining	O
Prim	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
with	O
Borůvka	O
's	O
.	O
</s>
<s>
A	O
faster	O
randomized	O
minimum	O
spanning	O
tree	O
algorithm	O
based	O
in	O
part	O
on	O
Borůvka	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
due	O
to	O
Karger	O
,	O
Klein	O
,	O
and	O
Tarjan	O
runs	O
in	O
expected	O
time	O
.	O
</s>
<s>
These	O
randomized	O
and	O
deterministic	O
algorithms	O
combine	O
steps	O
of	O
Borůvka	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
reducing	O
the	O
number	O
of	O
components	O
that	O
remain	O
to	O
be	O
connected	O
,	O
with	O
steps	O
of	O
a	O
different	O
type	O
that	O
reduce	O
the	O
number	O
of	O
edges	O
between	O
pairs	O
of	O
components	O
.	O
</s>
