<s>
Among	O
them	O
are	O
Prim	B-Algorithm
's	I-Algorithm
,	O
Kruskal	B-Algorithm
's	I-Algorithm
and	O
Borůvka	B-Algorithm
's	I-Algorithm
algorithms	I-Algorithm
,	O
each	O
utilising	O
different	O
properties	O
of	O
MSTs	O
.	O
</s>
<s>
One	O
option	O
of	O
improving	O
it	O
is	O
by	O
parallelising	B-Operating_System
known	O
MST	O
algorithms	O
.	O
</s>
<s>
This	O
selection	O
is	O
often	O
performed	O
using	O
a	O
priority	B-Application
queue	I-Application
(	O
PQ	O
)	O
.	O
</s>
<s>
Thus	O
using	O
Fibonacci	B-Application
heaps	I-Application
the	O
total	O
runtime	O
of	O
Prim	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
asymptotically	O
in	O
.	O
</s>
<s>
One	O
possible	O
idea	O
is	O
to	O
use	O
processors	O
to	O
support	O
PQ	O
access	O
in	O
on	O
an	O
EREW-PRAM	B-Operating_System
machine	O
,	O
thus	O
lowering	O
the	O
total	O
runtime	O
to	O
.	O
</s>
<s>
Kruskal	B-Algorithm
's	I-Algorithm
MST	O
algorithm	O
utilises	O
the	O
cycle	O
property	O
of	O
MSTs	O
.	O
</s>
<s>
The	O
subtrees	O
of	O
are	O
stored	O
in	O
union-find	B-Algorithm
data	I-Algorithm
structures	I-Algorithm
,	O
which	O
is	O
why	O
checking	O
whether	O
or	O
not	O
two	O
vertices	O
are	O
in	O
the	O
same	O
subtree	O
is	O
possible	O
in	O
amortised	O
where	O
is	O
the	O
inverse	O
Ackermann	O
function	O
.	O
</s>
<s>
Similarly	O
to	O
Prim	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
there	O
are	O
components	O
in	O
Kruskal	B-Algorithm
's	I-Algorithm
approach	O
that	O
can	O
not	O
be	O
parallelised	O
in	O
its	O
classical	O
variant	O
.	O
</s>
<s>
The	O
basic	O
idea	O
behind	O
Filter-Kruskal	O
is	O
to	O
partition	O
the	O
edges	O
in	O
a	O
similar	O
way	O
to	O
quicksort	B-Algorithm
and	O
filter	O
out	O
edges	O
that	O
connect	O
vertices	O
that	O
belong	O
to	O
the	O
same	O
tree	O
in	O
order	O
to	O
reduce	O
the	O
cost	O
of	O
sorting	O
.	O
</s>
<s>
The	O
main	O
idea	O
behind	O
Borůvka	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
edge	O
contraction	O
.	O
</s>
<s>
There	O
are	O
multiple	O
other	O
parallel	B-Operating_System
algorithms	I-Operating_System
that	O
deal	O
the	O
issue	O
of	O
finding	O
an	O
MST	O
.	O
</s>
