<s>
The	O
distributed	B-Operating_System
minimum	I-Operating_System
spanning	I-Operating_System
tree	I-Operating_System
(	O
MST	O
)	O
problem	O
involves	O
the	O
construction	O
of	O
a	O
minimum	O
spanning	O
tree	O
by	O
a	O
distributed	B-Operating_System
algorithm	I-Operating_System
,	O
in	O
a	O
network	O
where	O
nodes	O
communicate	O
by	O
message	B-Architecture
passing	I-Architecture
.	O
</s>
<s>
It	O
is	O
radically	O
different	O
from	O
the	O
classical	O
sequential	O
problem	O
,	O
although	O
the	O
most	O
basic	O
approach	O
resembles	O
Borůvka	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
One	O
important	O
application	O
of	O
this	O
problem	O
is	O
to	O
find	O
a	O
tree	O
that	O
can	O
be	O
used	O
for	O
broadcasting	B-Operating_System
.	O
</s>
<s>
The	O
message-passing	B-Architecture
model	O
is	O
one	O
of	O
the	O
most	O
commonly	O
used	O
models	O
in	O
distributed	B-Architecture
computing	I-Architecture
.	O
</s>
<s>
Two	O
commonly	O
used	O
algorithms	O
for	O
the	O
classical	O
minimum	O
spanning	O
tree	O
problem	O
are	O
Prim	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
and	O
Kruskal	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
However	O
,	O
it	O
is	O
difficult	O
to	O
apply	O
these	O
two	O
algorithms	O
in	O
the	O
distributed	O
message-passing	B-Architecture
model	O
.	O
</s>
<s>
Both	O
Prim	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
and	O
Kruskal	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
require	O
processing	O
one	O
node	O
or	O
vertex	O
at	O
a	O
time	O
,	O
making	O
it	O
difficult	O
to	O
make	O
them	O
run	O
in	O
parallel	O
.	O
</s>
<s>
For	O
example	O
,	O
Kruskal	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
processes	O
edges	O
in	O
turn	O
,	O
deciding	O
whether	O
to	O
include	O
the	O
edge	O
in	O
the	O
MST	O
based	O
on	O
whether	O
it	O
would	O
form	O
a	O
cycle	O
with	O
all	O
previously	O
chosen	O
edges	O
.	O
</s>
<s>
Both	O
Prim	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
and	O
Kruskal	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
require	O
processes	O
to	O
know	O
the	O
state	O
of	O
the	O
whole	O
graph	O
,	O
which	O
is	O
very	O
difficult	O
to	O
discover	O
in	O
the	O
message-passing	B-Architecture
model	O
.	O
</s>
<s>
Due	O
to	O
these	O
difficulties	O
,	O
new	O
techniques	O
were	O
needed	O
for	O
distributed	O
MST	O
algorithms	O
in	O
the	O
message-passing	B-Architecture
model	O
.	O
</s>
<s>
Some	O
bear	O
similarities	O
to	O
Borůvka	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
for	O
the	O
classical	O
MST	O
problem	O
.	O
</s>
<s>
The	O
GHS	O
algorithm	O
of	O
Gallager	O
,	O
Humblet	O
and	O
Spira	O
is	O
one	O
of	O
the	O
best-known	O
algorithms	O
in	O
distributed	B-Architecture
computing	I-Architecture
theory	O
.	O
</s>
<s>
This	O
algorithm	O
constructs	O
an	O
MST	O
in	O
the	O
asynchronous	O
message-passing	B-Architecture
model	O
.	O
</s>
<s>
Each	O
edge	O
delivers	O
messages	O
in	O
FIFO	B-Operating_System
order	O
.	O
</s>
<s>
This	O
algorithm	O
can	O
be	O
separated	O
into	O
three	O
stages	O
:	O
broadcast	B-Operating_System
,	O
convergecast	O
,	O
and	O
change	O
core	O
.	O
</s>
<s>
The	O
two	O
nodes	O
adjacent	O
to	O
the	O
core	O
broadcast	B-Operating_System
messages	O
to	O
the	O
rest	O
of	O
the	O
nodes	O
in	O
the	O
fragment	O
.	O
</s>
<s>
Each	O
broadcast	B-Operating_System
message	O
contains	O
the	O
ID	O
and	O
level	O
of	O
the	O
fragment	O
.	O
</s>
<s>
The	O
messages	O
sent	O
in	O
this	O
stage	O
are	O
in	O
the	O
opposite	O
direction	O
of	O
the	O
broadcast	B-Operating_System
stage	O
.	O
</s>
<s>
The	O
smallest	O
weight	O
will	O
be	O
sent	O
toward	O
the	O
branch	O
it	O
received	O
the	O
broadcast	B-Operating_System
from	O
.	O
</s>
<s>
As	O
discussed	O
above	O
,	O
every	O
node	O
needs	O
to	O
find	O
its	O
minimum	O
weight	O
outgoing	O
incident	O
edge	O
after	O
the	O
receipt	O
of	O
a	O
broadcast	B-Operating_System
message	O
from	O
the	O
core	O
.	O
</s>
<s>
If	O
node	O
receives	O
a	O
broadcast	B-Operating_System
,	O
it	O
will	O
pick	O
its	O
minimum	O
weight	O
basic	O
edge	O
and	O
send	O
a	O
message	O
to	O
the	O
node	O
on	O
the	O
other	O
side	O
with	O
its	O
fragment	O
's	O
ID	O
and	O
level	O
.	O
</s>
<s>
The	O
reason	O
is	O
that	O
the	O
two	O
nodes	O
may	O
belong	O
to	O
the	O
same	O
fragment	O
already	O
,	O
but	O
node	O
has	O
not	O
discovered	O
this	O
fact	O
yet	O
due	O
to	O
the	O
delay	O
of	O
a	O
broadcast	B-Operating_System
message	O
.	O
</s>
<s>
Node	O
has	O
received	O
broadcast	B-Operating_System
message	O
but	O
it	O
has	O
not	O
sent	O
a	O
convergecast	O
message	O
back	O
to	O
the	O
core	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
fragment	O
can	O
simply	O
join	O
the	O
broadcast	B-Operating_System
process	O
of	O
.	O
</s>
<s>
In	O
order	O
to	O
do	O
that	O
,	O
node	O
can	O
initiate	O
a	O
broadcast	B-Operating_System
to	O
to	O
update	O
the	O
fragment	O
ID	O
of	O
each	O
node	O
in	O
and	O
collect	O
minimum	O
weight	O
outgoing	O
edge	O
in	O
.Node	O
has	O
already	O
sent	O
a	O
convergecast	O
message	O
back	O
to	O
the	O
core	O
.	O
</s>
