<s>
Verification-based	O
message-passing	O
algorithms	O
(	O
VB-MPAs	O
)	O
in	O
compressed	O
sensing	O
(	O
CS	O
)	O
,	O
a	O
branch	O
of	O
digital	B-General_Concept
signal	I-General_Concept
processing	I-General_Concept
that	O
deals	O
with	O
measuring	O
sparse	O
signals	O
,	O
are	O
some	O
methods	O
to	O
efficiently	O
solve	O
the	O
recovery	O
problem	O
in	O
compressed	O
sensing	O
.	O
</s>
<s>
Here	O
is	O
an	O
example	O
of	O
a	O
binary	B-Algorithm
sparse	O
measurement	O
matrix	O
where	O
the	O
weights	O
of	O
the	O
edges	O
are	O
either	O
zero	O
or	O
one	O
.	O
</s>
<s>
The	O
basic	O
idea	O
behind	O
message	O
passing	O
algorithms	O
in	O
CS	O
is	O
to	O
transmit	O
appropriate	O
messages	O
between	O
variable	O
nodes	O
and	O
check	O
nodes	O
in	O
an	O
iterative	B-Algorithm
manner	I-Algorithm
in	O
order	O
to	O
efficiently	O
find	O
signal	O
.	O
</s>
<s>
In	O
each	O
iteration	B-Algorithm
,	O
every	O
variable	O
node	O
and	O
check	O
node	O
produce	O
a	O
new	O
message	O
to	O
be	O
transmitted	O
to	O
all	O
of	O
its	O
neighbors	O
based	O
on	O
the	O
messages	O
that	O
they	O
have	O
received	O
from	O
their	O
own	O
neighbors	O
.	O
</s>
<s>
This	O
local	O
property	O
of	O
the	O
message	O
passing	O
algorithms	O
enables	O
them	O
to	O
be	O
implemented	O
as	O
parallel	B-Operating_System
processing	I-Operating_System
algorithms	O
and	O
makes	O
the	O
time	O
complexity	O
of	O
these	O
algorithm	O
so	O
efficient	O
.	O
</s>
<s>
If	O
the	O
non-zero	O
elements	O
of	O
the	O
measurement	O
matrix	O
are	O
binary	B-Algorithm
,	O
then	O
this	O
algorithm	O
cannot	O
be	O
implemented	O
efficiently	O
and	O
the	O
locality	O
of	O
the	O
algorithm	O
will	O
be	O
violated	O
.	O
</s>
<s>
The	O
pseudo	B-Language
code	I-Language
of	O
the	O
VB-MPAs	O
is	O
as	O
follows	O
.	O
</s>
<s>
is	O
also	O
used	O
to	O
keep	O
the	O
set	O
of	O
verified	O
variable	O
nodes	O
in	O
the	O
previous	O
iteration	B-Algorithm
.	O
</s>
<s>
In	O
order	O
to	O
show	O
that	O
it	O
is	O
enough	O
to	O
show	O
that	O
all	O
of	O
the	O
verification	O
rules	O
work	O
perfectly	O
and	O
without	O
false	B-Error_Name
verification	I-Error_Name
.	O
</s>
<s>
Moreover	O
,	O
the	O
second	O
part	O
of	O
the	O
ECN	O
rule	O
is	O
not	O
necessary	O
to	O
be	O
implemented	O
as	O
the	O
non-zero	O
verified	O
variable	O
node	O
in	O
the	O
ECN	O
rule	O
will	O
be	O
removed	O
from	O
the	O
bipartite	O
graph	O
in	O
the	O
next	O
iteration	B-Algorithm
and	O
ZCN	O
rule	O
will	O
be	O
enough	O
to	O
verify	O
all	O
the	O
zero	O
valued	O
variable	O
nodes	O
remained	O
from	O
those	O
equations	O
with	O
the	O
same	O
right	O
hand	O
side	O
.	O
</s>
<s>
Every	O
minor	O
loop	O
in	O
the	O
main	O
loop	O
of	O
the	O
algorithm	O
can	O
be	O
executed	O
in	O
parallel	B-Operating_System
processors	I-Operating_System
,	O
if	O
we	O
consider	O
each	O
variable	O
and	O
check	O
node	O
as	O
a	O
separate	O
processor	O
.	O
</s>
<s>
Moreover	O
,	O
since	O
the	O
algorithm	O
will	O
terminate	O
when	O
there	O
is	O
no	O
progress	O
in	O
verification	O
of	O
the	O
variable	O
nodes	O
then	O
the	O
if	O
in	O
the	O
worst	O
case	O
in	O
each	O
iteration	B-Algorithm
of	O
the	O
main	O
loop	O
there	O
is	O
only	O
one	O
variable	O
node	O
to	O
be	O
verified	O
,	O
then	O
the	O
maximum	O
number	O
of	O
times	O
that	O
the	O
main	O
loop	O
will	O
be	O
executed	O
is	O
.	O
</s>
