<s>
In	O
computer	B-General_Concept
science	I-General_Concept
and	O
graph	O
theory	O
,	O
the	O
term	O
color-coding	B-Algorithm
refers	O
to	O
an	O
algorithmic	O
technique	O
which	O
is	O
useful	O
in	O
the	O
discovery	O
of	O
network	O
motifs	O
.	O
</s>
<s>
The	O
traditional	O
color-coding	B-Algorithm
algorithm	O
is	O
probabilistic	B-General_Concept
,	O
but	O
it	O
can	O
be	O
derandomized	O
without	O
much	O
overhead	O
in	O
the	O
running	O
time	O
.	O
</s>
<s>
Color-coding	B-Algorithm
also	O
applies	O
to	O
the	O
detection	O
of	O
cycles	O
of	O
a	O
given	O
length	O
,	O
and	O
more	O
generally	O
it	O
applies	O
to	O
the	O
subgraph	O
isomorphism	O
problem	O
(	O
an	O
NP-complete	O
problem	O
)	O
,	O
where	O
it	O
yields	O
polynomial	O
time	O
algorithms	O
when	O
the	O
subgraph	O
pattern	O
that	O
it	O
is	O
trying	O
to	O
detect	O
has	O
bounded	O
treewidth	O
.	O
</s>
<s>
The	O
color-coding	B-Algorithm
method	O
was	O
proposed	O
and	O
analyzed	O
in	O
1994	O
by	O
Noga	O
Alon	O
,	O
Raphael	O
Yuster	O
,	O
and	O
Uri	O
Zwick	O
.	O
</s>
<s>
The	O
following	O
results	O
can	O
be	O
obtained	O
through	O
the	B-Algorithm
method	I-Algorithm
of	I-Algorithm
color-coding	I-Algorithm
:	O
</s>
<s>
To	O
solve	O
the	O
problem	O
of	O
finding	O
a	O
subgraph	O
in	O
a	O
given	O
graph	O
,	O
where	O
can	O
be	O
a	O
path	O
,	O
a	O
cycle	O
,	O
or	O
any	O
bounded	O
treewidth	O
graph	O
where	O
,	O
the	B-Algorithm
method	I-Algorithm
of	I-Algorithm
color-coding	I-Algorithm
begins	O
by	O
randomly	O
coloring	O
each	O
vertex	O
of	O
with	O
colors	O
,	O
and	O
then	O
tries	O
to	O
find	O
a	O
colorful	O
copy	O
of	O
in	O
colored	O
.	O
</s>
<s>
The	O
derandomization	O
of	O
color-coding	B-Algorithm
involves	O
enumerating	O
possible	O
colorings	O
of	O
a	O
graph	O
,	O
such	O
that	O
the	O
randomness	O
of	O
coloring	O
is	O
no	O
longer	O
required	O
.	O
</s>
<s>
To	O
achieve	O
this	O
,	O
enumerating	O
a	O
-perfect	O
family	O
of	O
hash	O
functions	O
from	O
to	O
is	O
sufficient	O
.	O
</s>
<s>
By	O
definition	O
,	O
is	O
-perfect	O
if	O
for	O
every	O
subset	O
of	O
where	O
,	O
there	O
exists	O
a	O
hash	O
function	O
in	O
such	O
that	O
is	O
perfect	B-Algorithm
.	O
</s>
<s>
can	O
be	O
obtained	O
by	O
first	O
building	O
a	O
-perfect	O
family	O
that	O
maps	O
to	O
followed	O
by	O
building	O
another	O
-perfect	O
family	O
that	O
maps	O
to	O
In	O
the	O
first	O
step	O
,	O
it	O
is	O
possible	O
to	O
construct	O
such	O
a	O
family	O
with	O
random	O
bits	O
that	O
are	O
almost	O
-wise	O
independent	O
,	O
and	O
the	O
sample	O
space	O
needed	O
for	O
generating	O
those	O
random	O
bits	O
can	O
be	O
as	O
small	O
as	O
.	O
</s>
<s>
In	O
the	O
second	O
step	O
,	O
it	O
has	O
been	O
shown	O
by	O
Jeanette	O
P	O
.	O
Schmidt	O
and	O
Alan	O
Siegel	O
that	O
the	O
size	O
of	O
such	O
-perfect	O
family	O
can	O
be	O
.	O
</s>
<s>
Consequently	O
,	O
by	O
composing	O
the	O
-perfect	O
families	O
from	O
both	O
steps	O
,	O
a	O
-perfect	O
family	O
of	O
size	O
that	O
maps	O
from	O
to	O
can	O
be	O
obtained	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
derandomizing	O
well-coloring	O
,	O
where	O
each	O
vertex	O
on	O
the	O
subgraph	O
is	O
colored	O
consecutively	O
,	O
a	O
-perfect	O
family	O
of	O
hash	O
functions	O
from	O
to	O
is	O
needed	O
.	O
</s>
<s>
A	O
sufficient	O
-perfect	O
family	O
which	O
maps	O
from	O
to	O
can	O
be	O
constructed	O
in	O
a	O
way	O
similar	O
to	O
the	O
approach	O
3	O
above	O
(	O
the	O
first	O
step	O
)	O
.	O
</s>
<s>
In	O
particular	O
,	O
it	O
is	O
done	O
by	O
using	O
random	O
bits	O
that	O
are	O
almost	O
independent	O
,	O
and	O
the	O
size	O
of	O
the	O
resulting	O
-perfect	O
family	O
will	O
be	O
.	O
</s>
<s>
The	O
derandomization	O
of	O
color-coding	B-Algorithm
method	O
can	O
be	O
easily	O
parallelized	O
,	O
yielding	O
efficient	O
NC	O
algorithms	O
.	O
</s>
<s>
Recently	O
,	O
color-coding	B-Algorithm
has	O
attracted	O
much	O
attention	O
in	O
the	O
field	O
of	O
bioinformatics	O
.	O
</s>
<s>
However	O
,	O
by	O
exploiting	O
the	O
color-coding	B-Algorithm
method	O
,	O
the	O
motifs	O
or	O
signaling	O
pathways	O
with	O
vertices	O
in	O
a	O
network	O
with	O
vertices	O
can	O
be	O
found	O
very	O
efficiently	O
in	O
polynomial	O
time	O
.	O
</s>
