<s>
Dynamic	B-Algorithm
programming	I-Algorithm
is	O
both	O
a	O
mathematical	O
optimization	O
method	O
and	O
a	O
computer	O
programming	O
method	O
.	O
</s>
<s>
Likewise	O
,	O
in	O
computer	O
science	O
,	O
if	O
a	O
problem	O
can	O
be	O
solved	O
optimally	O
by	O
breaking	O
it	O
into	O
sub-problems	O
and	O
then	O
recursively	O
finding	O
the	O
optimal	O
solutions	O
to	O
the	O
sub-problems	O
,	O
then	O
it	O
is	O
said	O
to	O
have	O
optimal	B-Algorithm
substructure	I-Algorithm
.	O
</s>
<s>
If	O
sub-problems	O
can	O
be	O
nested	O
recursively	O
inside	O
larger	O
problems	O
,	O
so	O
that	O
dynamic	B-Algorithm
programming	I-Algorithm
methods	O
are	O
applicable	O
,	O
then	O
there	O
is	O
a	O
relation	O
between	O
the	O
value	O
of	O
the	O
larger	O
problem	O
and	O
the	O
values	O
of	O
the	O
sub-problems	O
.	O
</s>
<s>
In	O
terms	O
of	O
mathematical	O
optimization	O
,	O
dynamic	B-Algorithm
programming	I-Algorithm
usually	O
refers	O
to	O
simplifying	O
a	O
decision	O
by	O
breaking	O
it	O
down	O
into	O
a	O
sequence	O
of	O
decision	O
steps	O
over	O
time	O
.	O
</s>
<s>
This	O
is	O
done	O
by	O
defining	O
a	O
sequence	O
of	O
value	B-Algorithm
functions	I-Algorithm
V1	O
,	O
V2	O
,	O
...	O
,	O
Vn	O
taking	O
y	O
as	O
an	O
argument	O
representing	O
the	O
state	O
of	O
the	O
system	O
at	O
times	O
i	O
from	O
1	O
to	O
n	O
.	O
The	O
definition	O
of	O
Vn(y )	O
is	O
the	O
value	O
obtained	O
in	O
state	O
y	O
at	O
the	O
last	O
time	O
n	O
.	O
The	O
values	O
Vi	O
at	O
earlier	O
times	O
i	O
=	O
n1	O
,	O
n2	O
,...,	O
2	O
,	O
1	O
can	O
be	O
found	O
by	O
working	O
backwards	O
,	O
using	O
a	O
recursive	O
relationship	O
called	O
the	O
Bellman	O
equation	O
.	O
</s>
<s>
The	O
solution	O
to	O
this	O
problem	O
is	O
an	O
optimal	O
control	O
law	O
or	O
policy	O
,	O
which	O
produces	O
an	O
optimal	O
trajectory	O
and	O
a	O
cost-to-go	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
The	O
latter	O
obeys	O
the	O
fundamental	O
equation	O
of	O
dynamic	B-Algorithm
programming	I-Algorithm
:	O
</s>
<s>
In	O
practice	O
,	O
this	O
generally	O
requires	O
numerical	B-Algorithm
techniques	I-Algorithm
for	O
some	O
discrete	O
approximation	O
to	O
the	O
exact	O
optimization	O
relationship	O
.	O
</s>
<s>
The	O
dynamic	B-Algorithm
programming	I-Algorithm
approach	O
to	O
solve	O
this	O
problem	O
involves	O
breaking	O
it	O
apart	O
into	O
a	O
sequence	O
of	O
smaller	O
decisions	O
.	O
</s>
<s>
To	O
do	O
so	O
,	O
we	O
define	O
a	O
sequence	O
of	O
value	B-Algorithm
functions	I-Algorithm
,	O
for	O
which	O
represent	O
the	O
value	O
of	O
having	O
any	O
amount	O
of	O
capital	O
at	O
each	O
time	O
.	O
</s>
<s>
There	O
are	O
two	O
key	O
attributes	O
that	O
a	O
problem	O
must	O
have	O
in	O
order	O
for	O
dynamic	B-Algorithm
programming	I-Algorithm
to	O
be	O
applicable	O
:	O
optimal	B-Algorithm
substructure	I-Algorithm
and	O
overlapping	O
sub-problems	O
.	O
</s>
<s>
If	O
a	O
problem	O
can	O
be	O
solved	O
by	O
combining	O
optimal	O
solutions	O
to	O
non-overlapping	O
sub-problems	O
,	O
the	O
strategy	O
is	O
called	O
"	O
divide	B-Algorithm
and	I-Algorithm
conquer	I-Algorithm
"	O
instead	O
.	O
</s>
<s>
This	O
is	O
why	O
merge	B-Algorithm
sort	I-Algorithm
and	O
quick	B-Algorithm
sort	I-Algorithm
are	O
not	O
classified	O
as	O
dynamic	B-Algorithm
programming	I-Algorithm
problems	O
.	O
</s>
<s>
Optimal	B-Algorithm
substructure	I-Algorithm
means	O
that	O
the	O
solution	O
to	O
a	O
given	O
optimization	O
problem	O
can	O
be	O
obtained	O
by	O
the	O
combination	O
of	O
optimal	O
solutions	O
to	O
its	O
sub-problems	O
.	O
</s>
<s>
Such	O
optimal	B-Algorithm
substructures	I-Algorithm
are	O
usually	O
described	O
by	O
means	O
of	O
recursion	O
.	O
</s>
<s>
For	O
example	O
,	O
given	O
a	O
graph	O
G	O
=(	O
V	O
,	O
E	O
)	O
,	O
the	O
shortest	O
path	O
p	O
from	O
a	O
vertex	O
u	O
to	O
a	O
vertex	O
v	O
exhibits	O
optimal	B-Algorithm
substructure	I-Algorithm
:	O
take	O
any	O
intermediate	O
vertex	O
w	O
on	O
this	O
shortest	O
path	O
p	O
.	O
If	O
p	O
is	O
truly	O
the	O
shortest	O
path	O
,	O
then	O
it	O
can	O
be	O
split	O
into	O
sub-paths	O
p1	O
from	O
u	O
to	O
w	O
and	O
p2	O
from	O
w	O
to	O
v	O
such	O
that	O
these	O
,	O
in	O
turn	O
,	O
are	O
indeed	O
the	O
shortest	O
paths	O
between	O
the	O
corresponding	O
vertices	O
(	O
by	O
the	O
simple	O
cut-and-paste	O
argument	O
described	O
in	O
Introduction	O
to	O
Algorithms	O
)	O
.	O
</s>
<s>
Hence	O
,	O
one	O
can	O
easily	O
formulate	O
the	O
solution	O
for	O
finding	O
shortest	O
paths	O
in	O
a	O
recursive	O
manner	O
,	O
which	O
is	O
what	O
the	O
Bellman	B-Algorithm
–	I-Algorithm
Ford	I-Algorithm
algorithm	I-Algorithm
or	O
the	O
Floyd	B-Algorithm
–	I-Algorithm
Warshall	I-Algorithm
algorithm	I-Algorithm
does	O
.	O
</s>
<s>
For	O
example	O
,	O
consider	O
the	O
recursive	O
formulation	O
for	O
generating	O
the	O
Fibonacci	B-Algorithm
series	I-Algorithm
:	O
Fi	O
=	O
Fi−1	O
+	O
Fi−2	O
,	O
with	O
base	O
case	O
F1	O
=	O
F2	O
=	O
1	O
.	O
</s>
<s>
Dynamic	B-Algorithm
programming	I-Algorithm
takes	O
account	O
of	O
this	O
fact	O
and	O
solves	O
each	O
sub-problem	O
only	O
once	O
.	O
</s>
<s>
Scheme	B-Language
,	O
Common	B-Language
Lisp	I-Language
,	O
Perl	B-Language
or	O
D	B-Application
)	O
.	O
</s>
<s>
Some	O
languages	O
have	O
automatic	O
memoization	O
built	O
in	O
,	O
such	O
as	O
tabled	O
Prolog	B-Language
and	O
J	B-Language
,	O
which	O
supports	O
memoization	O
with	O
the	O
M	O
.	O
adverb	O
.	O
</s>
<s>
Memoization	O
is	O
also	O
encountered	O
as	O
an	O
easily	O
accessible	O
design	O
pattern	O
within	O
term-rewrite	O
based	O
languages	O
such	O
as	O
Wolfram	B-Language
Language	I-Language
.	O
</s>
<s>
Dynamic	B-Algorithm
programming	I-Algorithm
is	O
widely	O
used	O
in	O
bioinformatics	O
for	O
tasks	O
such	O
as	O
sequence	B-Algorithm
alignment	I-Algorithm
,	O
protein	O
folding	O
,	O
RNA	O
structure	O
prediction	O
and	O
protein-DNA	O
binding	O
.	O
</s>
<s>
The	O
first	O
dynamic	B-Algorithm
programming	I-Algorithm
algorithms	O
for	O
protein-DNA	O
binding	O
were	O
developed	O
in	O
the	O
1970s	O
independently	O
by	O
Charles	O
DeLisi	O
in	O
USA	O
and	O
Georgii	O
Gurskii	O
and	O
Alexander	O
Zasedatelev	O
in	O
USSR	O
.	O
</s>
<s>
From	O
a	O
dynamic	B-Algorithm
programming	I-Algorithm
point	O
of	O
view	O
,	O
Dijkstra	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
for	O
the	O
shortest	O
path	O
problem	O
is	O
a	O
successive	O
approximation	O
scheme	B-Language
that	O
solves	O
the	O
dynamic	B-Algorithm
programming	I-Algorithm
functional	O
equation	O
for	O
the	O
shortest	O
path	O
problem	O
by	O
the	O
Reaching	O
method	O
.	O
</s>
<s>
Using	O
dynamic	B-Algorithm
programming	I-Algorithm
in	O
the	O
calculation	O
of	O
the	O
nth	O
member	O
of	O
the	O
Fibonacci	B-Algorithm
sequence	I-Algorithm
improves	O
its	O
performance	O
greatly	O
.	O
</s>
<s>
Notice	O
that	O
if	O
we	O
call	O
,	O
say	O
,	O
fib(5 )	O
,	O
we	O
produce	O
a	O
call	O
tree	B-Data_Structure
that	O
calls	O
the	O
function	O
on	O
the	O
same	O
value	O
many	O
different	O
times	O
:	O
</s>
<s>
Now	O
,	O
suppose	O
we	O
have	O
a	O
simple	O
map	B-Application
object	O
,	O
m	O
,	O
which	O
maps	O
each	O
value	O
of	O
fib	O
that	O
has	O
already	O
been	O
calculated	O
to	O
its	O
result	O
,	O
and	O
we	O
modify	O
our	O
function	O
to	O
use	O
it	O
and	O
update	O
it	O
.	O
</s>
<s>
This	O
method	O
also	O
uses	O
O(n )	O
time	O
since	O
it	O
contains	O
a	O
loop	O
that	O
repeats	O
n	O
−	O
1	O
times	O
,	O
but	O
it	O
only	O
takes	O
constant	O
(O(1 )	O
)	O
space	O
,	O
in	O
contrast	O
to	O
the	O
top-down	O
approach	O
which	O
requires	O
O(n )	O
space	O
to	O
store	O
the	O
map	B-Application
.	O
</s>
<s>
There	O
are	O
at	O
least	O
three	O
possible	O
approaches	O
:	O
brute	B-Algorithm
force	I-Algorithm
,	O
backtracking	B-Algorithm
,	O
and	O
dynamic	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
Brute	B-Algorithm
force	I-Algorithm
consists	O
of	O
checking	O
all	O
assignments	O
of	O
zeros	O
and	O
ones	O
and	O
counting	O
those	O
that	O
have	O
balanced	O
rows	O
and	O
columns	O
(	O
zeros	O
and	O
ones	O
)	O
.	O
</s>
<s>
Backtracking	B-Algorithm
for	O
this	O
problem	O
consists	O
of	O
choosing	O
some	O
order	O
of	O
the	O
matrix	O
elements	O
and	O
recursively	O
placing	O
ones	O
or	O
zeros	O
,	O
while	O
checking	O
that	O
in	O
every	O
row	O
and	O
column	O
the	O
number	O
of	O
elements	O
that	O
have	O
not	O
been	O
assigned	O
plus	O
the	O
number	O
of	O
ones	O
or	O
zeros	O
are	O
both	O
at	O
least	O
.	O
</s>
<s>
While	O
more	O
sophisticated	O
than	O
brute	B-Algorithm
force	I-Algorithm
,	O
this	O
approach	O
will	O
visit	O
every	O
solution	O
once	O
,	O
making	O
it	O
impractical	O
for	O
larger	O
than	O
six	O
,	O
since	O
the	O
number	O
of	O
solutions	O
is	O
already	O
116,963,796,250	O
for	O
=	O
8	O
,	O
as	O
we	O
shall	O
see	O
.	O
</s>
<s>
Dynamic	B-Algorithm
programming	I-Algorithm
makes	O
it	O
possible	O
to	O
count	O
the	O
number	O
of	O
solutions	O
without	O
visiting	O
them	O
all	O
.	O
</s>
<s>
Imagine	O
backtracking	B-Algorithm
values	O
for	O
the	O
first	O
row	O
–	O
what	O
information	O
would	O
we	O
require	O
about	O
the	O
remaining	O
rows	O
,	O
in	O
order	O
to	O
be	O
able	O
to	O
accurately	O
count	O
the	O
solutions	O
obtained	O
for	O
each	O
first	O
row	O
value	O
?	O
</s>
<s>
Links	O
to	O
the	O
MAPLE	O
implementation	O
of	O
the	O
dynamic	B-Algorithm
programming	I-Algorithm
approach	O
may	O
be	O
found	O
among	O
the	O
external	O
links	O
.	O
</s>
<s>
Consider	O
a	O
checkerboard	O
with	O
n	O
×	O
n	O
squares	O
and	O
a	O
cost	O
function	O
c(i, j )	O
which	O
returns	O
a	O
cost	O
associated	O
with	O
square	O
(	O
i	O
,	O
j	B-Language
)	O
(	O
i	O
being	O
the	O
row	O
,	O
j	B-Language
being	O
the	O
column	O
)	O
.	O
</s>
<s>
This	O
problem	O
exhibits	O
optimal	B-Algorithm
substructure	I-Algorithm
.	O
</s>
<s>
q(i, j )	O
=	O
the	O
minimum	O
cost	O
to	O
reach	O
square	O
(	O
i	O
,	O
j	B-Language
)	O
.	O
</s>
<s>
The	O
function	O
q(i, j )	O
is	O
equal	O
to	O
the	O
minimum	O
cost	O
to	O
get	O
to	O
any	O
of	O
the	O
three	O
squares	O
below	O
it	O
(	O
since	O
those	O
are	O
the	O
only	O
squares	O
that	O
can	O
reach	O
it	O
)	O
plus	O
c(i, j )	O
.	O
</s>
<s>
Now	O
,	O
let	O
us	O
define	O
q(i, j )	O
in	O
somewhat	O
more	O
general	O
terms	O
:	O
</s>
<s>
It	O
represents	O
the	O
A	O
,	O
B	O
,	O
C	O
,	O
D	B-Application
terms	O
in	O
the	O
example	O
.	O
</s>
<s>
From	O
this	O
definition	O
we	O
can	O
derive	O
straightforward	O
recursive	O
code	O
for	O
q(i, j )	O
.	O
</s>
<s>
In	O
the	O
following	O
pseudocode	O
,	O
n	O
is	O
the	O
size	O
of	O
the	O
board	O
,	O
c(i, j )	O
is	O
the	O
cost	O
function	O
,	O
and	O
min( )	O
returns	O
the	O
minimum	O
of	O
a	O
number	O
of	O
values	O
:	O
</s>
<s>
However	O
,	O
we	O
can	O
compute	O
it	O
much	O
faster	O
in	O
a	O
bottom-up	O
fashion	O
if	O
we	O
store	O
path	O
costs	O
in	O
a	O
two-dimensional	O
array	O
q[ i	O
,	O
j ]	O
rather	O
than	O
using	O
a	O
function	O
.	O
</s>
<s>
This	O
avoids	O
recomputation	O
;	O
all	O
the	O
values	O
needed	O
for	O
array	O
q[ i	O
,	O
j ]	O
are	O
computed	O
ahead	O
of	O
time	O
only	O
once	O
.	O
</s>
<s>
Precomputed	O
values	O
for	O
(	O
i	O
,	O
j	B-Language
)	O
are	O
simply	O
looked	O
up	O
whenever	O
needed	O
.	O
</s>
<s>
To	O
do	O
this	O
,	O
we	O
use	O
another	O
array	O
p[ i	O
,	O
j ]	O
;	O
a	O
predecessor	O
array	O
.	O
</s>
<s>
This	O
array	O
records	O
the	O
path	O
to	O
any	O
square	O
s	O
.	O
The	O
predecessor	O
of	O
s	O
is	O
modeled	O
as	O
an	O
offset	O
relative	O
to	O
the	O
index	O
(	O
in	O
q[ i	O
,	O
j ]	O
)	O
of	O
the	O
precomputed	O
path	O
cost	O
of	O
s	O
.	O
To	O
reconstruct	O
the	O
complete	O
path	O
,	O
we	O
lookup	O
the	O
predecessor	O
of	O
s	O
,	O
then	O
the	O
predecessor	O
of	O
that	O
square	O
,	O
then	O
the	O
predecessor	O
of	O
that	O
square	O
,	O
and	O
so	O
on	O
recursively	O
,	O
until	O
we	O
reach	O
the	O
starting	O
square	O
.	O
</s>
<s>
In	O
genetics	O
,	O
sequence	B-Algorithm
alignment	I-Algorithm
is	O
an	O
important	O
application	O
where	O
dynamic	B-Algorithm
programming	I-Algorithm
is	O
essential	O
.	O
</s>
<s>
The	O
partial	O
alignments	O
can	O
be	O
tabulated	O
in	O
a	O
matrix	O
,	O
where	O
cell	O
(	O
i	O
,	O
j	B-Language
)	O
contains	O
the	O
cost	O
of	O
the	O
optimal	O
alignment	O
of	O
1	O
..	O
i	O
to	O
1	O
..	O
j	B-Language
.	O
</s>
<s>
The	O
cost	O
in	O
cell	O
(	O
i	O
,	O
j	B-Language
)	O
can	O
be	O
calculated	O
by	O
adding	O
the	O
cost	O
of	O
the	O
relevant	O
operations	O
to	O
the	O
cost	O
of	O
its	O
neighboring	O
cells	O
,	O
and	O
selecting	O
the	O
optimum	O
.	O
</s>
<s>
Different	O
variants	O
exist	O
,	O
see	O
Smith	B-Algorithm
–	I-Algorithm
Waterman	I-Algorithm
algorithm	I-Algorithm
and	O
Needleman	B-Algorithm
–	I-Algorithm
Wunsch	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
Tower	B-Language
of	I-Language
Hanoi	I-Language
or	O
Towers	B-Language
of	I-Language
Hanoi	I-Language
is	O
a	O
mathematical	O
game	O
or	O
puzzle	O
.	O
</s>
<s>
If	O
the	O
objective	O
is	O
to	O
maximize	O
the	O
number	O
of	O
moves	O
(	O
without	O
cycling	O
)	O
then	O
the	O
dynamic	B-Algorithm
programming	I-Algorithm
functional	O
equation	O
is	O
slightly	O
more	O
complicated	O
and	O
3n1	O
moves	O
are	O
required	O
.	O
</s>
<s>
Matrix	B-Algorithm
chain	I-Algorithm
multiplication	I-Algorithm
is	O
a	O
well-known	O
example	O
that	O
demonstrates	O
utility	O
of	O
dynamic	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
At	O
this	O
point	O
,	O
we	O
have	O
several	O
choices	O
,	O
one	O
of	O
which	O
is	O
to	O
design	O
a	O
dynamic	B-Algorithm
programming	I-Algorithm
algorithm	O
that	O
will	O
split	O
the	O
problem	O
into	O
overlapping	O
problems	O
and	O
calculate	O
the	O
optimal	O
arrangement	O
of	O
parenthesis	O
.	O
</s>
<s>
The	O
dynamic	B-Algorithm
programming	I-Algorithm
solution	O
is	O
presented	O
below	O
.	O
</s>
<s>
Let	O
's	O
call	O
m[i,j]	O
the	O
minimum	O
number	O
of	O
scalar	O
multiplications	O
needed	O
to	O
multiply	O
a	O
chain	O
of	O
matrices	O
from	O
matrix	O
i	O
to	O
matrix	O
j	B-Language
(	O
i.e.	O
</s>
<s>
i	O
<=	O
j	B-Language
)	O
.	O
</s>
<s>
We	O
split	O
the	O
chain	O
at	O
some	O
matrix	O
k	O
,	O
such	O
that	O
i	O
<=	O
k	O
<	O
j	B-Language
,	O
and	O
try	O
to	O
find	O
out	O
which	O
combination	O
produces	O
minimum	O
m[i,j]	O
.	O
</s>
<s>
is	O
the	O
column	O
dimension	O
of	O
matrix	O
j	B-Language
.	O
</s>
<s>
s[ i	O
,	O
j ]	O
=	O
k	O
//	O
Record	O
which	O
k	O
to	O
split	O
on	O
,	O
i.e.	O
</s>
<s>
So	O
far	O
,	O
we	O
have	O
calculated	O
values	O
for	O
all	O
possible	O
,	O
the	O
minimum	O
number	O
of	O
calculations	O
to	O
multiply	O
a	O
chain	O
from	O
matrix	O
i	O
to	O
matrix	O
j	B-Language
,	O
and	O
we	O
have	O
recorded	O
the	O
corresponding	O
"	O
split	O
point	O
"	O
.	O
</s>
<s>
This	O
algorithm	O
will	O
produce	O
"	O
tables	O
"	O
m[ 	O
,	O
]	O
and	O
s[ 	O
,	O
]	O
that	O
will	O
have	O
entries	O
for	O
all	O
possible	O
values	O
of	O
i	O
and	O
j	B-Language
.	O
</s>
<s>
The	O
term	O
dynamic	B-Algorithm
programming	I-Algorithm
was	O
originally	O
used	O
in	O
the	O
1940s	O
by	O
Richard	O
Bellman	O
to	O
describe	O
the	O
process	O
of	O
solving	O
problems	O
where	O
one	O
needs	O
to	O
find	O
the	O
best	O
decisions	O
one	O
after	O
another	O
.	O
</s>
<s>
Bellman	O
's	O
contribution	O
is	O
remembered	O
in	O
the	O
name	O
of	O
the	O
Bellman	O
equation	O
,	O
a	O
central	O
result	O
of	O
dynamic	B-Algorithm
programming	I-Algorithm
which	O
restates	O
an	O
optimization	O
problem	O
in	O
recursive	O
form	O
.	O
</s>
<s>
Bellman	O
explains	O
the	O
reasoning	O
behind	O
the	O
term	O
dynamic	B-Algorithm
programming	I-Algorithm
in	O
his	O
autobiography	O
,	O
Eye	O
of	O
the	O
Hurricane	O
:	O
An	O
Autobiography	O
:	O
</s>
<s>
This	O
usage	O
is	O
the	O
same	O
as	O
that	O
in	O
the	O
phrases	O
linear	B-Algorithm
programming	I-Algorithm
and	O
mathematical	O
programming	O
,	O
a	O
synonym	O
for	O
mathematical	O
optimization	O
.	O
</s>
<s>
Quoting	O
Kushner	O
as	O
he	O
speaks	O
of	O
Bellman	O
:	O
"	O
On	O
the	O
other	O
hand	O
,	O
when	O
I	O
asked	O
him	O
the	O
same	O
question	O
,	O
he	O
replied	O
that	O
he	O
was	O
trying	O
to	O
upstage	O
Dantzig	O
's	O
linear	B-Algorithm
programming	I-Algorithm
by	O
adding	O
dynamic	O
.	O
</s>
<s>
Many	O
algorithmic	O
problems	O
on	O
graphs	O
can	O
be	O
solved	O
efficiently	O
for	O
graphs	O
of	O
bounded	O
treewidth	O
or	O
bounded	O
clique-width	O
by	O
using	O
dynamic	B-Algorithm
programming	I-Algorithm
on	O
a	O
tree	B-Data_Structure
decomposition	O
of	O
the	O
graph	O
.	O
</s>
<s>
Some	O
methods	O
for	O
solving	O
the	O
travelling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
,	O
either	O
exactly	O
(	O
in	O
exponential	O
time	O
)	O
or	O
approximately	O
(	O
e.g.	O
</s>
