<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
the	O
largest	B-Algorithm
differencing	I-Algorithm
method	I-Algorithm
is	O
an	O
algorithm	O
for	O
solving	O
the	O
partition	B-Algorithm
problem	I-Algorithm
and	O
the	O
multiway	B-Algorithm
number	I-Algorithm
partitioning	I-Algorithm
.	O
</s>
<s>
It	O
is	O
also	O
called	O
the	O
Karmarkar	B-Algorithm
–	I-Algorithm
Karp	I-Algorithm
algorithm	I-Algorithm
after	O
its	O
inventors	O
,	O
Narendra	O
Karmarkar	O
and	O
Richard	O
M	O
.	O
Karp	O
.	O
</s>
<s>
Using	O
backtracking	B-Algorithm
,	O
compute	O
the	O
partition	O
.	O
</s>
<s>
Step	O
3	O
constructs	O
the	O
subsets	O
in	O
the	O
partition	O
by	O
backtracking	B-Algorithm
.	O
</s>
<s>
On	O
random	O
instances	O
,	O
this	O
approximate	O
algorithm	O
performs	O
much	O
better	O
than	O
greedy	B-Algorithm
number	I-Algorithm
partitioning	I-Algorithm
.	O
</s>
<s>
Simulation	O
experiments	O
show	O
that	O
,	O
when	O
the	O
numbers	O
are	O
uniformly	O
random	O
in	O
[0,1],	O
LDM	O
always	O
performs	O
better	O
(	O
i.e.	O
,	O
produces	O
a	O
partition	O
with	O
a	O
smaller	O
largest	O
sum	O
)	O
than	O
greedy	B-Algorithm
number	I-Algorithm
partitioning	I-Algorithm
.	O
</s>
<s>
It	O
performs	O
better	O
than	O
the	O
multifit	B-Algorithm
algorithm	I-Algorithm
when	O
the	O
number	O
of	O
items	O
n	O
is	O
sufficiently	O
large	O
.	O
</s>
<s>
Several	O
variants	O
of	O
LDM	O
were	O
developed	O
for	O
the	O
balanced	B-Algorithm
number	I-Algorithm
partitioning	I-Algorithm
problem	O
,	O
in	O
which	O
all	O
subsets	O
must	O
have	O
the	O
same	O
cardinality	O
(	O
up	O
to	O
1	O
)	O
.	O
</s>
<s>
Using	O
backtracking	B-Algorithm
,	O
compute	O
the	O
partition	O
.	O
</s>
<s>
RLDM	O
(	O
Restricted	O
Largest	B-Algorithm
Differencing	I-Algorithm
Method	I-Algorithm
)	O
works	O
as	O
follows	O
.	O
</s>
<s>
BLDM	O
(	O
Balanced	O
Largest	B-Algorithm
Differencing	I-Algorithm
Method	I-Algorithm
)	O
works	O
as	O
follows	O
.	O
</s>
<s>
The	O
complete	O
Karmarkar	B-Algorithm
–	I-Algorithm
Karp	I-Algorithm
algorithm	I-Algorithm
(	O
CKK	O
)	O
finds	O
an	O
optimal	O
solution	O
by	O
constructing	O
a	O
tree	O
of	O
degree	O
.	O
</s>
<s>
Korf	O
reports	O
that	O
CKK	O
can	O
optimally	O
partition	O
40	O
15-digit	O
double-precision	O
numbers	O
in	O
about	O
3	O
hours	O
,	O
while	O
CGA	O
requires	O
about	O
9	O
hours	O
.	O
</s>
<s>
In	O
practice	O
,	O
with	O
k	O
=	O
2	O
,	O
problems	O
of	O
arbitrary	O
size	O
can	O
be	O
solved	O
by	O
CKK	O
if	O
the	O
numbers	O
have	O
at	O
most	O
12	O
significant	B-Architecture
digits	I-Architecture
;	O
with	O
k	O
=3	O
,	O
at	O
most	O
6	O
significant	B-Architecture
digits	I-Architecture
.	O
</s>
<s>
CKK	O
can	O
also	O
run	O
as	O
an	O
anytime	B-Algorithm
algorithm	I-Algorithm
:	O
it	O
finds	O
the	O
KK	O
solution	O
first	O
,	O
and	O
then	O
finds	O
progressively	O
better	O
solutions	O
as	O
time	O
allows	O
(	O
possibly	O
requiring	O
exponential	O
time	O
to	O
reach	O
optimality	O
,	O
for	O
the	O
worst	O
instances	O
)	O
.	O
</s>
<s>
Combining	O
CKK	O
with	O
the	O
balanced-LDM	O
algorithm	O
(	O
BLDM	O
)	O
yields	O
a	O
complete	O
anytime	B-Algorithm
algorithm	I-Algorithm
for	O
solving	O
the	O
balanced	B-Algorithm
partition	I-Algorithm
problem	I-Algorithm
.	O
</s>
