<s>
Guillotine	B-Algorithm
partition	I-Algorithm
is	O
the	O
process	O
of	O
partitioning	O
a	O
rectilinear	O
polygon	O
,	O
possibly	O
containing	O
some	O
holes	O
,	O
into	O
rectangles	O
,	O
using	O
only	O
guillotine-cuts	O
.	O
</s>
<s>
Guillotine	B-Algorithm
partition	I-Algorithm
is	O
particularly	O
common	O
in	O
designing	O
floorplans	O
in	O
microelectronics	O
.	O
</s>
<s>
Guillotine	B-Algorithm
partitions	I-Algorithm
are	O
also	O
the	O
underlying	O
structure	O
of	O
binary	O
space	O
partitions	O
.	O
</s>
<s>
There	O
are	O
various	O
optimization	O
problems	O
related	O
to	O
guillotine	B-Algorithm
partition	I-Algorithm
,	O
such	O
as	O
:	O
minimizing	O
the	O
number	O
of	O
rectangles	O
or	O
the	O
total	O
length	O
of	O
cuts	O
.	O
</s>
<s>
These	O
are	O
variants	O
of	O
polygon	O
partitioning	O
problems	O
,	O
where	O
the	O
cuts	O
are	O
constrained	O
to	O
be	O
guillotine	B-Algorithm
cuts	I-Algorithm
.	O
</s>
<s>
A	O
related	O
but	O
different	O
problem	O
is	O
guillotine	B-Algorithm
cutting	I-Algorithm
.	O
</s>
<s>
The	O
algorithm	O
uses	O
dynamic	B-Algorithm
programming	I-Algorithm
based	O
on	O
the	O
following	O
observation	O
:	O
there	O
exists	O
a	O
minimum-length	O
guillotine	O
rectangular	O
partition	O
in	O
which	O
every	O
maximal	O
line	O
segment	O
contains	O
a	O
vertex	O
of	O
the	O
boundary	O
.	O
</s>
<s>
Therefore	O
,	O
in	O
each	O
iteration	O
,	O
there	O
are	O
possible	O
choices	O
for	O
the	O
next	O
guillotine	B-Algorithm
cut	I-Algorithm
,	O
and	O
there	O
are	O
altogether	O
subproblems	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
guillotine	B-Algorithm
partition	I-Algorithm
provides	O
a	O
constant-factor	O
approximation	O
to	O
the	O
general	O
problem	O
,	O
which	O
is	O
NP-hard	O
.	O
</s>
<s>
Arora	O
and	O
Mitchell	O
used	O
the	O
guillotine-partitioning	O
technique	O
to	O
develop	O
polynomial-time	B-Algorithm
approximation	I-Algorithm
schemes	I-Algorithm
for	O
various	O
geometric	O
optimization	O
problems	O
.	O
</s>
<s>
Besides	O
the	O
computational	O
problems	O
,	O
guillotine	B-Algorithm
partitions	I-Algorithm
were	O
also	O
studied	O
from	O
a	O
combinatorial	O
perspective	O
.	O
</s>
<s>
Suppose	O
a	O
given	O
rectangle	O
should	O
be	O
partitioned	O
into	O
smaller	O
rectangles	O
using	O
guillotine	B-Algorithm
cuts	I-Algorithm
only	O
.	O
</s>
<s>
However	O
,	O
the	O
number	O
of	O
structurally-different	O
guillotine	B-Algorithm
partitions	I-Algorithm
is	O
bounded	O
.	O
</s>
<s>
Asinowski	O
,	O
Barequet	O
,	O
Mansour	O
and	O
Pinter	O
also	O
study	O
the	O
number	O
of	O
cut-equivalence	O
classes	O
of	O
guillotine	B-Algorithm
partitions	I-Algorithm
.	O
</s>
<s>
Aigner-Horev	O
,	O
Katz	O
,	O
Krakovski	O
and	O
Loffler	O
proved	O
that	O
,	O
in	O
the	O
special	O
sub-case	O
in	O
which	O
the	O
graph	O
represents	O
a	O
guillotine	B-Algorithm
partition	I-Algorithm
,	O
a	O
strong	O
polychromatic	O
4-coloring	O
always	O
exists	O
.	O
</s>
<s>
Keszegh	O
extended	O
this	O
result	O
to	O
d-dimensional	O
guillotine	B-Algorithm
partitions	I-Algorithm
,	O
and	O
provided	O
an	O
efficient	O
coloring	O
algorithm	O
.	O
</s>
