<s>
Reverse-search	B-Algorithm
algorithms	I-Algorithm
are	O
a	O
class	O
of	O
algorithms	O
for	O
generating	O
all	O
objects	O
of	O
a	O
given	O
size	O
,	O
from	O
certain	O
classes	O
of	O
combinatorial	O
objects	O
.	O
</s>
<s>
They	O
work	O
by	O
organizing	O
the	O
objects	O
to	O
be	O
generated	O
into	O
a	O
spanning	O
tree	O
of	O
their	O
state	O
space	O
,	O
and	O
then	O
performing	O
a	O
depth-first	B-Algorithm
search	I-Algorithm
of	O
this	O
tree	O
.	O
</s>
<s>
Reverse-search	B-Algorithm
algorithms	I-Algorithm
were	O
introduced	O
by	O
David	O
Avis	O
and	O
Komei	O
Fukuda	O
in	O
1991	O
,	O
for	O
problems	O
of	O
generating	O
the	O
vertices	O
of	O
convex	O
polytopes	O
and	O
the	O
cells	O
of	O
arrangements	O
of	O
hyperplanes	O
.	O
</s>
<s>
A	O
reverse-search	B-Algorithm
algorithm	I-Algorithm
generates	O
the	O
combinatorial	O
objects	O
in	O
a	O
state	O
space	O
,	O
an	O
implicit	B-Data_Structure
graph	I-Data_Structure
whose	O
vertices	O
are	O
the	O
objects	O
to	O
be	O
listed	O
and	O
whose	O
edges	O
represent	O
certain	O
"	O
local	O
moves	O
"	O
connecting	O
pairs	O
of	O
objects	O
,	O
typically	O
by	O
making	O
small	O
changes	O
to	O
their	O
structure	O
.	O
</s>
<s>
It	O
finds	O
each	O
objects	O
using	O
a	O
depth-first	B-Algorithm
search	I-Algorithm
in	O
a	O
rooted	O
spanning	O
tree	O
of	O
this	O
state	O
space	O
,	O
described	O
by	O
the	O
following	O
information	O
:	O
</s>
<s>
A	O
classical	O
depth-first	B-Algorithm
search	I-Algorithm
of	O
this	O
spanning	O
tree	O
would	O
traverse	O
the	O
tree	O
recursively	O
,	O
starting	O
from	O
the	O
root	O
,	O
at	O
each	O
node	O
listing	O
all	O
of	O
the	O
children	O
and	O
making	O
a	O
recursive	O
call	O
for	O
each	O
one	O
.	O
</s>
<s>
Unlike	O
a	O
depth-first	B-Algorithm
search	I-Algorithm
of	O
a	O
graph	O
with	O
cycles	O
,	O
it	O
is	O
not	O
necessary	O
to	O
maintain	O
the	O
set	O
of	O
already-visited	O
nodes	O
to	O
avoid	O
repeated	O
visits	O
;	O
such	O
repetition	O
is	O
not	O
possible	O
in	O
a	O
tree	O
.	O
</s>
<s>
However	O
,	O
this	O
recursive	O
algorithm	O
may	O
still	O
require	O
a	O
large	O
amount	O
of	O
memory	O
for	O
its	O
call	B-General_Concept
stack	I-General_Concept
,	O
in	O
cases	O
when	O
the	O
tree	O
is	O
very	O
deep	O
.	O
</s>
<s>
However	O
,	O
if	O
there	O
are	O
objects	O
to	O
be	O
listed	O
,	O
then	O
the	O
search	O
performs	O
steps	O
,	O
so	O
the	O
number	O
of	O
times	O
it	O
generates	O
neighbors	O
of	O
objects	O
is	O
within	O
a	O
factor	O
of	O
two	O
of	O
the	O
number	O
of	O
times	O
the	O
recursive	O
depth-first	B-Algorithm
search	I-Algorithm
would	O
do	O
the	O
same	O
thing	O
.	O
</s>
<s>
The	O
vertex	B-Algorithm
enumeration	I-Algorithm
problem	I-Algorithm
is	O
the	O
problem	O
of	O
listing	O
all	O
of	O
these	O
vertices	O
.	O
</s>
<s>
The	O
simplex	B-Algorithm
algorithm	I-Algorithm
from	O
the	O
theory	O
of	O
linear	B-Algorithm
programming	I-Algorithm
finds	O
a	O
vertex	O
maximizing	O
a	O
given	O
linear	O
function	O
of	O
the	O
coordinates	O
,	O
by	O
walking	O
from	O
vertex	O
to	O
vertex	O
,	O
choosing	O
at	O
each	O
step	O
a	O
vertex	O
with	O
a	O
greater	O
value	O
of	O
the	O
function	O
;	O
there	O
are	O
several	O
standard	O
choices	O
of	O
"	O
pivot	O
rule	O
"	O
that	O
specify	O
more	O
precisely	O
which	O
vertex	O
to	O
choose	O
.	O
</s>
<s>
A	O
similar	O
algorithm	O
can	O
also	O
enumerate	O
all	O
bases	O
of	O
a	O
linear	B-Algorithm
program	I-Algorithm
,	O
without	O
requiring	O
that	O
it	O
defines	O
a	O
polytope	O
that	O
is	O
simple	O
.	O
</s>
<s>
The	O
cells	O
form	O
a	O
connected	O
state	O
space	O
under	O
local	O
moves	O
that	O
change	O
a	O
single	O
sign	O
by	O
one	O
unit	O
,	O
and	O
it	O
is	O
possible	O
to	O
check	O
that	O
this	O
operation	O
produces	O
a	O
valid	O
cell	O
by	O
solving	O
a	O
linear	B-Algorithm
programming	I-Algorithm
feasibility	O
problem	O
.	O
</s>
<s>
The	O
triangulations	B-Algorithm
of	I-Algorithm
a	I-Algorithm
planar	I-Algorithm
point	I-Algorithm
set	I-Algorithm
are	O
connected	O
by	O
"	O
flip	O
"	O
moves	O
that	O
remove	O
one	O
diagonal	O
from	O
a	O
triangulation	O
and	O
replace	O
it	O
by	O
another	O
.	O
</s>
<s>
If	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
is	O
chosen	O
as	O
the	O
root	O
,	O
then	O
every	O
triangulation	O
can	O
be	O
flipped	O
to	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
by	O
steps	O
in	O
which	O
the	O
triangulation	O
of	O
some	O
subset	O
of	O
four	O
points	O
is	O
replaced	O
by	O
its	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
.	O
</s>
<s>
Topological	B-Algorithm
orderings	I-Algorithm
of	O
directed	O
acyclic	O
graphs	O
,	O
using	O
a	O
state	O
space	O
whose	O
local	O
moves	O
reverse	O
the	O
ordering	O
of	O
two	O
elements	O
.	O
</s>
<s>
Vertices	O
or	O
facets	O
of	O
the	O
Minkowski	B-Algorithm
sum	I-Algorithm
of	O
convex	O
polytopes	O
.	O
</s>
