<s>
In	O
distributed	B-Architecture
computing	I-Architecture
and	O
geometric	O
graph	O
theory	O
,	O
greedy	B-Protocol
embedding	I-Protocol
is	O
a	O
process	O
of	O
assigning	O
coordinates	O
to	O
the	O
nodes	O
of	O
a	O
telecommunications	O
network	O
in	O
order	O
to	O
allow	O
greedy	B-Algorithm
geographic	B-Protocol
routing	I-Protocol
to	O
be	O
used	O
to	O
route	O
messages	O
within	O
the	O
network	O
.	O
</s>
<s>
Although	O
greedy	B-Protocol
embedding	I-Protocol
has	O
been	O
proposed	O
for	O
use	O
in	O
wireless	B-Architecture
sensor	I-Architecture
networks	I-Architecture
,	O
in	O
which	O
the	O
nodes	O
already	O
have	O
positions	O
in	O
physical	O
space	O
,	O
these	O
existing	O
positions	O
may	O
differ	O
from	O
the	O
positions	O
given	O
to	O
them	O
by	O
greedy	B-Protocol
embedding	I-Protocol
,	O
which	O
may	O
in	O
some	O
cases	O
be	O
points	O
in	O
a	O
virtual	O
space	O
of	O
a	O
higher	O
dimension	O
,	O
or	O
in	O
a	O
non-Euclidean	O
geometry	O
.	O
</s>
<s>
In	O
this	O
sense	O
,	O
greedy	B-Protocol
embedding	I-Protocol
may	O
be	O
viewed	O
as	O
a	O
form	O
of	O
graph	O
drawing	O
,	O
in	O
which	O
an	O
abstract	O
graph	O
(	O
the	O
communications	O
network	O
)	O
is	O
embedded	B-Algorithm
into	O
a	O
geometric	O
space	O
.	O
</s>
<s>
The	O
idea	O
of	O
performing	O
geographic	B-Protocol
routing	I-Protocol
using	O
coordinates	O
in	O
a	O
virtual	O
space	O
,	O
instead	O
of	O
using	O
physical	O
coordinates	O
,	O
is	O
due	O
to	O
Rao	O
et	O
al	O
.	O
</s>
<s>
Subsequent	O
developments	O
have	O
shown	O
that	O
every	O
network	O
has	O
a	O
greedy	B-Protocol
embedding	I-Protocol
with	O
succinct	O
vertex	O
coordinates	O
in	O
the	O
hyperbolic	O
plane	O
,	O
that	O
certain	O
graphs	O
including	O
the	O
polyhedral	O
graphs	O
have	O
greedy	B-Protocol
embeddings	I-Protocol
in	O
the	O
Euclidean	O
plane	O
,	O
and	O
that	O
unit	O
disk	O
graphs	O
have	O
greedy	B-Protocol
embeddings	I-Protocol
in	O
Euclidean	O
spaces	O
of	O
moderate	O
dimensions	O
with	O
low	O
stretch	O
factors	O
.	O
</s>
<s>
In	O
greedy	B-Algorithm
routing	O
,	O
a	O
message	O
from	O
a	O
source	O
node	O
s	O
to	O
a	O
destination	O
node	O
t	O
travels	O
to	O
its	O
destination	O
by	O
a	O
sequence	O
of	O
steps	O
through	O
intermediate	O
nodes	O
,	O
each	O
of	O
which	O
passes	O
the	O
message	O
on	O
to	O
a	O
neighboring	O
node	O
that	O
is	O
closer	O
to	O
t	O
.	O
If	O
the	O
message	O
reaches	O
an	O
intermediate	O
node	O
x	O
that	O
does	O
not	O
have	O
a	O
neighbor	O
closer	O
to	O
t	O
,	O
then	O
it	O
cannot	O
make	O
progress	O
and	O
the	O
greedy	B-Algorithm
routing	O
process	O
fails	O
.	O
</s>
<s>
A	O
greedy	B-Protocol
embedding	I-Protocol
is	O
an	O
embedding	O
of	O
the	O
given	O
graph	O
with	O
the	O
property	O
that	O
a	O
failure	O
of	O
this	O
type	O
is	O
impossible	O
.	O
</s>
<s>
Thus	O
,	O
it	O
can	O
be	O
characterized	O
as	O
an	O
embedding	O
of	O
the	O
graph	O
with	O
the	O
property	O
that	O
for	O
every	O
two	O
nodes	O
x	O
and	O
t	O
,	O
there	O
exists	O
a	O
neighbor	O
y	O
of	O
x	O
such	O
that	O
d(x,t )	O
>d(y,t )	O
,	O
where	O
d	O
denotes	O
the	O
distance	O
in	O
the	O
embedded	B-Algorithm
space	O
.	O
</s>
<s>
Not	O
every	O
graph	O
has	O
a	O
greedy	B-Protocol
embedding	I-Protocol
into	O
the	O
Euclidean	O
plane	O
;	O
a	O
simple	O
counterexample	O
is	O
given	O
by	O
the	O
star	O
K1	O
,	O
6	O
,	O
a	O
tree	O
with	O
one	O
internal	O
node	O
and	O
six	O
leaves	O
.	O
</s>
<s>
Whenever	O
this	O
graph	O
is	O
embedded	B-Algorithm
into	O
the	O
plane	O
,	O
some	O
two	O
of	O
its	O
leaves	O
must	O
form	O
an	O
angle	O
of	O
60	O
degrees	O
or	O
less	O
,	O
from	O
which	O
it	O
follows	O
that	O
at	O
least	O
one	O
of	O
these	O
two	O
leaves	O
does	O
not	O
have	O
a	O
neighbor	O
that	O
is	O
closer	O
to	O
the	O
other	O
leaf	O
.	O
</s>
<s>
In	O
Euclidean	O
spaces	O
of	O
higher	O
dimensions	O
,	O
more	O
graphs	O
may	O
have	O
greedy	B-Protocol
embeddings	I-Protocol
;	O
for	O
instance	O
,	O
K1	O
,	O
6	O
has	O
a	O
greedy	B-Protocol
embedding	I-Protocol
into	O
three-dimensional	O
Euclidean	O
space	O
,	O
in	O
which	O
the	O
internal	O
node	O
of	O
the	O
star	O
is	O
at	O
the	O
origin	O
and	O
the	O
leaves	O
are	O
a	O
unit	O
distance	O
away	O
along	O
each	O
coordinate	O
axis	O
.	O
</s>
<s>
However	O
,	O
for	O
every	O
Euclidean	O
space	O
of	O
fixed	O
dimension	O
,	O
there	O
are	O
graphs	O
that	O
cannot	O
be	O
embedded	B-Algorithm
greedily	O
:	O
whenever	O
the	O
number	O
n	O
is	O
greater	O
than	O
the	O
kissing	O
number	O
of	O
the	O
space	O
,	O
the	O
graph	O
K1	O
,	O
n	O
has	O
no	O
greedy	B-Protocol
embedding	I-Protocol
.	O
</s>
<s>
Unlike	O
the	O
case	O
for	O
the	O
Euclidean	O
plane	O
,	O
every	O
network	O
has	O
a	O
greedy	B-Protocol
embedding	I-Protocol
into	O
the	O
hyperbolic	O
plane	O
.	O
</s>
<s>
In	O
contrast	O
,	O
there	O
exist	O
graphs	O
that	O
have	O
greedy	B-Protocol
embeddings	I-Protocol
in	O
the	O
Euclidean	O
plane	O
,	O
but	O
for	O
which	O
any	O
such	O
embedding	O
requires	O
a	O
polynomial	O
number	O
of	O
bits	O
for	O
the	O
Cartesian	O
coordinates	O
of	O
each	O
point	O
.	O
</s>
<s>
The	O
class	O
of	O
trees	O
that	O
admit	O
greedy	B-Protocol
embeddings	I-Protocol
into	O
the	O
Euclidean	O
plane	O
has	O
been	O
completely	O
characterized	O
,	O
and	O
a	O
greedy	B-Protocol
embedding	I-Protocol
of	O
a	O
tree	O
can	O
be	O
found	O
in	O
linear	O
time	O
when	O
it	O
exists	O
.	O
</s>
<s>
For	O
more	O
general	O
graphs	O
,	O
some	O
greedy	B-Protocol
embedding	I-Protocol
algorithms	O
such	O
as	O
the	O
one	O
by	O
Kleinberg	O
start	O
by	O
finding	O
a	O
spanning	O
tree	O
of	O
the	O
given	O
graph	O
,	O
and	O
then	O
construct	O
a	O
greedy	B-Protocol
embedding	I-Protocol
of	O
the	O
spanning	O
tree	O
.	O
</s>
<s>
The	O
result	O
is	O
necessarily	O
also	O
a	O
greedy	B-Protocol
embedding	I-Protocol
of	O
the	O
whole	O
graph	O
.	O
</s>
<s>
However	O
,	O
there	O
exist	O
graphs	O
that	O
have	O
a	O
greedy	B-Protocol
embedding	I-Protocol
in	O
the	O
Euclidean	O
plane	O
but	O
for	O
which	O
no	O
spanning	O
tree	O
has	O
a	O
greedy	B-Protocol
embedding	I-Protocol
.	O
</s>
<s>
conjectured	O
that	O
every	O
polyhedral	O
graph	O
(	O
a	O
3-vertex-connected	O
planar	O
graph	O
,	O
or	O
equivalently	O
by	O
Steinitz	O
's	O
theorem	O
the	O
graph	O
of	O
a	O
convex	O
polyhedron	O
)	O
has	O
a	O
greedy	B-Protocol
embedding	I-Protocol
into	O
the	O
Euclidean	O
plane	O
.	O
</s>
<s>
By	O
exploiting	O
the	O
properties	O
of	O
cactus	O
graphs	O
,	O
proved	O
the	O
conjecture	O
;	O
the	O
greedy	B-Protocol
embeddings	I-Protocol
of	O
these	O
graphs	O
can	O
be	O
defined	O
succinctly	O
,	O
with	O
logarithmically	O
many	O
bits	O
per	O
coordinate	O
.	O
</s>
<s>
However	O
,	O
the	O
greedy	B-Protocol
embeddings	I-Protocol
constructed	O
according	O
to	O
this	O
proof	O
are	O
not	O
necessarily	O
planar	O
embeddings	O
,	O
as	O
they	O
may	O
include	O
crossings	O
between	O
pairs	O
of	O
edges	O
.	O
</s>
<s>
For	O
maximal	O
planar	O
graphs	O
,	O
in	O
which	O
every	O
face	O
is	O
a	O
triangle	O
,	O
a	O
greedy	B-Algorithm
planar	O
embedding	O
can	O
be	O
found	O
by	O
applying	O
the	O
Knaster	O
–	O
Kuratowski	O
–	O
Mazurkiewicz	O
lemma	O
to	O
a	O
weighted	O
version	O
of	O
a	O
straight-line	O
embedding	O
algorithm	O
of	O
Schnyder	O
.	O
</s>
<s>
The	O
strong	O
Papadimitriou	O
–	O
Ratajczak	O
conjecture	O
,	O
that	O
every	O
polyhedral	O
graph	O
has	O
a	O
planar	O
greedy	B-Protocol
embedding	I-Protocol
in	O
which	O
all	O
faces	O
are	O
convex	O
,	O
remains	O
unproven	O
.	O
</s>
<s>
The	O
wireless	B-Architecture
sensor	I-Architecture
networks	I-Architecture
that	O
are	O
the	O
target	O
of	O
greedy	B-Protocol
embedding	I-Protocol
algorithms	O
are	O
frequently	O
modeled	O
as	O
unit	O
disk	O
graphs	O
,	O
graphs	O
in	O
which	O
each	O
node	O
is	O
represented	O
as	O
a	O
unit	O
disk	O
and	O
each	O
edge	O
corresponds	O
to	O
a	O
pair	O
of	O
disks	O
with	O
nonempty	O
intersection	O
.	O
</s>
<s>
For	O
this	O
special	O
class	O
of	O
graphs	O
,	O
it	O
is	O
possible	O
to	O
find	O
succinct	O
greedy	B-Protocol
embeddings	I-Protocol
into	O
a	O
Euclidean	O
space	O
of	O
polylogarithmic	O
dimension	O
,	O
with	O
the	O
additional	O
property	O
that	O
distances	O
in	O
the	O
graph	O
are	O
accurately	O
approximated	O
by	O
distances	O
in	O
the	O
embedding	O
,	O
so	O
that	O
the	O
paths	O
followed	O
by	O
greedy	B-Algorithm
routing	O
are	O
short	O
.	O
</s>
