<s>
In	O
computational	O
geometry	O
,	O
a	O
greedy	B-Algorithm
geometric	I-Algorithm
spanner	I-Algorithm
is	O
an	O
undirected	O
graph	O
whose	O
distances	O
approximate	O
the	O
Euclidean	O
distances	O
among	O
a	O
finite	O
set	O
of	O
points	O
in	O
a	O
Euclidean	O
space	O
.	O
</s>
<s>
The	O
edges	O
of	O
the	O
spanner	O
are	O
selected	O
by	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
that	O
includes	O
an	O
edge	O
whenever	O
its	O
two	O
endpoints	O
are	O
not	O
connected	O
by	O
a	O
short	O
path	O
of	O
shorter	O
edges	O
.	O
</s>
<s>
Greedy	B-Algorithm
geometric	I-Algorithm
spanners	I-Algorithm
have	O
bounded	O
degree	O
,	O
a	O
linear	O
total	O
number	O
of	O
edges	O
,	O
and	O
total	O
weight	O
close	O
to	O
that	O
of	O
the	O
Euclidean	O
minimum	O
spanning	O
tree	O
.	O
</s>
<s>
Although	O
known	O
construction	O
methods	O
for	O
them	O
are	O
slow	O
,	O
fast	O
approximation	B-Algorithm
algorithms	I-Algorithm
with	O
similar	O
properties	O
are	O
known	O
.	O
</s>
<s>
The	O
greedy	B-Algorithm
geometric	I-Algorithm
spanner	I-Algorithm
is	O
determined	O
from	O
an	O
input	O
consisting	O
a	O
set	O
of	O
points	O
and	O
a	O
parameter	O
.	O
</s>
<s>
It	O
may	O
be	O
constructed	O
by	O
a	O
greedy	B-Algorithm
algorithm	I-Algorithm
that	O
adds	O
edges	O
one	O
at	O
a	O
time	O
to	O
the	O
graph	O
,	O
starting	O
from	O
an	O
edgeless	O
graph	O
with	O
the	O
points	O
as	O
its	O
vertices	O
.	O
</s>
<s>
All	O
pairs	O
of	O
points	O
are	O
considered	O
,	O
in	O
sorted	O
(	O
ascending	O
)	O
order	O
by	O
their	O
distances	O
,	O
starting	O
with	O
the	O
closest	B-Algorithm
pair	I-Algorithm
.	O
</s>
<s>
By	O
construction	O
,	O
the	O
resulting	O
graph	O
is	O
a	O
geometric	B-Algorithm
spanner	I-Algorithm
with	O
stretch	O
factor	O
at	O
most	O
.	O
</s>
<s>
This	O
is	O
because	O
the	O
considerations	O
for	O
each	O
of	O
the	O
pairs	O
of	O
points	O
involve	O
an	O
instance	O
of	O
Dijkstra	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
to	O
find	O
a	O
shortest	O
path	O
in	O
a	O
graph	O
with	O
edges	O
.	O
</s>
<s>
The	O
greedy	B-Algorithm
geometric	I-Algorithm
spanner	I-Algorithm
in	O
any	O
metric	O
space	O
always	O
contains	O
the	O
minimum	O
spanning	O
tree	O
of	O
its	O
input	O
,	O
because	O
the	O
greedy	O
construction	O
algorithm	O
follows	O
the	O
same	O
insertion	O
order	O
of	O
edges	O
as	O
Kruskal	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
for	O
minimum	O
spanning	O
trees	O
.	O
</s>
<s>
If	O
the	O
greedy	O
spanner	O
algorithm	O
and	O
Kruskal	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
are	O
run	O
in	O
parallel	O
,	O
considering	O
the	O
same	O
pairs	O
of	O
vertices	O
in	O
the	O
same	O
order	O
,	O
each	O
edge	O
added	O
by	O
Kruskal	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
will	O
also	O
be	O
added	O
by	O
the	O
greedy	O
spanner	O
algorithm	O
,	O
because	O
the	O
endpoints	O
of	O
the	O
edge	O
will	O
not	O
already	O
be	O
connected	O
by	O
a	O
path	O
.	O
</s>
<s>
Greedy	B-Algorithm
geometric	I-Algorithm
spanners	I-Algorithm
in	O
bounded-dimension	O
Euclidean	O
spaces	O
also	O
have	O
total	O
weight	O
at	O
most	O
a	O
constant	O
times	O
the	O
total	O
weight	O
of	O
the	O
Euclidean	O
minimum	O
spanning	O
tree	O
.	O
</s>
