<s>
A	O
vantage-point	B-Data_Structure
tree	I-Data_Structure
(	O
or	O
VP	B-Data_Structure
tree	I-Data_Structure
)	O
is	O
a	O
metric	B-Data_Structure
tree	I-Data_Structure
that	O
segregates	O
data	O
in	O
a	O
metric	O
space	O
by	O
choosing	O
a	O
position	O
in	O
the	O
space	O
(	O
the	O
"	O
vantage	O
point	O
"	O
)	O
and	O
partitioning	O
the	O
data	O
points	O
into	O
two	O
parts	O
:	O
those	O
points	O
that	O
are	O
nearer	O
to	O
the	O
vantage	O
point	O
than	O
a	O
threshold	O
,	O
and	O
those	O
points	O
that	O
are	O
not	O
.	O
</s>
<s>
By	O
recursively	O
applying	O
this	O
procedure	O
to	O
partition	O
the	O
data	O
into	O
smaller	O
and	O
smaller	O
sets	O
,	O
a	O
tree	B-Application
data	I-Application
structure	I-Application
is	O
created	O
where	O
neighbors	O
in	O
the	O
tree	B-Application
are	O
likely	O
to	O
be	O
neighbors	O
in	O
the	O
space	O
.	O
</s>
<s>
One	O
generalization	O
is	O
called	O
a	O
multi-vantage-point	O
tree	O
(	O
or	O
MVP	B-Data_Structure
tree	I-Data_Structure
)	O
:	O
a	O
data	B-General_Concept
structure	I-General_Concept
for	O
indexing	O
objects	O
from	O
large	O
metric	O
spaces	O
for	O
similarity	B-Algorithm
search	I-Algorithm
queries	O
.	O
</s>
<s>
Peter	O
Yianilos	O
claimed	O
that	O
the	O
vantage-point	B-Data_Structure
tree	I-Data_Structure
was	O
discovered	O
independently	O
by	O
him	O
(	O
Peter	O
Yianilos	O
)	O
and	O
by	O
Jeffrey	O
Uhlmann	O
.	O
</s>
<s>
Uhlmann	O
called	O
the	O
data	B-General_Concept
structure	I-General_Concept
a	O
metric	B-Data_Structure
tree	I-Data_Structure
,	O
the	O
name	O
VP-tree	B-Data_Structure
was	O
proposed	O
by	O
Yianilos	O
.	O
</s>
<s>
Vantage-point	B-Data_Structure
trees	I-Data_Structure
have	O
been	O
generalized	O
to	O
non-metric	O
spaces	O
using	O
Bregman	O
divergences	O
by	O
Nielsen	O
et	O
al	O
.	O
</s>
<s>
This	O
iterative	O
partitioning	O
process	O
is	O
similar	O
to	O
that	O
of	O
a	O
-d	O
tree	B-Application
,	O
but	O
uses	O
circular	O
(	O
or	O
spherical	O
,	O
hyperspherical	O
,	O
etc	O
.	O
)	O
</s>
<s>
The	O
vantage-point	B-Data_Structure
tree	I-Data_Structure
is	O
particularly	O
useful	O
in	O
dividing	O
data	O
in	O
a	O
non-standard	O
metric	O
space	O
into	O
a	O
metric	B-Data_Structure
tree	I-Data_Structure
.	O
</s>
<s>
The	O
way	O
a	O
vantage-point	B-Data_Structure
tree	I-Data_Structure
stores	O
data	O
can	O
be	O
represented	O
by	O
a	O
circle	O
.	O
</s>
<s>
First	O
,	O
understand	O
that	O
each	O
node	B-Data_Structure
of	O
this	O
tree	B-Application
contains	O
an	O
input	O
point	O
and	O
a	O
radius	O
.	O
</s>
<s>
All	O
the	O
left	O
children	O
of	O
a	O
given	O
node	B-Data_Structure
are	O
the	O
points	O
inside	O
the	O
circle	O
and	O
all	O
the	O
right	O
children	O
of	O
a	O
given	O
node	B-Data_Structure
are	O
outside	O
of	O
the	O
circle	O
.	O
</s>
<s>
The	O
tree	B-Application
itself	O
does	O
not	O
need	O
to	O
know	O
any	O
other	O
information	O
about	O
what	O
is	O
being	O
stored	O
.	O
</s>
<s>
A	O
vantage-point	B-Data_Structure
tree	I-Data_Structure
can	O
be	O
used	O
to	O
find	O
the	O
nearest	O
neighbor	O
of	O
a	O
point	O
.	O
</s>
<s>
At	O
any	O
given	O
step	O
we	O
are	O
working	O
with	O
a	O
node	B-Data_Structure
of	O
the	O
tree	B-Application
that	O
has	O
a	O
vantage	O
point	O
and	O
a	O
threshold	O
distance	O
.	O
</s>
<s>
If	O
that	O
distance	O
is	O
less	O
than	O
then	O
use	O
the	O
algorithm	O
recursively	O
to	O
search	O
the	O
subtree	B-Application
of	O
the	O
node	B-Data_Structure
that	O
contains	O
the	O
points	O
closer	O
to	O
the	O
vantage	O
point	O
than	O
the	O
threshold	O
;	O
otherwise	O
recurse	O
to	O
the	O
subtree	B-Application
of	O
the	O
node	B-Data_Structure
that	O
contains	O
the	O
points	O
that	O
are	O
farther	O
than	O
the	O
vantage	O
point	O
than	O
the	O
threshold	O
.	O
</s>
<s>
If	O
the	O
recursive	O
use	O
of	O
the	O
algorithm	O
finds	O
a	O
neighboring	O
point	O
with	O
distance	O
to	O
that	O
is	O
less	O
than	O
then	O
it	O
cannot	O
help	O
to	O
search	O
the	O
other	O
subtree	B-Application
of	O
this	O
node	B-Data_Structure
;	O
the	O
discovered	O
node	B-Data_Structure
is	O
returned	O
.	O
</s>
<s>
Otherwise	O
,	O
the	O
other	O
subtree	B-Application
also	O
needs	O
to	O
be	O
searched	O
recursively	O
.	O
</s>
<s>
In	O
the	O
recursion	O
,	O
the	O
other	O
subtree	B-Application
is	O
searched	O
for	O
nearest	O
neighbors	O
of	O
the	O
point	O
whenever	O
only	O
of	O
the	O
nearest	O
neighbors	O
found	O
so	O
far	O
have	O
distance	O
that	O
is	O
less	O
than	O
.	O
</s>
<s>
Updating	O
a	O
vantage-point	B-Data_Structure
tree	I-Data_Structure
is	O
relatively	O
easy	O
compared	O
to	O
the	O
FastMap	O
approach	O
.	O
</s>
<s>
The	O
time	O
cost	O
to	O
build	O
a	O
vantage-point	B-Data_Structure
tree	I-Data_Structure
is	O
approximately	O
.	O
</s>
<s>
For	O
each	O
element	O
,	O
the	O
tree	B-Application
is	O
descended	O
by	O
levels	O
to	O
find	O
its	O
placement	O
.	O
</s>
<s>
However	O
there	O
is	O
a	O
constant	O
factor	O
where	O
is	O
the	O
number	O
of	O
vantage	O
points	O
per	O
tree	B-Application
node	B-Data_Structure
.	O
</s>
<s>
The	O
time	O
cost	O
to	O
search	O
a	O
vantage-point	B-Data_Structure
tree	I-Data_Structure
to	O
find	O
a	O
single	O
nearest	O
neighbor	O
is	O
.	O
</s>
<s>
There	O
are	O
levels	O
,	O
each	O
involving	O
distance	O
calculations	O
,	O
where	O
is	O
the	O
number	O
of	O
vantage	O
points	O
(	O
elements	O
)	O
at	O
that	O
position	O
in	O
the	O
tree	B-Application
.	O
</s>
<s>
The	O
time	O
cost	O
to	O
search	O
a	O
vantage-point	B-Data_Structure
tree	I-Data_Structure
for	O
a	O
range	O
,	O
which	O
may	O
be	O
the	O
most	O
important	O
attribute	O
,	O
can	O
vary	O
greatly	O
depending	O
on	O
the	O
specifics	O
of	O
the	O
algorithm	O
used	O
and	O
parameters	O
.	O
</s>
<s>
The	O
space	O
cost	O
for	O
a	O
vantage-point	B-Data_Structure
tree	I-Data_Structure
is	O
approximately	O
.	O
</s>
<s>
Each	O
element	O
is	O
stored	O
,	O
and	O
each	O
tree	B-Application
element	O
in	O
each	O
non-leaf	O
node	B-Data_Structure
requires	O
a	O
pointer	O
to	O
its	O
descendant	O
nodes	O
.	O
</s>
<s>
The	O
parameter	O
for	O
number	O
of	O
elements	O
at	O
each	O
node	B-Data_Structure
plays	O
a	O
factor	O
.	O
)	O
</s>
<s>
Note	O
that	O
some	O
metric	O
space	O
tools	O
require	O
a	O
matrix	O
of	O
pair-wise	O
distance	O
values	O
,	O
costing	O
,	O
but	O
that	O
is	O
not	O
required	O
with	O
vantage-point	B-Data_Structure
trees	I-Data_Structure
.	O
</s>
