<s>
A	O
kinetic	B-Algorithm
triangulation	I-Algorithm
data	O
structure	O
is	O
a	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
that	O
maintains	O
a	O
triangulation	B-Algorithm
of	O
a	O
set	O
of	O
moving	O
points	O
.	O
</s>
<s>
Maintaining	O
a	O
kinetic	B-Algorithm
triangulation	I-Algorithm
is	O
important	O
for	O
applications	O
that	O
involve	O
motion	O
planning	O
,	O
such	O
as	O
video	O
games	O
,	O
virtual	O
reality	O
,	O
dynamic	O
simulations	O
and	O
robotics	O
.	O
</s>
<s>
The	O
efficiency	O
of	O
a	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
is	O
defined	O
based	O
on	O
the	O
ratio	O
of	O
the	O
number	O
of	O
internal	O
events	O
to	O
external	O
events	O
,	O
thus	O
good	O
runtime	O
bounds	O
can	O
sometimes	O
be	O
obtained	O
by	O
choosing	O
to	O
use	O
a	O
triangulation	B-Algorithm
scheme	O
that	O
generates	O
a	O
small	O
number	O
of	O
external	O
events	O
.	O
</s>
<s>
For	O
simple	O
affine	O
motion	O
of	O
the	O
points	O
,	O
the	O
number	O
of	O
discrete	O
changes	O
to	O
the	O
convex	O
hull	O
is	O
estimated	O
by	O
,	O
thus	O
the	O
number	O
of	O
changes	O
to	O
any	O
triangulation	B-Algorithm
is	O
also	O
lower	O
bounded	O
by	O
.	O
</s>
<s>
Finding	O
any	O
triangulation	B-Algorithm
scheme	O
that	O
has	O
a	O
near-quadratic	O
bound	O
on	O
the	O
number	O
of	O
discrete	O
changes	O
is	O
an	O
important	O
open	O
problem	O
.	O
</s>
<s>
The	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
seems	O
like	O
a	O
natural	O
candidate	O
,	O
but	O
a	O
tight	O
worst-case	O
analysis	O
of	O
the	O
number	O
of	O
discrete	O
changes	O
that	O
will	O
occur	O
to	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
(	O
external	O
events	O
)	O
was	O
considered	O
an	O
open	O
problem	O
until	O
2015	O
;	O
it	O
has	O
now	O
been	O
bounded	O
to	O
be	O
between	O
and	O
.	O
</s>
<s>
There	O
is	O
a	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
that	O
efficiently	O
maintains	O
the	O
Delaunay	B-Algorithm
triangulation	I-Algorithm
of	O
a	O
set	O
of	O
moving	O
points	O
,	O
in	O
which	O
the	O
ratio	O
of	O
the	O
total	O
number	O
of	O
events	O
to	O
the	O
number	O
of	O
external	O
events	O
is	O
.	O
</s>
<s>
developed	O
a	O
randomized	B-General_Concept
triangulation	B-Algorithm
scheme	O
that	O
experiences	O
an	O
expected	O
number	O
of	O
external	O
events	O
,	O
where	O
is	O
the	O
maximum	O
number	O
of	O
times	O
each	O
triple	O
of	O
points	O
can	O
become	O
collinear	O
,	O
,	O
and	O
is	O
the	O
maximum	O
length	O
of	O
a	O
Davenport-Schinzel	O
sequence	O
of	O
order	O
s	O
+	O
2	O
on	O
n	O
symbols	O
.	O
</s>
<s>
There	O
is	O
a	O
kinetic	B-Data_Structure
data	I-Data_Structure
structure	I-Data_Structure
(	O
due	O
to	O
Agarwal	O
et	O
al	O
.	O
)	O
</s>
<s>
which	O
maintains	O
a	O
pseudo-triangulation	O
in	O
events	O
total	O
.	O
</s>
