<s>
Algorithms	O
that	O
construct	O
convex	O
hulls	O
of	O
various	O
objects	O
have	O
a	O
broad	O
range	O
of	O
applications	O
in	O
mathematics	O
and	O
computer	B-General_Concept
science	I-General_Concept
.	O
</s>
<s>
In	O
computational	O
geometry	O
,	O
numerous	O
algorithms	O
are	O
proposed	O
for	O
computing	O
the	O
convex	O
hull	O
of	O
a	O
finite	O
set	O
of	O
points	O
,	O
with	O
various	O
computational	B-General_Concept
complexities	I-General_Concept
.	O
</s>
<s>
Computing	O
the	O
convex	O
hull	O
means	O
that	O
a	O
non-ambiguous	O
and	O
efficient	O
representation	B-General_Concept
of	O
the	O
required	O
convex	O
shape	O
is	O
constructed	O
.	O
</s>
<s>
Its	O
most	O
common	O
representation	B-General_Concept
is	O
the	O
list	O
of	O
its	O
vertices	O
ordered	O
along	O
its	O
boundary	O
clockwise	O
or	O
counterclockwise	O
.	O
</s>
<s>
For	O
a	O
finite	O
set	O
of	O
points	O
in	O
the	O
plane	O
the	O
lower	O
bound	O
on	O
the	O
computational	O
complexity	O
of	O
finding	O
the	O
convex	O
hull	O
represented	O
as	O
a	O
convex	O
polygon	O
is	O
easily	O
shown	O
to	O
be	O
the	O
same	O
as	O
for	O
sorting	B-Algorithm
using	O
the	O
following	O
reduction	B-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
in	O
the	O
general	O
case	O
the	O
convex	O
hull	O
of	O
n	O
points	O
cannot	O
be	O
computed	O
more	O
quickly	O
than	O
sorting	B-Algorithm
.	O
</s>
<s>
The	O
standard	O
Ω(n log n )	O
lower	O
bound	O
for	O
sorting	B-Algorithm
is	O
proven	O
in	O
the	O
decision	B-Application
tree	I-Application
model	I-Application
of	O
computing	O
,	O
in	O
which	O
only	O
numerical	O
comparisons	O
but	O
not	O
arithmetic	O
operations	O
can	O
be	O
performed	O
;	O
however	O
,	O
in	O
this	O
model	O
,	O
convex	O
hulls	O
cannot	O
be	O
computed	O
at	O
all	O
.	O
</s>
<s>
Sorting	B-Algorithm
also	O
requires	O
Ω(n log n )	O
time	O
in	O
the	O
algebraic	B-Application
decision	I-Application
tree	I-Application
model	O
of	O
computation	O
,	O
a	O
model	O
that	O
is	O
more	O
suitable	O
for	O
convex	O
hulls	O
,	O
and	O
in	O
this	O
model	O
convex	O
hulls	O
also	O
require	O
Ω(n log n )	O
time	O
.	O
</s>
<s>
However	O
,	O
in	O
models	O
of	O
computer	O
arithmetic	O
that	O
allow	O
numbers	O
to	O
be	O
sorted	O
more	O
quickly	O
than	O
O(n log n )	O
time	O
,	O
for	O
instance	O
by	O
using	O
integer	B-Algorithm
sorting	I-Algorithm
algorithms	O
,	O
planar	O
convex	O
hulls	O
can	O
also	O
be	O
computed	O
more	O
quickly	O
:	O
the	O
Graham	B-Algorithm
scan	I-Algorithm
algorithm	O
for	O
convex	O
hulls	O
consists	O
of	O
a	O
single	O
sorting	B-Algorithm
step	O
followed	O
by	O
a	O
linear	O
amount	O
of	O
additional	O
work	O
.	O
</s>
<s>
However	O
,	O
the	O
complexity	O
of	O
some	O
convex	B-Algorithm
hull	I-Algorithm
algorithms	I-Algorithm
can	O
be	O
characterized	O
in	O
terms	O
of	O
both	O
input	O
size	O
n	O
and	O
the	O
output	O
size	O
h	O
(	O
the	O
number	O
of	O
points	O
in	O
the	O
hull	O
)	O
.	O
</s>
<s>
Such	O
algorithms	O
are	O
called	O
output-sensitive	B-Language
algorithms	I-Language
.	O
</s>
<s>
The	O
lower	O
bound	O
on	O
worst-case	O
running	O
time	O
of	O
output-sensitive	B-Language
convex	B-Algorithm
hull	I-Algorithm
algorithms	I-Algorithm
was	O
established	O
to	O
be	O
Ω(n log h )	O
in	O
the	O
planar	O
case	O
.	O
</s>
<s>
There	O
are	O
several	O
algorithms	O
which	O
attain	O
this	O
optimal	O
time	B-General_Concept
complexity	I-General_Concept
.	O
</s>
<s>
The	O
earliest	O
one	O
was	O
introduced	O
by	O
Kirkpatrick	O
and	O
Seidel	O
in	O
1986	O
(	O
who	O
called	O
it	O
"	O
the	O
ultimate	B-Algorithm
convex	I-Algorithm
hull	I-Algorithm
algorithm	I-Algorithm
"	O
)	O
.	O
</s>
<s>
A	O
much	O
simpler	O
algorithm	O
was	O
developed	O
by	O
Chan	O
in	O
1996	O
,	O
and	O
is	O
called	O
Chan	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Known	O
convex	B-Algorithm
hull	I-Algorithm
algorithms	I-Algorithm
are	O
listed	O
below	O
,	O
ordered	O
by	O
the	O
date	O
of	O
first	O
publication	O
.	O
</s>
<s>
Time	B-General_Concept
complexity	I-General_Concept
of	O
each	O
algorithm	O
is	O
stated	O
in	O
terms	O
of	O
the	O
number	O
of	O
inputs	O
points	O
n	O
and	O
the	O
number	O
of	O
points	O
on	O
the	O
hull	O
h	O
.	O
Note	O
that	O
in	O
the	O
worst	O
case	O
h	O
may	O
be	O
as	O
large	O
as	O
n	O
.	O
</s>
<s>
Gift	B-Algorithm
wrapping	I-Algorithm
,	O
a.k.a.	O
</s>
<s>
Jarvis	B-Algorithm
march	I-Algorithm
—	O
O(nh )	O
One	O
of	O
the	O
simplest	O
(	O
although	O
not	O
the	O
most	O
time	O
efficient	O
in	O
the	O
worst	O
case	O
)	O
planar	O
algorithms	O
.	O
</s>
<s>
It	O
has	O
O(nh )	O
time	B-General_Concept
complexity	I-General_Concept
,	O
where	O
n	O
is	O
the	O
number	O
of	O
points	O
in	O
the	O
set	O
,	O
and	O
h	O
is	O
the	O
number	O
of	O
points	O
in	O
the	O
hull	O
.	O
</s>
<s>
Graham	B-Algorithm
scan	I-Algorithm
—	O
O(n log n )	O
A	O
slightly	O
more	O
sophisticated	O
,	O
but	O
much	O
more	O
efficient	O
algorithm	O
,	O
published	O
by	O
Ronald	O
Graham	O
in	O
1972	O
.	O
</s>
<s>
Quickhull	B-Algorithm
Created	O
independently	O
in	O
1977	O
by	O
W	O
.	O
Eddy	O
and	O
in	O
1978	O
by	O
A	O
.	O
Bykat	O
.	O
</s>
<s>
Just	O
like	O
the	O
quicksort	B-Algorithm
algorithm	O
,	O
it	O
has	O
the	O
expected	O
time	B-General_Concept
complexity	I-General_Concept
of	O
O(n log n )	O
,	O
but	O
may	O
degenerate	O
to	O
O(n2 )	O
in	O
the	O
worst	O
case	O
.	O
</s>
<s>
The	O
algorithm	O
can	O
be	O
seen	O
as	O
a	O
variant	O
of	O
Graham	B-Algorithm
scan	I-Algorithm
which	O
sorts	O
the	O
points	O
lexicographically	O
by	O
their	O
coordinates	O
.	O
</s>
<s>
Incremental	O
convex	B-Algorithm
hull	I-Algorithm
algorithm	I-Algorithm
—	O
O(n log n )	O
Published	O
in	O
1984	O
by	O
Michael	O
Kallay	O
.	O
</s>
<s>
Kirkpatrick	B-Algorithm
–	I-Algorithm
Seidel	I-Algorithm
algorithm	I-Algorithm
—	O
O(n log h )	O
The	O
first	O
optimal	O
output-sensitive	B-Language
algorithm	I-Language
.	O
</s>
<s>
Chan	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
—	O
O(n log h )	O
A	O
simpler	O
optimal	O
output-sensitive	B-Language
algorithm	I-Language
created	O
by	O
Chan	O
in	O
1996	O
.	O
</s>
<s>
It	O
combines	O
gift	B-Algorithm
wrapping	I-Algorithm
with	O
the	O
execution	O
of	O
an	O
O(n log n )	O
algorithm	O
(	O
such	O
as	O
Graham	B-Algorithm
scan	I-Algorithm
)	O
on	O
small	O
subsets	O
of	O
the	O
input	O
.	O
</s>
<s>
The	O
following	O
simple	O
heuristic	O
is	O
often	O
used	O
as	O
the	O
first	O
step	O
in	O
implementations	O
of	O
convex	B-Algorithm
hull	I-Algorithm
algorithms	I-Algorithm
to	O
improve	O
their	O
performance	O
.	O
</s>
<s>
It	O
is	O
based	O
on	O
the	O
efficient	O
convex	B-Algorithm
hull	I-Algorithm
algorithm	I-Algorithm
by	O
Selim	O
Akl	O
and	O
G	O
.	O
T	O
.	O
Toussaint	O
,	O
1978	O
.	O
</s>
<s>
If	O
the	O
points	O
are	O
random	O
variables	O
,	O
then	O
for	O
a	O
narrow	O
but	O
commonly	O
encountered	O
class	O
of	O
probability	O
density	O
functions	O
,	O
this	O
throw-away	O
pre-processing	O
step	O
will	O
make	O
a	O
convex	B-Algorithm
hull	I-Algorithm
algorithm	I-Algorithm
run	O
in	O
linear	O
expected	O
time	O
,	O
even	O
if	O
the	O
worst-case	O
complexity	O
of	O
the	O
convex	B-Algorithm
hull	I-Algorithm
algorithm	I-Algorithm
is	O
quadratic	O
in	O
n	O
.	O
</s>
<s>
Dynamic	B-Algorithm
convex	I-Algorithm
hull	I-Algorithm
maintenance	O
:	O
The	O
input	O
points	O
may	O
be	O
sequentially	O
inserted	O
or	O
deleted	O
,	O
and	O
the	O
convex	O
hull	O
must	O
be	O
updated	O
after	O
each	O
insert/delete	O
operation	O
.	O
</s>
<s>
A	O
later	O
simplification	O
by	O
and	O
uses	O
only	O
a	O
single	O
stack	B-Application
data	I-Application
structure	I-Application
.	O
</s>
<s>
As	O
it	O
does	O
,	O
it	O
stores	O
a	O
convex	O
sequence	O
of	O
vertices	O
on	O
the	O
stack	B-Application
,	O
the	O
ones	O
that	O
have	O
not	O
yet	O
been	O
identified	O
as	O
being	O
within	O
pockets	O
.	O
</s>
<s>
At	O
each	O
step	O
,	O
the	O
algorithm	O
follows	O
a	O
path	O
along	O
the	O
polygon	O
from	O
the	O
stack	B-Application
top	O
to	O
the	O
next	O
vertex	O
that	O
is	O
not	O
in	O
one	O
of	O
the	O
two	O
pockets	O
adjacent	O
to	O
the	O
stack	B-Application
top	O
.	O
</s>
<s>
Then	O
,	O
while	O
the	O
top	O
two	O
vertices	O
on	O
the	O
stack	B-Application
together	O
with	O
this	O
new	O
vertex	O
are	O
not	O
in	O
convex	O
position	O
,	O
it	O
pops	O
the	O
stack	B-Application
,	O
before	O
finally	O
pushing	O
the	O
new	O
vertex	O
onto	O
the	O
stack	B-Application
.	O
</s>
<s>
When	O
the	O
clockwise	O
traversal	O
reaches	O
the	O
starting	O
point	O
,	O
the	O
algorithm	O
returns	O
the	O
sequence	O
of	O
stack	B-Application
vertices	O
as	O
the	O
hull	O
.	O
</s>
<s>
Chan	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
used	O
for	O
dimensions	O
2	O
and	O
3	O
,	O
and	O
Quickhull	B-Algorithm
is	O
used	O
for	O
computation	O
of	O
the	O
convex	O
hull	O
in	O
higher	O
dimensions	O
.	O
</s>
<s>
Its	O
representation	B-General_Concept
is	O
not	O
so	O
simple	O
as	O
in	O
the	O
planar	O
case	O
,	O
however	O
.	O
</s>
<s>
The	O
size	O
of	O
the	O
output	O
face	O
information	O
may	O
be	O
exponentially	O
larger	O
than	O
the	O
size	O
of	O
the	O
input	O
vertices	O
,	O
and	O
even	O
in	O
cases	O
where	O
the	O
input	O
and	O
output	O
are	O
both	O
of	O
comparable	O
size	O
the	O
known	O
algorithms	O
for	O
high-dimensional	O
convex	O
hulls	O
are	O
not	O
output-sensitive	B-Language
due	O
both	O
to	O
issues	O
with	O
degenerate	O
inputs	O
and	O
with	O
intermediate	O
results	O
of	O
high	O
complexity	O
.	O
</s>
