<s>
In	O
computational	O
geometry	O
,	O
a	O
bitonic	B-Algorithm
tour	I-Algorithm
of	O
a	O
set	O
of	O
point	O
sites	O
in	O
the	O
Euclidean	O
plane	O
is	O
a	O
closed	O
polygonal	O
chain	O
that	O
has	O
each	O
site	O
as	O
one	O
of	O
its	O
vertices	O
,	O
such	O
that	O
any	O
vertical	O
line	O
crosses	O
the	O
chain	O
at	O
most	O
twice	O
.	O
</s>
<s>
The	O
optimal	B-Algorithm
bitonic	I-Algorithm
tour	I-Algorithm
is	O
a	O
bitonic	B-Algorithm
tour	I-Algorithm
of	O
minimum	O
total	O
length	O
.	O
</s>
<s>
It	O
is	O
a	O
standard	O
exercise	O
in	O
dynamic	B-Algorithm
programming	I-Algorithm
to	O
devise	O
a	O
polynomial	O
time	O
algorithm	O
that	O
constructs	O
the	O
optimal	B-Algorithm
bitonic	I-Algorithm
tour	I-Algorithm
.	O
</s>
<s>
The	O
problem	O
of	O
constructing	O
optimal	B-Algorithm
bitonic	I-Algorithm
tours	I-Algorithm
is	O
often	O
credited	O
to	O
Jon	O
L	O
.	O
Bentley	O
,	O
who	O
published	O
in	O
1990	O
an	O
experimental	O
comparison	O
of	O
many	O
heuristics	O
for	O
the	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
;	O
however	O
,	O
Bentley	O
's	O
experiments	O
do	O
not	O
include	O
bitonic	B-Algorithm
tours	I-Algorithm
.	O
</s>
<s>
The	O
first	O
publication	O
that	O
describes	O
the	O
bitonic	B-Algorithm
tour	I-Algorithm
problem	O
appears	O
to	O
be	O
a	O
different	O
1990	O
publication	O
,	O
the	O
first	O
edition	O
of	O
the	O
textbook	O
Introduction	O
to	O
Algorithms	O
by	O
Thomas	O
H	O
.	O
Cormen	O
,	O
Charles	O
E	O
.	O
Leiserson	O
,	O
and	O
Ron	O
Rivest	O
,	O
which	O
lists	O
Bentley	O
as	O
the	O
originator	O
of	O
the	O
problem	O
.	O
</s>
<s>
The	O
optimal	B-Algorithm
bitonic	I-Algorithm
tour	I-Algorithm
has	O
no	O
self-crossings	O
,	O
because	O
any	O
two	O
edges	O
that	O
cross	O
can	O
be	O
replaced	O
by	O
an	O
uncrossed	O
pair	O
of	O
edges	O
with	O
shorter	O
total	O
length	O
due	O
to	O
the	O
triangle	O
inequality	O
.	O
</s>
<s>
the	O
optimal	B-Algorithm
bitonic	I-Algorithm
tour	I-Algorithm
is	O
the	O
one	O
that	O
minimizes	O
the	O
total	O
amount	O
of	O
horizontal	O
motion	O
,	O
with	O
ties	O
broken	O
by	O
Euclidean	O
distance	O
.	O
</s>
<s>
For	O
points	O
in	O
the	O
plane	O
with	O
distinct	O
integer	O
-coordinates	O
and	O
with	O
real-number	O
-coordinates	O
that	O
lie	O
within	O
an	O
interval	O
of	O
length	O
or	O
less	O
,	O
the	O
optimal	B-Algorithm
bitonic	I-Algorithm
tour	I-Algorithm
is	O
an	O
optimal	O
traveling	O
salesperson	O
tour	O
.	O
</s>
<s>
The	O
same	O
dynamic	B-Algorithm
programming	I-Algorithm
algorithm	O
that	O
finds	O
the	O
optimal	B-Algorithm
bitonic	I-Algorithm
tour	I-Algorithm
may	O
be	O
used	O
to	O
solve	O
other	O
variants	O
of	O
the	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
that	O
minimize	O
lexicographic	O
combinations	O
of	O
motion	O
in	O
a	O
fixed	O
number	O
of	O
coordinate	O
directions	O
.	O
</s>
<s>
At	O
the	O
5th	O
International	O
Olympiad	O
in	O
Informatics	O
,	O
in	O
Mendoza	O
,	O
Argentina	O
in	O
1993	O
,	O
one	O
of	O
the	O
contest	O
problems	O
involved	O
bitonic	B-Algorithm
tours	I-Algorithm
:	O
the	O
contestants	O
were	O
to	O
devise	O
an	O
algorithm	O
that	O
took	O
as	O
input	O
a	O
set	O
of	O
sites	O
and	O
a	O
collection	O
of	O
allowed	O
edges	O
between	O
sites	O
and	O
construct	O
a	O
bitonic	B-Algorithm
tour	I-Algorithm
using	O
those	O
edges	O
that	O
included	O
as	O
many	O
sites	O
as	O
possible	O
.	O
</s>
<s>
As	O
with	O
the	O
optimal	B-Algorithm
bitonic	I-Algorithm
tour	I-Algorithm
,	O
this	O
problem	O
may	O
be	O
solved	O
by	O
dynamic	B-Algorithm
programming	I-Algorithm
.	O
</s>
