Chapter
The
Jurek
Aquarium
51
Keeper’s
Czyzowiczt
Peter Egyed$
Hazel EvereW
David RappaportS
Thomas SherrnerT
Diane Souvainell
Godfried
Toussaint$
Abstract
Heron
We solve the problem
work
inside
a given
(Aquarium
of computing
polygon
which
Z{eeper’s
a linear-time
and shortest-path
by using
Tour).
algorithm
hulls
closed path
edge at least
polygons,
uses the
We then
convex
which
every
For convex
maps.
relative
for polygons
the shortest
visits
which
Problem*
we present
reflection
generalize
to provide
a linear
method
1985,
Klotzler
infinite
algorithm
given
variously
Steiner
one
hundred
years
with
one
he also
vertex
case, there
principle
an
on
to show
acute
perimeter
inscribed
with
earliest
was even
two
i.e.
the
on one side
points
via
vertex
of it,
the
with
find
line.
the
This
any
reelection
altitudes
minimum
the
minimum
by
twice
realized
with
of the
two
input
the reflection
the triangle:
but
In
of the
triangles,
is
the
triangle
of the
vertices
and
over
triangle
used the
points
it is degenerate
solved
than
solved
[4] ,[8] ,[12] ,[13].
obtuse
obtuse
problem
simpler
the
only
of a given
foot
triangle
altitude,
coinciding
points
For
that,
perimeter
Schwarz
the
are
polygon.
shortest
that
that
triangle
edge
problem
[11]
insist
not
minimum
each
the
Fagnano
sources
Schwarz
the
is consensus
inscribed
the
ago,
posed
to
many
of computing
that
The
attributed
[5] ,[9] ,[15] ,[14],
problem
of
ple
given
the
[13]
rithm,
triangle.
principle
a line
and two
shortest
path
between
problem
was solved
by
*Pm-t of the work was carried out when the authom were
participants
of the Workshop on Illuminating
Sets at the Bellairs
Research Institute of McGill University.
tDepartment
d’Inforrnzkique,
Universit6
du Quebech Hull,
Quebec, CANADA
H3S 2A7.
$D~pw@ent
of computing
University,
and Information
Montreal,
Science, Queen’s
path
University, Kingston, Ontario, CANADA
K7L 3N6.
11S&X-J of Computing Science, Simon Fraser University,
V5A 1 S6.
Col-bi.,
CANADA
IIDepartment
of Computer Science, Rutgers
Burn-
University,
New
Brunswick, New Jersey, USA 08903. Supported in part by NSF
Grant CCR-8S-03549.
**Department
of Computer
Science, University
of Ottawa,
Ottawa, Ontario, CANADA
KIN 6N5.
of correct
it
is unclear
the
workshop,
every
then
polygon
which
optimum
uses
We
using
convex
hulls
relative
polygons
which
used
a somewhat
principle
Problem.
In
and
Zoo
edges
of P,
in P and not
ending
contains
log~ n)
algorithm
cages
for
time
of P
glass
plates
n cages,
then
the
re-
a related
Keeper’s
Keeper’s
given
polygons
point
a sim(cages)
x on
the
perime-
of any cage, starting
cage.
algorithm
(i. e., the
to an-
Zoo
the minimum
every
of a series
zoo
Ntafos
of the
is related
the
an entry
the problem
as edges
the front
459
and
is to find
visits
and
solve
Problem,
k convex
in the interior
at z, that
an O(n
[2],
Keeper’s
of P, the goal
by
algorithm
[3].
Problem
Ntafos
P of n vertices,
to
boundary
the
Chin
to
and
method
a linear
shortest-paths
the
a linear-
principle
combination
Keeper’s
Chin
in fact,
our
convex.
polygons
Aquarium
is,
reflection
to provide
not
If the poly-
we present
generalize
similm
on rectilinear
ple polygon
the
are
and
problem
the
Keeper’s
once.
path
and
also
for the shortest
(Aquarium
at least
inpolygon,
maps.
algorithm
[161 Posed the
he asked
edge
the
perimeter
his
Toussaint
a simple
visits
algorithm
and
four
algo-
solution.
differently:
inside
solutions
whether
correct
shortest-path
O(nz)
aby, British
and
detect
is convex,
ter tour
He presents
In
attached
Science, McGill
descent.
de-
princi-
one
somewhat
other
a finite
reflection
omits
a recent
problem
using
Schwarz’
analysis.
type
flection
for
complexity
which
have
in-
an algorithm
no formal
Tour)
for
of a
provides
closed
time
edge
perimeter
of the
minimum
the
each
performance
in fact
At
gon
vertices
coordinate-wise
a semiminimum
efficient
realizable
problem
on
minumum
inpolygon,
from
his classification
would
used
the
[5] presented
optimum
Euclid’s
to illustrate
but
addition,
[9]
vertex
the
Focke
extending
determine
one
or
resulting
and
The
Hull, Quebec, CANADA.
of Computer
$s&ool
1986,
method
examples
with
while
Rudolph
to
polygon,
In
computing
Introduction
Although
polygon
convex
scent
1
and
simplex-method
polygon.
are not convex.
in 100 AD
[8].
perimeter
principle
that
of Alexandria
on optics
In
once
Jorge Urrutia**
The
and
paper
refers
[2]
to a
as well.
If we consider
edges
of P represent
of aquariums)
problem
and have
reduces
tO a
CZYZOWICZ
460
simplified
version
fixed
starting
this
restriction,
either
O(n
2
of the
Aquarium
point
is given.
our
complexity
logn
Problem
Note
that
in which
even
a
without
is appreciably
less than
,q3( v,)
n) or 0(n2).
Optimum
ET AL.
Keeper’s
Tour
in
a
Convex
Aquarium
Given
a convex
polygon
given
in
ented
in clockwise
is to
determine
such
that
the
ei
clockwise
let
order
least
such
which
of
the
with
a vertex
vj
tinct
edges,
there
can be no advantage
once,
ori-
The
goal
wo, W1, . . . . wk.
lies
on
of
the
closed
length
that
edge
Wi
Since
of P and
it is possible
the
at vi.
points,
the
. . . ~wk - 1, w(I is a minimum.
coincide
VO, VI, . . . . v~-1
originates
of
one
that
vertices
ei represent
a sequence
at
and
W(), wl,
P with
order,
a point
thus
in visiting
of
path
w~ may
lie on two
k < n. Since
-l
each
dis-
P is convex,
an edge more
Figure
than
1:
then
the
polygon
tour
2.1.
If
the
optimum
aquarium
keeper’s
in order.
is exactly
In
the
polygon
tour
other
visits
words,
optimum
is
the
the
Proof.
Assume
edges
in
Wj Wj+l
order,
a convex
every
triangle
the
optimum
of the
tour
as follows,
keeper’s
in
and
the
of
that
the
sum
of the lengths
the
of
Wlj ..., WiJWj,
of
the
its
then
be optimal,
does
visit
a side
LEMMA
the
interior
angle
Vj+l
Proof.
line
P
original
and
its
triangulated
points
not
lie on ej,
of the
to
in equal
on ei+l
points
are
. ..>wk.
wl
then
v2i_l
and
rotate
the
of the
then
lies
are
disjoint.
wi+l.
vzi
Q2i_l.
until
vertices
and
v2i+1
in
vzi
appropriate
it
Qn-l
of
V.
so. If the
tour
fOllOwS:
qi
=
i =
edge
tour
lies
must
wi–lwit.j
[n/2j
ellipse
q which
wi_ 1 and
coincide
then
Wi
of tangency
=
with
foci
to
and
If q does
with
the
endpoint
of
q.
But
the
radii
of
the
tangent
intersect
V1 as
Vz~+l
of tour
in the
triangle
identically
Thus,
triangle
for
equal
it
a point
on ei and
Avivi+l
to
is possible
vi+z,
vi+l
or
a
whether
the
for the first
but
it
on the
polygon
does
not
starts
to unwrap
Proof.
the
image
Vn+l.
Thus,
be
ion
=
Thus,
the
in
a chain
we have
The
optimum
and
ends
at a point
Every
qLn/2J
it
qLr@J
qn+l q.
L? if
n is
of Q, except
R
+1
which
which
2),
We focus
given.
on Vovl
from
The
(see fig.
keeper’s
the corresponding
path
directed
of triangles:
is or not.
been
2.3.
on qLn/ZJ 9Ln/ZJ+l
of Q.
on
as
vz(~-i+l)
of edges
be non-simple
whether
path from
the
identified
qi
pair
and
second
have two such sides each.
Q may
matter
triangular
to its ima9e
ey triangle
its
these
1).
are
perform
Avn _ 1Vovl
9Lfl/2j91n/2j+l)
which
and
Vzi+l
of
can
of
and
with
Label
O... [n/2];
composed
reflect
translate
and
Finally,
and
to the
(resP.
and last
LE~NIA
two
i =
position
aligned
has one side on the boundary
the shortest
between
Vn
order
+ 1 (see fig.
and qLn/2J+l~Ln/2J
even(resp.
odd).
which
[8].
Q.
VI
until
translate
Vzi
Q2i.
at
Q2i,
Qzi+l,
of
form
clockwise
+ l...n
triangulated
minimizes
Wi+l.
build
VOW in P corresponds
every
L represent
To
transformation
to
current
build
Vzi are
Vzi+z
instances
for
v2i_ 1 and
reflect
Q1
AvzvavA
vzi - 1V2i and
in
Q in
the
the
to
Avz; vz;+l
with
to
rotate
general,
edge
vertices
of
Let
a unique
from
w~ will
a point
until
vertices
then
In
around
this
optimum
angle
it
Qz.
inst antes,
interior,
Wi of the
rotate
attach
consists
with
1vzivzi+l
aligned
of reflection.
to L at a point
distances
segment
both
in its
angle
Then
ej.
angles
point
Thus
Av2i_
V3 are aligned
Q2,
it
Q
AVOV1V2
build
attach
and
V3 in
structure
triangle
To
P.
V1V2 and
Q3, translate
Q is a polygon
to q. If q ~ ej,
an ellipse
The
polygon
the
sides.
wj+2,
necessarily
the
it follows
a contradiction.
wi– 1 and
wi _ 1 and Wi is tangent
ej closer
wj+l,
the
exceeds
of opposite
tour
edge ej,
containing
sum
sides,
than
in
polygonal
represent
in
edge
V2 and
V2 and
Q1
position
around
vertices
of
a new
Let
V2. To build
W~W~+l.
Fix
the
two
m’any
not
and
that
less
diagonals
Wi+l,
a point
of
is
In
but
If
side
VI.
equals
2.2.
the fact
one candidate
of the
of incidence
one
providing
vI, v3, v5, . . .. Vn_I.
will
from
the
..., Wi+2,
Wi Wi+l
and
Wi wj Wi+I wj+l
and
pairs
tour,
does not visit
edges
Then
other
of
of both
Wj-~,
If n is even,
tour
that
of
lengths
a shorter
vertices
points
thus
tour
for i < j.
length
lengths
sum
describes
equal
the optimum
producing
of n + 2 vertices.
inpo!ygon.
quadrilateral
sum
angle
that
cross each other,
define
line
polygon
convex,
sides
AV1V2V3
the
convex
Q.
so k ~ n.
LEMMA
in
A
unfolding
tour
point
passes
a point
passes
in
corresponds
on
P
to
on qn+lqo
through
ev-
qn+l q.
through
to
every
THE
AQUARIUM
KEEPER’S
461
PROBLEM
~3
‘ .<
q~
q~
%
Figure
----------
3:
from
In
q. and
unfolded
q~+l
and the induced
P’
and
Figure
2:
A
obtained
vertices
can
polygon
P’
will
non-simple
from
the
triangulated
triangle
be placed
whose
P.
in the
polygon
Any
odd
marked
associated
Q
polygon
qn+l
S( Q(q~+l,
ward
Q’
to
be non-simple.
of
P.
Divide
R
crosses.
Reflect
portion
of the
to
Av2i_
path
1v2iv2i+1
the
(corresponding
tour
the
a path
tour
R in
+
the
some
Q of exactly
to
of these
– t)9in/2J+l)
tour
t,
the
and
the
in P.
right-
finishing
same
+
length
between
correspond
(resp.
tours
and
to the
polygon
2.1.
The
optimum
P of n vertices
keeper’s
point
compute
cici+l
Vn Vn+l,
the
anchors
of the
a
in O(n)
the
in
first
minimizes
to tdi
Q,
Proof.
scribed
the
Create
above
shortest
in
the
linear
path
triangulated
time.
inside
polygon
In
linear
Q from
Q
time
qo to
as de[7],
qln/zJ
if C’i and Di
and
from
qn+l)
or
let
the first
(see fig.
on the
by
cor-
9Ln/zj +1,
3).
left-
and
Melissaratos
minimum
and
length
perimeter
segments.
entirely
the
Di
areatriangle
Ci to Di
or
find
distance
square
time,
orientation,
joins
value
each
of
In
t that
tci+(1 –t)c~+l
from
distance
can be
t which
can
to note
then
two
we can
visible.
It is interesting
either
right-
from
partially
of
on
and
visible
the
segment
didi+l
left-
of this
function
time.
have opposite
the
=
In constant
is to
The
each
Di
are entirely
invisible,
goal
from
image
examining
in constant
segment
(resp.
z on the
equal
path
its
as a quadratic
cre-
On !lLn/zj
used
shortest
by
+ (1 – i!)di+l.
minimized
Di
area
the straight-line
represented
a point
the
VOV1 to
Ci and
case,
to
polygon.
the
two
path
left-anchor(C’~)
analysis
to that
begin
whether
z on a single
shortest
for
S(Q(z,
of case
minimum
of
we
decide
time.
ate
To
Given
path
the
=
C’i
(resp.
in a simple
in-
the exception
of the straight
point
to the first vertex on
in computing
or
inscribed
qo)
similar
[10]
points
same
right-anchor(.lli)
a type
as the
so as to subdivide
All
segment
(resp.
as well
chains
the
and
[7], it is also possible
chains
Similarly,
oriented
right-anchors
in
time
is denoted
)).
on S(Q(Z,
(resp.
tour
can be computed
are in-
In linear
vertex
we use
other
THEOREM
That
Below
R.
convex
))
of qoqn+ 1 share
vertex
separator
paths,
qi~lzj
paths
of the
lie
to
shortest
versa
qo qn+l
These
tangents
The
at a point
odd).
segments
S(Q(qo,
of these
/eft-am%or(Di)
Souvaine
from
paths
vice
denoted
edges
anchor(Ci
every
(1 – ‘)qLn/2J
shortest
respectively.
and qin/zJ qinlzj+l.
Ci
responding
has exactly
process,
chains.
qoqn+l
the path.
the
t < 1 corresponds
O s
if n is even
T must
at
its endpoints
ezi+l
pieces
reversing
tqLn/2J+l
1 – 1 correspondence
optimum
until
T in P starting
for
lie
of the line
9in/ZJ (resP. qi@~ +1) with
segment
from the particular
the
e2~ in P. Translate
. .
wlthm
Aq~-i+l
q~qn-i
on e2i and
By
rotate
(corresponding
endpoints
R
union
as R.
it
+1)),
the
segment
and
of
points
(1 – t)qO
tqLm/zJ +(1
its
tour
triangles
and
Aqiqn-iqi+l
on e2i-1
by the
all
translate
that
portion
length
two + (1 – t)vl
tqn+l
so
keeper’s
of
to Av2ivz~+lv2i+z)
T formed
candidate
then
within
)
corresponding
same
boundaries
and
points
rotate
at the
the
first,
corresponding
and
at
to a candidate
9Ln/2J+l,
common
edges
of Q corresponds
to
qi@j
convex
extend
ner
triangle
subdivisions
the
qin/2J +1 and
qLrz/2J qLra/2J+l.
from
of
to createa
triangulated
Q,
and
is
number
region
polygon
to qin/z]
be
that
the shortest
endpoints
or is
CZYZOWICZ
462
perpendicular
lines
to the bisector
cent aining
Ci
In the second
c E G’i to
least
one
of starting
points,
rotate
aligned
C;.
with
at an endpoint
constant
adopted
in this
thus
and
D~ is
sides
new
of
figure
of segments
easily
shorter
and
(wj,
Z).
the
sum
of the
obtaining
in
in
case, first
the
second
left-
constraint
chosen
on the
other
by either
case,
andjor
use the same
of the
perform
same
possibly
that
both
and
Keep
left-anchors
that
the
using
for
as in the first
segment
procedure
twice,
right-anchors
procedure
side.
the
C
csse with
lie on one side
both
right-anchors
optimum
path
lies
determined
technique.
There
are
at
most
a linear
Di), an optimum
(Ci,
constant
corresponds
number
path
time,
and
the
to the
optimum
can
of
be found
shortest
of
all
for
these
Proof.
each
Let
Optimum
Keeper’s
Tour
in
a
tains
Non-Convex
Given
a simple,
Vo, vi,...
but
, Vn-l
given
the edge oriented
vi.
non-convex,
Certainly,
in clockwise
in clockwise
all
reflex
identified
in linear
time.
rj
among
the
all
O ~ j
original
< h -1.
of points,
wo,
closed
point
Wi may
on two
P
The
of the
more
the polygon.
cross
Proof.
there
that
such
a vertex
is an optimum
the
that
reflex
to
order
two
edges
length
of the
edges
Since
thus
all
a
lie
Since
may
be
edges
of
path
at a single
subset
points
in its
keeper’s
tour
than
We
thus
follows
from
a to b keeps
demonstrated
tour
Let
wj
of v,
and
immediately
orientation.
3.1 that
the
Q.
wj
of Q to P that
the
path
Since
It
part
a, . . .. b in
any
tour
that
leaves
of
Thus
Wi, . . . . Wj
the
path
rz)
that
to denote
passes from
reflex
intermediary
denoted
right
of Q to its right.
in
we
radial
between
to the
the
rz, two adj scent
the
then
line
seg-
Q, we have
a reflex
vertex
be suboptimal.
We use Q(rl,
keeper’s
case,
than
that
must
1, the
in P, con-
a and
by replacing
wi, . . ..a. b, . . . . wj
path
ab is shorter
unvisited
before
interior
tour
Assume,
a suitable
of contact
of lemma
the
the
lying
Q in a clockwise
the corollary
we can shorten
the
points
traversing
v.
segment
the
and
once.
optimum
of intersection
immediately
b when
line
P by
left
visits
v unvisited.
through
is not
the
an
v and
points
tour
vertex
b, respectively.
the
be di-
P at least
exists
the
If this
a and
can
and the interior
keeper’s
passing
to
of a
equiva-
and an open
tour
aquarium
that
exist
tour
two
traversal.
it so by rotating
there
into
interior
a reflex
line
denote
the part
rl
of the optimum
to (the
vertices
edges
as l(P(rl,
LEMMA
first
occurrence
of a simple
aquarium.
of P between
rl
and
r2 be
r2)).
keeper’s
contradiction
constrained
Q of a simple
Proof
that
aquar-
I(P(rl,
3.3.
Q(rl,
every
and S(P(rl,
of
z)
z).
keeper’s
there
leaves
interior.
are encountered
touches
of an optimum
sake
is
(wi,
the shortest
P.
cross.
the
(wj+l,
of the
of L containing
make
respectively
Let
is shorter
and
Wi+l
wj))
edges
x)
optimum
of generality,
we label
ment
within
from
Wj+l))
lies to the lefi
that
Q that
1 immediately
of)
edges
Suppose
Thus,
by
bounded
of a simple
Q and
after
of the
plane
and
Every
angle.
Q from
one of the
interiors
b)) to denote
for
a sequence
vertex
two
of
rj,
k < n.
touch
P and
lies inside
for
=
of a polygonal
in their
polygon
two
can be
Vj of P and
in
aquarium
Assume
that
at
lies
to be optimum,
The
of a clockwise
v in its
which
index
wo is a minimum.
a single
No
Vij
at least
once
a to b that
originates
the
i.e.
then
a simple
of a simple
represent
is possible
intersect
3.1.
which
of P,
a and b, we use S(P(a,
from
LEMMA
tour
than
it
vertices
let ei represents
is to determine
ei and
with
P with
?’0, rq, . . . . ?%- 1
ij
such
_ 1
edges,
edges
Given
interior
path
Let
We say that
if the
point.
vertices
goal
coincide
distinct
is non-convex,
visited
order
Wo, WI, . . . . wn_~,
path
order,
vertices
. . . . wn
WI,
Wi lies on each
polygon
and
S(P(Wi,
optimal
the
vertex
loss
can easily
Aquarium
Every
L be a horizontal
connected
3
Q’ is a keeper’s
Therefore,
partitions
the exterior
tour
without
tour.
that
edges
tour
and
w3+l))
Wi is visible
we assumed
3.1.
3.2.
reflex
keeper’s
paths
that
of (wi+l,
a closed
LEMMA
segment
Observe
S(P(Wi+I,
exterior,
so that
every
keeper’s
of the lengths
lengths
to the right
lie
wj+l)
crossing
wj )) is convex
wj+l.
a tour
classes,
rected
the
z), because
Similarly
unbounded
the
from
Q’.
sum
aquarium
lence
D.
(wj,
the
a contradiction.
simple
the
and
wj+l)
from
shortened
and
S(P(toi, Wj)) and S’(P(Wi+l,
wj,
the
Wi+l)
Eliminate
S(P(zo~,
A(wi,
than
(wi,
~.
tour
path
Wj is visible
have
joined
can be computed
edges
point
a new
The
the triangle
translate
are on opposite
or a pair
tour.
To determine
and then
C to D
to obtain
to
is independent
d).
the
the
(wi, wi+l)
and (wj,
replace
them with
at
closest
with
at
COROLLARY
added
in
D
a point
through
Di and D until
by
D
from
segment
third
appropriate
pairs
C and
from
vertex
to
on c (resp.
of c, and
P
time.
In the
Then
C
if necessary,
path
a straight
path
Q passes
be the
defined
and
shortest
is either
(D)
point
triangle
Ci
shortest
from
reflect,
the
The
C
path
(ending)
two
and
Let
The
ium
by the
Di.
d G D~ inside
vertex.
d).
formed
cross
case, every
a point
c (resp.
these
and
of the angle
ET AL.
r2)
to
edge
lie
r2)
of
in
of
I(P(rl,
a region
a
simple
r2))
in
bounded
aquarium
order,
P,
and
by I(P(rl,
is
r2))
rz)).
Assume
are
not
that
visited
one
by
or
Q(rl,
more
r2).
edges
Let
of
e denote
THE
AQUARIUM
KEEPER’S
463
PROBLEM
a)
Figure
4: a) A simple
structure
Ql
one such
of Q(rl,
edge.
rz)
Let
and
succeeding
traversed
there
is at
edge
rl
then
Thus
rl
sees rz,
that
Q(rl,
every
rz)
other
hand
rl
path
Q(rl,
P,
must
lie
the
intermediary
the
path
even
between
replaced
path
may
The
we can
compute
Q
the
assume
rz))
visit
of
times.
even
in order
step,
then,
the
shortest
of
the
by
must
r2))
rz)
reflex
vertices
rl
and
rz.
b) The
thus
ackle
path
tree
and
the
from
be
in
Tz)).
naively
reflex
The
of the reflex
joining
and
can
be non-simple
to the
polygon
geodesic
of
P
but
right
adjacent
be computed
as
each
vertices
region
of the
time
the
algorithm
follows
if
the
is dominated
be triangulated
The
global
Aquarium
tour
the
shortest
rj+l
that
of
of finite
oriented
edges
path
joining
because
S
consists
in
rj
of
[17]
by
problem
of
reduces
and
[17].
In
of
The
can
one
a set
S of
y of
anywhere
in
vertices
of
of
simplifies
considerably
P.
fixed
the
vertex
of the
rj+ 1 without
rj
new
optimum
of finding
to
boundary
crossing
P
Since
Chazelle’s
computing
an
opt imalit
lie
with
reflex
[17],
to h subproblems
edge
convezof
is presented
hull
a set
time
follows.
every
set
triangulating
result
from
relative-
the
algorithm
convex
in O(n)
now
touches
space
P.
only
[1], the
P
polygon
algorithm
and
algorithm
S
the
of
to
relative
s simple
can
chain
O(n)
in
then
huil)
respect
and
However,
R is in fact
convex
with
computing
points
mUSt
boundaries
every
paths
be repeated,
by R lies
case.
R consisting
shortest
may
worst
it.
log n)
the
rz)
edge
the problem
n
P,
the entire
and
R
may
(also
O(n
rz))
can
S(P(TI,
time.
vertices
P.
Q(rl,
-v
of l’,
is to compute
Tot
of
the
in the
in P has size U(n)
defined
hull
for
and
S(P(rl,
making
~2)
that
Q(rl, m) crosses
visit
boundary
by Q(r,,
the
rz))
crossings
The polygon
boundary
Proof.
to lie within
parts
3.4.
and
O (n2)
rz),
On
We claim
time
vertices
O(n)
the
take
of P
bounding
As shown
cross
must
in
to rz visiting
S(P(rl,
If
The
(rl,
reflex
area
rz)).
in order.
see r2.
e
Q,
shortest
edge
polygon.
rz)).
on the
bounded
visit
rl
could
LEMMA
T’2)
must
is the
from
numbered
of S(P(rl,
Q(rl,
by consecutive
vertices
of Q(rl,
that
edges
it
and
interior
it is constrained
and
Thus
defined
vertex
of contact
edge of I(P(rl,
path
then
for all of the subproblems.
and
Since
e in
I(P(rl,
r2))
P1
preceding
part
bounded
edges
odd
points
of the
necessarily
since
region
traversal.
every
S(P(rl,
a region
by parts
first
of
not
number
rz))
b.
must
r2),
in
lie in the region
the
edge in a convex
path
shorter.
I(p(rl,
a to
point
rz visits
is,
S(P(rl,
the
left
is the shortest
2, this
the
an
and
intermediary
in sect.
b denote
e in a clockwise
one
between
shaded
T2) immediately
from
least
a contradiction.
If
a and
with
P1.
all of e lies to the
when
path
from
l(P(T1,
the
Thus
aquarium
obtained
another
convex
a second
464
CZYZOWICZ
convex
chain
us say that
(where
Pj
we
P~
described
only
Av/vi+l
may
rri+z,
but
need
a greater
on the
chain.
As
all
polygon
Pj
polygonal
linear
For
composed
structure
THEOREM
simple
of two
Q? can
in the size
Vi+l
[10]
to
~ ))
visible
Qj
(see fig.
can
4).
h subproblerns
convex
be formed
on
chains,
in time
The
optimum
keeper’s
in
a
a
in O(n)
time.
of sect.
vertex
If P should
2 apply.
which
problem
image
of
lie
reflex
vertex
can
be done
sum
of the
be computed
rj
linear
of all
to
in d(n)
Qj
one
tour.
path
the
equal
Q~.
in the
the
procedures
at least
shortest
structure
in time
sizes
the
(possibly
polygonal
the
shortest
vertex
~j+l
then
P has
on the
to finding
each
triangulated
can
Otherwise,
must
reduces
succeeding
be convex,
size
reflex
Thus
the
from
the
image
of
to
~j !)
in
For
each
j,
of Q~.
is O(n).
Thus,
a Simple
Polygon
the
the
this
But
the
the
tour
time.
References
[1] B.
Chazelle,
Time.”
May
[2] W.
“Triangulating
Tech.
Report
CS-TR-264-90,
in Linear
Princeton
Univ.,
1990.
Chin,
S. Ntafos,
“Optimum
Zoo-Keeper
Routes.”
Congresses Numerantium,
1989.
S. Ntafos,
“Optimum
Watchman
Routes.”
[3] W. Chin,
Prac. of the ACM Syrnp. on Comput.
Geometry,
Yorktown
[4]
ford
[5]
Heights,
R. Courant,
Univ.
J. Focke,
Problem
1986, 24-33.
H. Robbins,
Press,
“A
Finite
of In polygons
Optimization,
[6] A. Fournier,
What
is Mathematics?,
Ox-
1941, 346-352.
Descent
with
Method
Minimal
for
STEINER’s
Circumference.”
17 (1986), pp. 355-366.
D. Y. Montuno,
“Triangulating
Polygons
and Equivalent
Problems,”
tions on Gravhics 3 (1984). 153-74.
ACM
Simple
Z’ransac-
J. Steiner.”
Optimization,
16
Problems
of the A CM Symp.
“On
Using
Solving
Shortest
on Comput.
Geo-
Paths.”
Geometry,
Berke-
H.
Rademacher
[13]
Schwarz,
lungen.
2.Bd.,
[14]
J. Steiner,
[15]
R.
and
Gesammelte
Berlin
Maxima
und angewande
Inst.
‘An
Optimal
[17] —,
Relative
Werke.
Convex
Processing
of EURASIP-86,
Abhand-
2. Bd. Berlin
und
Univ.,
III:
Thwries
2, The
for
Sets, Bel-
Computing
the
in a Polygon.”
and Applications,
Hague,
reine
Feb. 1990.
of a Set of Points
Part
Steiners
fir
36-77.
on Illumh.sting
Algorithm
1882, 45.
zu
Journal
96 (1884),
of McGill
Hull
Zusi.tze
und Minima.”
Workshop
Research
of
1957.
Mathematische
Mathematik,
G. T. Toussaint,
Enjoyment
Press,
1890, 344-345.
“Bemerkungen
Au fsiizen iiber
The
University
Gesammelte
Sturm,
lairs
O. Toeplitz,
Princeton
H. A.
Signal
Proofi
“Zur analytischen
und aleines geometrischen
Opti-
[12]
space
tour
can be computed
and Optimal
1985, p. 60.
I. Niven, Maxima
and Minima
Without
Calculus, The
Mathematical
Association
of America,
1981, p. 163.
[16]
P of n vertices
Inc.,
[11]
the
and
Mathematics
D. L. Souvaine,
Optimization
Mathematics,
from
and
Simple
209-233.
ley, 1990, 350-359.
convex
polygon
E. A. Melissaratos,
Prac.
focus
Triangulated
Books,
H. Rudolph,
Behandlung
von
M. Sharir,
R.
Visibility
and
pp. 833-848.
metric
~f~- 1)~~~ n to
are
the
nested
(1985),
of Pj.
3.1.
polygon
polygons
of the
need
individual
of creating
each
to
American
mierungsproblems
we may
~, v(i+I)~Od
from
each
in this
process
a point
Inside
2 (1987),
A. Tromba,
Scientific
[9] R. Klotzler,
gorithmischen
by ei_ 1, ei and
lj~~d
path
along
triangulate
3.5.
we
Form,
triangle
vi-1
Problems
Algorithmic,
[8] S. Hildebrandt,
polygonal
the
from
fact,
S(P(v(i_
vertices
vi
in
Path
Polygons.”
of
and translate
between
determined
from
in the
LEMMA
In
i
each
of the polygon,
shortest
every
P~
procedure
the
lies
diagonal
interior
chain
the
for
Wi, diagonals
be added
of tour
the
same
reflect
still
ei+l
solve
create
segment
polygons
represents
V(i+ ~)mOd ~
To
restriction.
convex
the
on
since
starshaped
inward
which
2.
lie in the
a polygon
can
we no longer
The
[7] L. Guibas,
J. Hershberger,
D. Leven,
Tarjan, “Linear
Time
Algorithms
for
Let
vertices.
Shortest
we
essentially
section
a point
no longer
two
define
polygon).
convex,
using
triangles.
same
chains
a spiml
nearly
in
and
the
convex
Q1, however,
ei
the
is
subproblems
structure
on
joins
two
is in fact
Since
these
which
these
ET AL.
September
Proc.
1986.