d ia g ra m
O n th e g eo d e s i c V o r o n o i
of
p o in t
s i te s
in
a
sim pl e pol ygon
B oris Aronov
Co u r a n t
In s t i t u t e of Ma t he ma t i ca l
S c i e n c es
N e w $ork Un i v e rs i ty
25 1 Me r c e r S t re e t N e w $o rk N$ 1 001 2
,
,
AB STRACT
Gi ven a si m p l e po l ygon with n sides in th e p l ane
an d a set o f k point
sites in its interior or o n the
bound ar y co m p u te th e Voronoi di agr am of the set
o f sites
usin g the intern al
geo desic dist ance
inside the polygon as the m etric We d e s cribe an
o (( u + k ) l o g (n + k ) l o gn ) ti m e al gorith m for solving
this
problem
an d
sketch
f aster
a
al gorithm for the cas e when
the set of sites includes al l reflex vertices o f the
polygon in question
$
$
,
$
$
$
.
.
.
O n th e g e o d e s ic Vo r o n o i
d i a gr a m
p o i nt s i t e s
of
in
a
s im p l e p ol y g o n
.
Boris Ar o n o v r
’
Co u ra n t
1
.
In tro d u cti o n
Ins ti tu t e of Ma the ma t i c a l S c i e n c e s
.
Rec ently there h as been a significant upsurge o f results co ncern ing
g e o metry inside a si mple po l ygon incl uding an i mproved tri ang u l ation algo
efficient al gori th ms fo r cal cul ating the geo des ic ce nter [PSR] an d
ri thm [TV]
the geo desic di am eter [82 ] o f a polygon a nu mb er o f new sho rtest p ath and
visi b ili ty rel ated al gorithm s th at require line ar amount of ti me beyond a tri
an d al gorithms fo r link dist ance pro b lems [ S]
an gu l a tion [ GHLST]
[Le a]
So m e o f the work concentr ates on intern al dist ance an alogs of fund am ent al
problem s for point sets in the Euclidean pl an e For ex ample Touss ai nt [T]
developed an al gorithm for co mputing the rel ative co nvex hull of a set o f
points whi ch is the shortest cycle co nt ai ning all given points an d cont ai n e d in
a given si mple polygon
an d Suri [8 3 ] d es cribed an al l geodesic furthest
neighbors al gori thm for si mple polygons Ou r pres ent work extends another
fund am ent al problem in co mput ation al geo m etry n am ely th at of cal cul ating
the ( nearest neigh b or ) Voronoi di agr am of a set o f point sites ( see for ex am
ple
to the cas e of points in a si mple polygon under the interior geo
desic m etri c (Figure
,
,
-
,
-
.
,
,
.
,
$
$
,
-
-
,
.
,
-
,
In sections th at follow we present an algorithm w hich cal cul at e s the geo
desic Voronoi di agr am of a set S of k points
in a si mple polygon
with n vertices using the length of the shortest intern al p ath betw een two
points in the polygon as the measure of dist ance Our al gorithm runs in ti m e
whi ch is not far fro m being opti mal an d w hi ch is an
order of m agni tude f as ter th an a previous al gorithm o f [ AA] ( see below )
Our work can be considered as an other gener ali z ation o f Voronoi di agr am s
using a new kin d o f m etric Recent gener al i z ations o f si mi l ar n ature include
[A] [AB ] [CD] [F ] [HML [LS] [LW] [L2 ] [ O S$ 1] [ 0 33 0 ]
On e not able property o f the geodesic m etri c is th at cal cul ati on o f the
sho rtest p ath betwee n two points is not an e asy oper ati on an d without
preproce ssin g must t ake 9 02 ) ti m e in the worst cas e Also a single geo
d e sic bis e ctor betwee n a p air of sit e s can be the concaten ation o f @ (n ) dis
tinct s tr aight an d hyperbolic arcs whi ch m ay at first sight sugges t th at the
worst ca se over al l co mplexity of the geodesic Voronoi di agr am is
Fort u n ately thi s is not the ca se an d we show th at the tot al size of the
,
.
.
.
,
,
,
,
,
,
,
,
,
$
,
.
,
$
,
-
-
,
1
'
PhD
W o rk
Schol
ars
on
hip
,
th is p p
a
at
er
New $
o
w as
rk
p fo
er
rm e d
t
U n iv e rs i y
.
w
h il
e
th e a u
tho h l d
r
e
AT & T B e
ll
La
bo to
ra
ri e s
F ig
i c V o o o i di
l
u re
o
G e de s
r
n
.
a g ra m o f
th
t
re e s i e s
.
is onl y @ ( n + k ) where onl y 0 ( k ) of the di agr am verti ces ar e true
b
co n
a
a
m
ca
n
e
degre
ver
ices
Roughly
spe
king
the
di
gr
be
describ
d
a
t
e
3
y
(
)
s tru cti n g th e set of segm ents th at extend the ed ges of the short est p ath tree
[ GHLST] fro m e ach site s ES but only within the Voronoi cell of s ; the au g
me nted di agr am can then be reg ar ded as the uni on o f al l thes e sets of exten
sion segments trun ca ted to their corr esponding Voronoi ce lls and sep ar ated
fro m e ach other by Voronoi ed ges
Th ere has been so m e work done previ ously in des ign of an algori thm for
constructing the geodesic Voronoi di agr am of a set o f point sites inside a s im
ple polygon Th e best r e s u lt o f whi ch we are currently aw are is th e al go ri thm
o f [ AA]
whi ch runs in ti me 0 ( n k + n l o g l o g n + k l o g k ) The portion o f
their al gori thm th at necessit ates qu adr atic ti m e is the ex pli cit co nstructi on of
the short es t p ath tree [ GHLS T] for the full polygon fro m e a ch site As thes e
trees ar e easily see n to h ave a tot al o f n k distinct ed ges in the si mple cas e o f
a co nvex polygon
th e ( worst ca se ) qu adr atic bound follows We h ave cir
cu mv ented thi s proble m by never building the fu l l shortest p ath tree of the
polygon for e ach o f the sites inste ad the tree ( in f act a different b u t closely
rel ated extension seg m ent stru ct u re) fro m e ach site is construct ed only for
the p art of the polygon th at can conce iv ably lie in the Voronoi ce ll o f the
site
di agr am
$
$
,
.
,
,
,
.
v
.
.
,
.
-
.
,
,
$
,
$
.
Ou r al gori thm
uses a f amili ar divide an d co nquer str ategy for obt aini ng
the geodesic Voronoi di agr am However pecul i ar iti es of the geodesic metric
necessit ate a so mewh at non st an d ard i mplement ation of this str ategy The
-
.
-
,
-
.
G o d s ic Vo o oi d i
e
e
r
n
a g ra
m
2
B A o o
.
r
n
v
rel atively st an d ard m erge step for ex am p l e m ust b e prece ded b y a step th at
ex tends a recursively co mputed di agr am o f the su b set o f sites inside h al f the
polygon to the full po l ygon The ex tension ph ase is the le as t co nvention al
p art o f o ur al gorithm an d requires sw e eping the tri angul ation of the po l ygon
by a polygon al scan line
Re ma rk : If the set S of sites co nt ains al l reflex vertice s o f P a si mpler
al go rithm can be used to co m pute the geodes ic Voronoi di agr am in ti m e
see discussion in Section 5
0 ( ( n + k ) l o g (n
Possible appli ca tions of our al gorith m include the closest p air problem
the neares t post office pro bl em and al l ne ar est neigh b ors pro b lem i n the co n
text o f a polygon al universe such as a ( polygon al ) isl and with no interior
l ak es o r a polygon al f actory floor It is likely th at other pl an ar point loca tion
and proximi ty problems whose solutions employ Euclide an Voronoi di agr am
co ul d be gener al ized to ques tions about intern al m etric in a si mple polygon
F o r ex am ple one mi ght wish to
an d co u ld t ake a dv ant age o f o u r al gorith m
investig ate how the an al ogues of a De l aun ay tri angul ation an d mini mum
1
sp anni ng tree beh ave in the co ntext of the geodesic m etric
The p aper is org anized as fo ll o w s : Section 2 describes the gener al strue
tur e of o ur algo rithm Section 3 ex amines so m e o f the m ore i m port an t
geo metri c properties o f the geodesic Voronoi di agr am Section 4 gives a
m ore det ai led d e scription o f o u r al gorithm Section 5 outlines a si mplifi e d
version o f the algorithm whi ch produces the di agr am for a set o f sites th at
includes all reflex vertice s Section 6 mentions so m e rel at e d open problems
,
,
.
-
.
,
.
,
-
-
,
.
.
,
$
$
$
$
.
w
,
.
.
,
.
2
.
Ge n e r al
Our
ou
tl i n e
of
.
the a l g o rit h m
.
ppro ach to solving the problem is the following : Le t P be a si mple
polygon wi th n sides an d S b e a set of k point sites inside P Th e g e o de s i c
Vo ro no i di a g ra m of S in P denoted as Va rp (S ) is the p ar titioni ng of P into k
ce lls Vp ( s 1 )
Vp (S k ) such th at
a
.
,
,
,
,
k}
{x
where dp is the geod e sic dist ance inside P So m e initi al proces sing includ e s
tri an gul ation of P [TV] and loca ting e ach site o f S in the r e sulting tri angles
[ST] Then a b al ance d d e co mposition [Ch ] o f the tri an gul ation tree is co m
The algori thm pro ceeds by cutting P by a chord
pu ted ( see al so
into tw o p arts P 1 an d P 2 o f roughl y the s am e n u mber o f edges an d recur
s i v e l y co m puting the di agr am s V0 r
where 5 1 (S 2 ) is the
p l (S l ) an d Vo rp 2 ( S 2 )
subset o f S co nt ai ned in P 1 (P 2 respectively ) At thi s point the Voronoi
m ust be ex tended to the full polygon P Th e tw o extended
di agr am
VP (S I )
3
.
.
,
,
.
,
.
l
t o of D l
O n e g e n e ra i z a i
ti g a te d b y Le e a n d Li n
1
n
[ LL]
e au n a y
,
f
t
l tio
ri a n g u a
w h o d e i n e d i t di re
n
( b r i e fl y
ct l y
,
di s
c
us se d
a n d n o t a s th e
i n Se
ct i o
du a
o
l
n
5 ) w as i n v e s
f the V
o o oi
r
n
di
agram .
G e od e s ic Vo o noi d i
r
a g ra m
3
B A o o
.
r
n
v
di agr ams thereb y o b t ained ar e nex t m erged i n a m ann er s i mi l ar to th e usu al
Th e
S h am os Hoey scan [ SH] ( or its m odific ation due to Ki rk p atrick
m ai n novel fe atu re o f our al gorith m is the di agr am ex tension step whi ch
itself t akes o ( (h + k ) l o g (n + k )) ti me Th e extension is done b y prop ag ating
s ay V0 rp ( S l ) fro m the di agon al cu t e sep ar ating P I an d P 2 thr ough e ach of
1
the tri angl es in P 2 in a preorder tr avers al o f the tri angul ati on ( sub ) tree o f
P 2 As th e construction proceeds deeper into P 2 tw o typ e s of events t ak e
pl ace : ( i ) a Voronoi ce ll o f V0 rp (S l ) dies out or ( ii ) a vert ex of P 2 is
re ached as si gned to th e V oronoi ce ll of Va r, (S 1 ) co nt ai ning it an d is inco r
p o rated into the short est p ath tree withi n th at ce l l A w ave front is m ai n
tai n ed an d upd ated at e ach event At ( i ) a Voronoi cell dis appe ar s fro m the
w ave front an d new pred icti ons are m ade as for the ti m es o f dis appe ar an ce of
the two newly adj acent ce lls At ( ii ) the w ave front is sp l i t in two o n e for
e ach o f th e u nvi sited tri an gul ation subtrees rooted at the current tri angle
Progres s is achieved b y repe ated l y selecting the nex t clos est event to t ak e
pl ace an d mak i n g appropri ate ch an ges in the w avefront A m ore det ai led
descri p tion can be found in the rem ai nder of the p aper A sketch of a si mpli
fi ed al gorithm for the cas e when S includes all reflex vertices of P is given in
S ection 5
-
,
.
,
,
,
.
$
$
,
»
,
,
$
$
.
.
.
,
.
.
.
.
g eo d e s i c Vo r o n o i d i a gr a m
On e is h ard press ed to efficiently co m pute the geo desic V oronoi di agr am
b efore h aving understood so m e o f its properties an d h avin g determin ed its
worst cas e co mplexi ty I n thi s s ection the defi nition of the di agr am is given
followed by a list o f its b asic properti es and a proof th at it h as li ne ar co m
3
Pr o p e r t i es
.
of
th e
.
-
.
pl exi ty
,
,
.
De fi niti o ns
st ru ctur al pr o p e rt i e s
Let P be a co mp act region in the pl ane whose b ound ary
n go n wi th set of vertices V = { v 1
v }
Le t S = { s l
set o f k point sites
a nd
.
-
t
,
$
,
n
.
is a si mple
s k } C P be a
6P
,
,
$
.
De fi nit i o n
t
For
p a th
two points v an d w o f P let h ( v w ) b e the s hor
w entirely cont ai ned in P Th e l as t vertex ( o r v if
o n h ( v w ) is referred to as th e a n c ho r of w
with
(
any
,
betw een v an d
there is none) before w
respect to v)
No t e
As P is closed an d bounded
exists an d is indeed unique for every p air ( v
h ( v w ) is pi ece wise line ar with bre ak points
s ee for ex ample
[LP]
te s
,
.
,
.
$
$
,
,
length
,
a
,
,
.
,
.
.
,
De fin iti o n
si mple polygon h ( v w )
w ) o f points o f P Moreover
at ve rtices of 6 P
F o r a proof
by
For v
,
w EP ,
let
the
d is t a n c e
from v
be the
to w ,
of
No t e
by definition d ( v w ) is a metric on P Moreo ver
d ( v w ) is continuous in both v an d w ( with respect to Euclide an met ric on
Essenti al ly
,
,
.
,
,
G e o de s ic Vo o n oi d i
r
a g ra m
4
B A o o
.
r
n
v
P)
.
De fi n i t i o n
The Vo ro n o i
c e ll
of
a
site
s
ES
{ x EP I V t ES
( )
P
Tri vi al ly L
J VP (s )
QS
Vp
No t e
is
S
s
.
,
obj e ctive o f o u r al gorithm is to o b t ai n the deco mposition of P by
V oronoi cells O b serve th at applying k nown techniques for pl an ar point loca
tion to such a deco mposition al lows efficient co mput ation o f { s I x EVP (s )
for an arbitr ary query point x 6 P
Th e s ho r t es t p a th t re e o f P fro m a site s
is the
De fi n i t i o n
uni on o f the shortest p aths fro m s to vertices of P It is known th at T (P s ) is
indee d a pl an ar tree root e d at s with str aight line edg e s and Vu {s } as the set
of vertice s ( s ee for ex am ple
Assu mi ng s is not a vertex T (P s ) has
ex actly n ed ges an d n + 1 vertice s and e ach of its e dges is either a side o r an
interior chord o f P ( Figure
De fi n i t i o n
Th e s ho r t es t p a t h p a r t i t i o n o f P fro m s is the p artiti on o f
P into m ax im al sets e ach cont aining points x al l h aving the s am e anchor with
respect to s ( thus al l the p aths h (s x ) p as s through the s ame sequence o f ver
ti ce s of P )
Le t e be an ed ge o f T (P s ) an d let its endpoint further fro m s be v Le t
r b e the open h al f line co lline ar with e an d ex tending fro m v in the dir e ction
o f incre as ing dist an ce fro m s
If so m e initi al section o f r is cont ai ned in the
Th e
.
.
,
.
,
-
,
,
,
,
‘
,
.
.
,
-
.
Sho t t p th t
r es
G o de s ic Vo o n oi d i
e
r
a g ra m
a
F ig
re e
T( P
,
s
)
of
u re
P
2
.
f om
r
5
s su
p
e ri m
po
se d o n
P
.
B A o o
.
r
n
v
P we will refer to the max i m al such initi al section as the ext e ns io n
O therwise e h as no
s e g me n t of e ( or th e extension seg m ent e ma n a t i ng fro m v)
ex tens ion seg ment
D e fi n i t i o n
Le t the col l ection o f ex tension seg ments o f edges of
We will often abuse the not ati on an d wri te
T (P s ) b e denoted b y
Note th a t E (P s )
E (P s ) fo r the u n i o n o f the ex tension seg ments of
—
co nt ains at m ost n segm ents one per vertex of P
N o te
It w as shown in [ GHLS T] th at the p art ition form ed by split
i s a t ri a ng u l a t i o n in which
ting P along the seg ments o f
points o f each tri angle sh are an an chor Such tri an gles however are not
neces s ari ly maxima l sets with this property If f act it is e as y to s ee th at split
ting P al ong the seg ments o f E (P s ) al one produces m axi m al sets an d thus
co rresponds to the short est p ath p artition o f P In p arti cu l ar it is a po l yg o
n al p ar tition o f P ( s ee Figure
De fin i t i o n
If v w EP an d v =1 w we will fo l lowing [PSR] define
the d i re c t i o n fro m v to w
as the u nit vect or at v direct ed al ong the
first segment of h
interi or
of
,
.
.
,
,
,
.
,
,
.
,
,
.
1
,
,
:
,
,
,
,
ii ( v , w )
continuous function o f v an d w for al l p ai rs of
(v w ) excep t when v w o r when there is a seg ment of E (P w ) eman ating
fro m v Moreover it is e as ily verified th at d ( v w ) is a continuously differen
ti able fun cti on o f w with the ex ce ption of points noted above an d in f act
is the gr ad i ent o f d ( v w ) with respect to w
No t e
is
a
,
,
.
,
,
,
.
,
F ig
G o d s ic Vo o noi d i
e
e
r
a g ra m
u re
3
.
Sho t t p th p t i t i o n o f P f o m
r es
,
a
ar
6
r
s
.
B A o o
.
r
n
v
The bi s e c t o r of
b (s t )
{x EP
De fin i t i o n
,
two ( distinct ) sites
I d (s x )
s an d t
is
,
the set of points equidist ant fro m s and t
A set Q C P is s t a r s hap e d a ro u nd x 6 P ( with respect to
D e fin i t i o n
the geodesic metric) if Vy 6 Q : h (x y ) C Q
If s and t are tw o distinct sites let
Defin i t i o n
H (s t)
{ x 6 P [ h (s x ) fl o (s t ) Q }
In other words H (s t) is the max ima l su b set o f P b (s r) st ar sh ap e d
ar ound s Note th at tri vi ally s EH
Lemm a
is a p ar tition of P Moreover
fo r a point x EP
d (x s )
x EH (s t)
an d
x Eb (s t) e: d (x s )
d (x s )
x
It is sufficient to demonstr ate the equiv al ence o f o u r origin al
P ro o f
defini tion o f H ( s r) and the al tern ative ch ar acteri z ation given above
We must show th at
Ass um e x
so h (s x ) fl b (s r) Q
Co ntinuously p aram etrize h (s x ) by
such
d
= x Note th at
= s an d
th at
is a continuous
function o f 1 o n
It st ar ts off at
i
.
e
.
.
,
-
.
,
,
.
,
,
,
-
,
,
,
.
.
,
,
,
,
,
,
,
.
.
,
.
,
,
,
.
It never reaches zero
b
it
Thus
d (s , t)
0
.
th at would indica te an intersection of h (s x ) an d
is
positive
throughout
In
p ar ticu l ar
as desired
O i e d (x t)
a
Conversely
suppose x EH
so
there
exists
point
d (y s ) an d d ( s y )
But
Thus d (x s )
d (x y )
y Eh (x s ) f) b
then by the tri an gle inequ ality
d (x J ) S d (Jay )
d ( w)
d (m )
d (L s )
d (x s )
: 1
as des ired which co mpletes the proof
Co ro l l ar y
By a sim ple continuity argu m ent b ( s t) actu al ly
s ep ara te s H ( s t ) fro m H ( t s ) in the sense th at any p ath fro m a po i nt of the
form er to a po i nt o f the l atter m ust intersect
In p articu l ar H (s t)
co uld as wel l h ave b ee n defined as the p a th c onn e c t e d ( r ather th an st ar
sh aped aro u nd s ) co mponent of P b (s i ) cont aining s ( Figure
H ( s r) U b ( s i ) is st ar sh aped around s
Le mm a
Pro o f Suppose h ( s x )
Ub
i e h (s x )
Le t
95 Q
Then
y Eh ( s x )
d (s x )
d (s y )
d (y x )
d (t y )
d (y x )
so at
as desired D
,
as
,
.
.
,
.
,
.
,
,
,
,
,
,
,
,
a
.
,
,
,
,
,
,
,
-
,
-
,
.
,
.
.
,
.
,
.
,
,
,
,
,
,
,
.
C o ro l l a ry
nected subset o f P
G e ode s ic Vo on o i di
r
H (s
t
,
)
U b ( s , t)
is
a
p ath connected
-
an d
si mply
co n
.
a g ra m
7
B A o o
.
r
n
v
an d
S
C o r ol l a ry
VP (s )
n
t6S
ol i d l i
n e s re
H(t
p
,
s
)
re s e n
Fig
to g th
e
t
u re
4
.
th th ho t t p th p t i t i o
h d t n io n gm t
wi
er
d as
e
-
e s
r es
ex e
s
a
ar
en s
se
,
,
.
.
an d
By definition of the Voronoi cel l
In p articul ar lemm a
(H (s t ) U
n
lemm a
implies th at
{s }
is
st
a
r
sh
a
pe
a
round
s
d
( )
F o r an al tern ate proof of the f act th at in gener al the Voronoi ce ll C o f
a ( point ) site s is alw ays st ar sh aped around s in the m etri c us ed to defin e C
see for ex ample [P rS]
De fi niti o n
P an d S are i n g e n era l p os i t i o n if no vertex o f P is
equi dist ant fro m tw o distinct sites i e n o bisector b (s i ) co nt ai ns a vertex
of P
Le mm a
Under the assu mption o f gener al position b (s t ) is a
s mooth curve co nn ecting tw o points o n 6 P an d h aving no other points in
co mm on with 6 P It is the co nca ten ation o f 0 ( n ) str ai ght an d hyperbolic
arcs
an d th e po i nts along b ( s t ) where adj acent p airs o f th es e ar cs m eet are
ex actly the in tersec tions o f b ( s t ) with seg ments o f E (P s ) o r E
More
over the t angent to b (s t) at point x bisects the angle betw ee n u (x s ) and
V,
»
s
-
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,
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,
,
,
,
,
,
,
into polygon al regions by rem oving fro m it
E (P s )
Let Q be the closure of one p f the resulting regions By
co nstruction there are tw o a n cho r vertice s Ean d t such th at for an y point x in
—x
—
d
Q
| an
Ii x I where points are
Pr o o f
Parti tion
.
P
,
.
,
,
,
G e o d s ic Vo o oi d i
e
r
n
a g ra m
8
B A o o
.
r
n
v
co n
T
denotes
uclide
us
the
an norm
t
h
e
E
reg arded as vectors an d
h
I
diti on
is eq u iv al ent in Q to
It x l
I3
Th
e
a
o
f
a
is
in
gener
portion
hyperbol
hyper
a
l
a
i
b
s
i
n
herefore
T
Q
( )
bol a in ques tion has i and t as its two foci an d by elem ent ary an al ytical
geo me try its t an gent bisects the an gle betw een the directions to 3 an d to t
The following degener ate cas e s are possible :
i
hyperbol
a de ener a tes into a straight line ( in f act the perpendicul ar
e
T
h
()
g
d ( i t ) an d 3 $ 1 Note th at thi s is the o n ly
bisector o f } and t) if
possi bi lity for a po rtion o f b ( s t) to b e a str ai ght line seg m ent fo r P and
In p articul ar the str ai ght line portion o f b ( s, t)
S in gener al position
t
ca nn ot be co lline ar with either of the anchors as this woul d i mply 3
a use
t
m
and
s
ii
ere
no
point
equidist
ant
fro
either
bec
are
T
h
( )
s
a ( or str ai ght
si
ply
bec
use
the
hyperbol
m
a
o
r
d
d
I
line) in ques tion does not intersect Q
= t an d d
would
belong
to
of
a ll
iii
If
§
Q
( )
points
Th en however the vertex § = t would be equidist ant fro m s an d t
viol ati ng the gener al position assu mption
( iv) Th e i ntersecti on o f the hyperbol a with Q consists of discrete points
Thi s situ ation occurs onl y when the hyperbol a in question intersects the
bound ary o f Q b u t not its interior
The above f acts i mply th at b ( s i ) ca n be reg arded as the uni on o f ( rel a
tively clos ed ) s tr ai ght an d hyperbolic ar cs an d discrete points We now
proceed to prove a seri es of properti es of b (s i ) whi ch will allow us to co n
cl u de th at it is indeed a s mooth curve co nn ecting two points of BP an d co m
pl e tel y co nt ain ed in the interi or o f P otherwise
( 1) Every p ath co nnected co mponent of b (s t) m eets GP for suppose a co m
—
ponent C does not then there is a closed curve p ar ound C th at do e s not
in tersect ei ther b (s t) o r aP Th us p is co mpletely co nt ained in s ay
—
H
m
a
s
t
H
t
i plying th t b ( ) U ( s ) is not conn ected a co ntr adi ction
(2 ) There are no non trivi al cycles in b ( s t ) under the as su mption of gener al
positi on th at is there is no closed si mple curve co mpletely co nt ai ned in
b
Suppose there ex ists such a curve and consider a mini m al cycle C
i e
o n e whose interior cont ai ns ex actly o n e co nn ec ted co m ponent o f
s ay
P
But then C is a loop in H (s t) U b (s t) th at can
not be contr acted to a point contr adicting si mple connec tedn ess o f
H ( s t) U b (s i ) ( coroll ary
( 3) Thus we h ave proven th at b (s i ) is a fores t with e ach tree att ached to GP
in at l eas t o ne point W e will proce ed to prove th at all le aves o f b ( s t)
ar e points on 6 P Suppose it is not the cas e
Let x be a leaf o f b (s r) not
on 6 P Then a su ffici ently s mall open disk D ce ntered at i t does not meet
6 P an d i n tersects b ( s t) in a si mple curve co nn ecting i t to a point on 6 D
Therefore D
b (s t) is co mpletely co nt ained in either H (s r) o r H
.
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,
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,
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*
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$
,
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G e od s i c Vo ron o i dia g
e
ra m
,
9
B A o o
.
r
n
v
As
Without loss of gener al ity as s u me th at it is co nt ain e d i n
is st ar sh aped around t h (x t) must be co nt ai ned in
H
DD
b (s t ) fl D
h (x i ) N D C
Ub
an d
H (t s )
which is im possible in a non degener ate conf igu r ati on as it i mplies the
exi stence o f a str ai ght l ine portion o f b ( s t ) lyin g o n the lin e p as sing
through one o f the an chor points whi ch w as shown in ( iii ) to co n tr adict
the gener al position as su mption
f
o r suppose there were
here
no
isol
ated points o f b ( s t) o n GP
ar
4
T
e
( )
such a point x Th en there is a neigh b orhood D o f x such th at D {x } is
co mpletely cont ai ned in s ay H (s t ) which i mplies th at H ( t s ) U b (s t) is
n o t co n n e cted co ntr adicting co roll ary
a
i
as
a
No
non
t
v
subseg
ent
of
6
P
ca
n
be
cont
ned
in
th
t
5
r
i
i
a
l
m
( )
woul d i mply th at both anchors for a point o n this segm ent would h ave to
lie on the str ai ght line co nt aining the seg ment which according to ( iii )
co ntr adicts the gener al position assu mption
( 6 ) Th e facts proven so far i mply th at b ( s t) is a forest al l o f whose leaves
li e on BP an d whi ch h as no non tri vi al i nters e cti ons with (BF an d no iso
l ated points o n it H owever co roll ary
shows th at rem ov al of b (s t)
fro m P creates exa c tly two co nnected co mponents Th erefore the forest
i n question co nsists o f a single p ath th at does not m ee t 8 P o ther th an at
either o f its endpo i nts thus proving th at b (s t ) is a p ath co nn ecting tw o
po i nts o n 6 P and h aving no other intersecti ons with 6 P
Fin al ly the directions u (x s ) an d ii (x t ) v ary continuously al ong b (s t )
an d our as su m ption on gener al position )
so the an gle
( beca us e o f note
bisecti on property o f the individu al hyperbolic arcs al so holds at their joining
points i mplyi ng th at the individu al arcs are glued together s m oothl y D
Noti ce th at b (s t) ca nnot be t an gent to an extension seg ment of E
fo r it would i mply ( by the previous lemm a) th at the dir e cti ons fro m th e point
o f t an gency to s and to t coincide so th at 3
t contr adicting th e gener al posi
ti on as sum pti ons ( cf p art ( iii ) of the proof above) In p art icul ar e ach seg
m ent o f E (P s ) intersects b ( s r) in at m ost one point
No t e
Th e f act th at the bisector of two sites is a s m ooth curve
whose t an gent bisec ts the an gle between the directi ons to the two sites is
qui te gener al No t onl y does it ( obviously) hold for point sites in the
Euclide an metric but it w as also shown in [L8 ] to hold for arbitr ary convex
sites in the Euclidean pl ane
C o r o l l a ry
Under the as su mption o f gener al position
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$
$
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{x 6 P I
an d
VP (5 )
c l o s u re
—
V t ES {s }
( in te ri o r ( Vp
where both the interior an d the closure
are t ak en rel ative to P
In p art icul ar Vp (S ) is a pl an ar region bounded by
sections of 6 P an d o f bisectors o f the form b
for t ah s
.
,
.
o
G e de s
ic Vo o oi di
r
n
a g ra m
10
B A o o
.
r
n
v
In t e rio r ( VP
C o r o l l a ry
H ( s r) is
C o r o l l a ry
is st ar sh aped with res pect to
-
s
,
as
e ach
.
,
non degener ate configur ation
F) V; ( t )
i n t e ri o r ( VP
s aé t
$ for
C H ( s t ) an d V, ( t ) C H ( t s ) U b
i n teri o r ( VP
The previous co roll ary which excludes non trivi al overl ap
No t e
am ong Voronoi ce lls is the m ai n re ason for restricting our di s cu ssion to co n
fi gu r ati ons i n gener al position If there are two sites equidist ant fro m a sin
gle vertex o f P b (s t) m ay no longer be a cu rve and Voronoi ce lls ( cf
defini ti on
may overl ap n on trivi al ly ( see Figure
Le mm a
co nsists o f at m ost o ne
Le t v 6 P then
point ( under the as su mption of gener al position)
Pr o o f Suppose there ar e two such points x an d y W ithout loss o f gen
e ral i ty assu m e th a t y is the point further fro m s
Since both H ( s t) and
H ( s t) U b (s t ) are st ar sh aped with respect to s ( definiti on
an d lemm a
an d h ( s y ) p asses through x the s u bp ath p of h ( s y ) fro m x to y must
be co mpletely co nt ai n e d in
By choosing if necess ar y a di fferent x
we m ay assum e p is a s tr ai ght line seg ment an d the two an chor points st ay
const ant over p In p articul ar b (s t) cont ai ns a str aight line po rtion p
w hich force s p to be a p art of the perpendicul ar bisector o f the tw o anchor
points H owever p is a portion o f h
an d hence is co nt ai ned in the line
p as sing through one of the an chor points whi ch in turn force s the two an chor
points to co inci de However an chor points cann ot coincide o n the bis ector
under the gener al position as su mption by the proof of lemm a
Co ntrad
In
a
-
,
,
»
,
,
-
,
,
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,
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,
,
-
,
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,
-
,
,
,
,
,
,
,
,
-
-
.
.
,
.
,
,
.
i cti o n
,
,
D
F ig
co f i g t i o n n
P o i t i n th
h d d gi o
A
n s
G e ode s ic Vo r o n o i d ia g
ra
m
n
e s
u ra
a
e
re
u re
ot
5
.
n a re e u
q
11
l po i t i o n
idi t n t f o m
i n g e n e ra
s
s a
.
r
s an d
t
.
B A o o
.
r
n
v
Co r o l l a ry
If e is a seg m ent o f
m ost o ne point
not i ng t h at
Pr o o f
Appl y lemm a
b eing the endpo i nt of e fu rt h est fro m s
then
b (s
,
t
) fl e co nsists of
at
.
.
by
co nstru cti on
e
C h
y
.
Co m pl ex i ty o f the g e o d e s i c Vo r o n o i d i a gr a m
Since by corol l ary
V oronoi ce ll interiors do not overl ap the V oro
noi di agr am Varp (S ) o f a set o f sites S in P is co mpletely des crib ed by the
the b ound ary o f a ce l l b eing the set
u ni on o f th e V oronoi cell b ound aries
difference o f the cell an d its rel ative interior in P Note th at by coroll ary
a poin t on the b ound ary of a Voronoi ce ll nece ss aril y lies in at l eas t two ce lls
Co nsider the pl an ar
an d thus is closes t to ( at le as t ) two sites si mu l t an eous l y
m ap o n P induced by the union of Voronoi ce ll b ound ari es with bound ari es
t ak en rel ative to P A point in three or more Voronoi cells or on BP and in
M ax i m al si mple curves
tw o o r more Voronoi ce lls is cal l ed a Vo ro n o i v e rtex
cont ained i n the u nion o f the cell b ound aries an d not co nt ai ning V oronoi ver
tices are Vo ro no i e dg e s A maxi mal su bcu rv e of a Voronoi edge th at is a co n
ti gu o us portion o f a single hyper b ol a o r a str aight line is referred to as an
a rc
Endpoints o f arcs which are not V oronoi vert ices are ca ll ed bre a kp o i n ts
Note th at a break point neces s ari l y h as degree two as degr ee o n e is ex cl u ded
b y coroll ary
an d m ore th an thr ee arcs sh aring an endpoint woul d consti
tute a Voronoi vertex First let us consider the gr aph defined by Voronoi
edges an d verti ces an d ignore the individu al arcs an d bre akpoints By co n
structi on it is a pl an ar gr aph with vertice s of degree three an d above By
Eu ler s fo rmul a the co mplexity of t his map is line ar in the num ber o f its
f aces However for s 6 5 in t e ri or ( Vp
is st ar sh aped around s an d in p ar
ti cul ar is co nnected and cont ains s so Voronoi cells co rrespond o ne o ne to
f ace s o f the pl an ar map Th erefore there are onl y k f aces in this m ap proving
th at there ar e 0 (k ) Voronoi vertices an d ed ges
Recall th at 6 P has n edges thus adding n vert ices an d edges to th e co m
pl ex ity o f the above pl an ar map as soon as we explode into co nstituent
segments th e porti ons of 6 P which pl ay the role of the ex teri or boun d ary It
rem ains to es ti mate the tot al nu mb er of bre ak points on Voronoi ed ges We
cl ai m th at the num b er o f such points is 0 ( n ) thus bounding the tot al co m
pl exi ty o f the m ap in ques tion by 0 ( n + k ) an d proving th at the si z e o f the
desired Voronoi di agr am is line ar in the si ze of the input Aug ment the
pl an ar m ap i nduced by the V oronoi di agr am b y the foll ow i ng ed ges : For e ach
s ES
add to the m ap seg ments o f E (P s ) trunca ted to i n te ri o r V;
Se g
(
m ents o f E (P s ) th at do not intersect the interi or of VP ( s ) are si mply dis
car ded Note th at the o nly seg ments th at remai n are those em an ating fro m
vertice s o f P in the interi or of VP (s ) ( Figure
A Voronoi ed ge th at is a portion of bis ector b (s t) in the ori gin al m ap has
breakpoints p re c is e ly at the points of its intersection either with seg ments of
.
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,
$
$
.
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,
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,
o
G e de s
ic Vo o oi di
r
n
a g ra m
12
B A o o
.
r
n
v
F ig
ic V o o o i di
u re
Au gm e n
t
ed
o
ge de s
r
n
6
.
ag ra m
o f th
t
re e s i e s
o f Fi g
u re
1
.
co ming fro m
o r with seg m ents o f E (P t ) co mi ng fro m H ( t s )
Th es e ar e ex actly the portions o f seg m ents of E (P s ) an d
( lemm a
E (P t) th at are included in the newly cre ated structure On e sho u ld note th at
the intersection of the seg ment ( if an y)
by st ar sh ape dn es s o f i nte ri or ( VP
is i ndeed a
o f E (P s ) em an atin g fro m v 6 i n te ri o r ( VP ( s )) with i n teri o r ( VP
single segment eman ating fro m v Since such seg ments cann ot in ters ect the
bo u nd ary o f a Voronoi cell at more th an one point by coroll ary
an d
m or e over e ach seg ment is in o ne one correspondence with the vert ex it
em an ates fro m we conclude th at the tot al nu mber o f bre akpoints is bounded
above by the su m of the nu mber o f vertices occurring in v ari ous Voronoi cell
interi ors By coroll ary
the interiors of Voronoi ce lls are disjoint so the
num ber o f break points is bound ed above by n thus ensuring th at the sizes o f
the ori gin al m ap as well as the new ( aug mented ) m ap are linear in n + k Note
th at the aug m ented structure (we will refer to it as the a ug m e n t e d g e o de s i c
Voro n o i di a g ra m Va r; ( S ) ) is indee d a pl an ar m ap beca use ex tension seg m ents
ar e tru nca ted to their respect ive Voronoi cells and thus are ( openly ) disjoint
fro m the o ri gin al m ap an d am ong themselves Ou r al gorithm will actu ally
co mpute the l atter map Thus we obt ain :
Theo re m
Th e co mplexity of Varp (S ) is o (n + k ) where n is the
number of sides o f P and IS I k Moreover the aug ment e d di agr am Var; (S )
al so h as co mplexity 0 ( n + k )
E (P , s )
,
,
,
.
,
-
,
.
-
,
,
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,
,
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,
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,
.
G ode s ic Vo o n oi d i
e
r
a g ra m
13
B A o o
.
r
n
v
g o r i thm
We b egin thi s section with an overview o f o u r al gorithm for co mputing
the ( augm ented ) geodes ic Voronoi di agr am of a set of sites S in a si mple
polygon P Th e steps th at require more det ailed tre at m ent ar e di scuss e d at
greater length in the following subsections
Alg o ri th m A
Inp u t : A s impl e pol ygon P o f n s i des an d a set S o f k point sites in the inte
rior o r o n th e b ound ary o f P S an d P are as su m ed to b e in gener al posit i on
Ou tp u t : Th e p l an ar m ap ind u ced o n P b y Va r; (S ) ( cf S ection
( 1) Pr e pr o ce ssin g
a
m
ri
gul
te
P
in
n
o
log
n
ti
m
e
or
by
si
pler
T
V
i
a
0
l
T
a
n
(
g
)
[ ] (
()
0 (n l o g n ) algo ri th m of
a
l
b
r
ii
eter
ne
e
ch
site
in
wh
ch
tri
ng
e
it
lie
const
ucting
the
o
r
a
i
s
D
m
i
f
y
( )
pl an ar m ap induce d b y the tri angu l ation preprocessing it for point
loca tion queries [ST] an d performing a query for each site in tot al
0 ( (n
k ) log n ) ti m e
In 0 ( n l o gn ) ti m e co mpute a b al an ced deco mposition o f the tri an gu l a
( iii )
tion tree [ Ch] which will allow the m ai n body o f our al go ri thm to
r e cursively cut the polygon into two p arts e ach h avi ng at le as t one
qu art er o f the nu mber o f sid e s in const ant ti m e per cut ( s ee al so
4
.
F i n a l l y , th e
al
.
.
.
.
.
.
.
,
,
.
(2 ) Ma i n p a r t o f th e a l g o r i t h m (recursive)
If S consists o f onl y o n e site s Vo rp (S ) is a single ce ll
(i)
Var; (S ) is then co mputed by utili zing the line ar ti m e sho rtes t p ath
p artition co nstr uction of [ GHLST] ; otherwise
If P is a tri angle co mpute the Euclide an Voronoi di agr am of S in
( ii )
0 (k log k ) tim e tru nca te it to P in line ar ti m e reco rding the intersec
,
-
,
,
,
tions o f the Voronoi edges with (W an d return the r esult ; otherwise
Split P into two roughly equ al polygons PL an d PR by a cut and
divi de S i nto S, and SR such th at SL C PL and SR C PR Th is ca n be
perform ed in co ns t an t ti m e as a b al anced deco mpositi on of P has
bee n preco mputed and the sites are al re ady as soci ated with the tri an
gles in which they lie thereby m ak ing the l atter Oper ation i m plicit so
th at no processing is required to p artition S
Recursively co mpute Vo r h ( SL ) an d Va r} , (SR )
,
( iii )
.
_
,
,
.
( iv )
( v)
( vi )
.
Extend V0 271), (SR )
0 ( (n
k ) log ( n
to
Vo r; (SR )
an d
Vo rE
5
1
(
L
) to
Vo r} ( S L )
in ti m e
des cribed in Section
Co mpute Vo r; (S ) by merging Va r; (SL) and Va r; (SR ) in ti me
0 (n
k ) Th e det ai ls of this step can be found in Secti on
as
.
G e o de s i c Vo o n o i dia g
r
ra m
14
B A o o
.
r
n
v
F ig
u re
7. S
o l id l i
n e s re
p
re s e n
t G to th
ge
,
wi
er
th
d as
h
ed
—F
.
Pr o o f Suppose it cont ains a non trivi al cy cle R ecal l t h at we ident i fy a
gr aph wi th its emb e dding in P Therefore we may spe ak of a min ima l cycle
—
C o ne whi ch together with its interi or does not properly cont ain another
cycle As G ca nn ot h ave vertice s of degr e e one in the interi or o f P ( co roll ary
the interior o f C cont ains no point of G an d thus is a single conn ect e d
co mponent o f P G i e a ce ll of the Voronoi di agr am ( strictly speaking
the rel ative interior o f a Voronoi cell ) Thus there is a Voronoi cell whose
a V oronoi ce ll m us t
However b y co roll ar y
i nterior li es entirely in P 2
co nt ain i ts owner site in its interi or contr adicting the assu mption S C P I Cl
C o r o ll a r y
F O P ; is a forest with le aves o n 6 P 2
Proo f Sin ce in terms of its embedding in P F dif fers fro m G only by
th e extension segments truncated to their respective Voronoi ce lls it is suffi
cient to show th at extension seg ments can never close a cycle in G It is
2
cle arly enough to est ablish th at no co llecti on of extens ion seg ments co n
tai ned in o n e ce ll can together wi th th eir incident endpoints form a co n
n ected co mponent th at m eets G twice
Since the extension seg ments are
defined as porti ons o f shortest p aths fro m the owner v o f the ce ll an y two
such seg ments are neces s ari ly disjoint for otherwise either their co mm on
endpoint would h ave two shortest p aths to v o r the seg ments would overl ap
whi ch woul d i mply existence of more th an o ne ex tension seg ment em an ati n g
fro m a single an chor It is e as ily verifi ed th at the o nly other w ay in whi ch
clos ur es o f two extens ion seg ments can inters ect is for one o f them to end at
the anchor o f the other extension seg ment ( refer to Figure
I n thi s situ a tion it is conveni ent to co nsider the ex tens ion seg m ent endpoints
coinci di ng o n 6 P to be distinct verti ces ( le aves ) of F Note th at this situ ation
ca n only occu r in
degener ate configur ations when ei ther thr ee verti ce s of
P or two verti ces of P and a site of S are co lline ar Th us we may cons ider
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we wi
ll
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on
s e gm e n
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ru n
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B A o o
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r
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v
F ig
t io
( b)
u re
D e g e n e ra
In
bo th c
t
e e x e ns
as e s
.
t
n s e gm e n s o f
i s t h e d e g e n e ra
e
,
8
t
e s e gm e n
t
.
ex tension seg m ents effe ctivel y disjoint fro m each other N o w an ex tension
segm ent eman ates fro m a vertex v of P and ter min ates at the nex t intersection
with 6 P o r with a Voronoi edge Under the as su mption of gener al position v
ca nnot appear o n the bis ector of two sites an d thus o n an y Voronoi edge
All
Th erefore an ex tension seg m ent intersec ts G at m ost once as desired
le aves o f F 0 P 2 lie o n 6 P 2 since the s am e is true o f leav e s o f G 0 P 2 an d
extension segm ents cre ate le aves o n 8 P 2 onl y D
We now procee d to an alyze the structure of the forest F 0 P 2 in order to
justify the sweeping algorithm for co mputing it which is given at the end o f
this subs ection
Le mm a
Co nsider a tri angle A whi ch w as entered through its side f
in pre order tr avers al o f the tri an gul ation tr e e of P 2 fro m ( the t ri an gle
inci dent to ) e Le t x 6 i n t e ri o r ( A) b e a point on an edge g o f F 0 P 2 Th en
the line p t an gent to g at x crosses f
Pr o o f Th e lemm a is trivi al ly tru e for an extension seg ment g as g fo l
lows the shortest p ath fro m a site an d thus must enter A thr ough f there can
be no anchors in the interior of A
Co nsi der an arc g o f b (s t) ( see Figure
Since th e shortest p aths fro m x to s and t enter A through f an d there are no
anchors in the interior of A ii (x s ) an d ii (x t ) must be direct ed in such a w ay
as for the h alf lin e s cont ai ning them to cut f But by lemm a
p bisects the
angle between i2(x s ) and
so it must also m eet f D
L e mm a
Let A an d f be as above Then al l edges of G 0 P 2 th at
mee t f inters e ct it tr an svers al ly
Pr o o f Let f an d f
b e the two rem aining sides o f A Co nsider an arc
Note th at x is an interior point
g o f b ( s t ) non tr ansvers al ly m eeting f at x
o f f since
o n the as su mption o f gener al position b ( s t ) cannot cont ain ver
tice s o f P Si nce the shortes t p aths fro m x to s an d to t must not co m e fro m
f f o r the interi or of A both ii (x s ) and ii (x t) must lie in the clos e d
h alf pl an e bounded by f an d not cont ai ni ng A The onl y possible w ay o f
k e eping thi s co nsistent with lemm a
th at st at e s th at the t angent to g at x
i e f must bisect the angle betw een ii (x s ) an d ii (x t) is for the l atter tw o
vectors to co inci de and point al ong f But coinci dence o f ii (x s ) an d ii (x t) on
a bisector is i mpossible in gener al position by the proof of lemm a
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G e ode s i c Vo o no i d ia g
r
ra m
,
,
,
17
B A o o
.
r
n
v
F ig
u re
9
.
Contr adicti on
Th e o b vious ex tens ion o f the previous lemma to all edges of
No t e
F 0 P 2 does not nece ss ari ly hold Th e re as on is th at though ordi n ari ly ex ten
sion segm ents meet f tr ansvers ally in so me degener ate confi gur ations ( cf
Figur e 8 ) an extension seg ment mi ght co incide with i nte ri o r (f ) We will
here after refer to thes e as de g e n e ra te ext ens io n s e g m ents
We now proceed to define a n atur al orient ation on F 0 P 2 fo rmal iz
ing the i ntuitive notion aw ay fro m e th at tr an sforms F 0 P 2 into a root
direc ted for e st Ori ent edges of F 0 P 2 as fo ll ows : Ag ai n l e t A b e a tri angle
o f P 2 entered through f Gi ven an edge g a point x o n g in th e interi or o f A
o r o f f the t an gent p to g at x inters ects f tr ansvers ally Co nsider the dir ce
ti on o f tr avers al o f g for which the velocity v ector at x points alo ng p a w ay
fro m f n p ( Fi gu re 10 a) In cas e x Ef direct the velocity vector i nw a rd A ( Fig
ure l 0b)
A degener ate ex tension seg ment th at overl aps f is oriented so as to e ma na t e
fro m its an chor point We cl ai m th at thi s orient ation is well defi ned an d
indeed m akes F 0 P 2 a root directed forest To show thi s it is sufficient to
prove th at the above m ethod indeed assigns a u n iqu e orient ati on to e ach edge
independent o f the point x sel ected to define the orient ation an d th at no ver
tex h as o u t degree gre ater th an o ne
Le mm a
Th e orient ation of an edge g is well defined i e there is a
tr avers al o f g co nsistent with the above require ments on the direction of the
velocity vector at every point of g
P r oo f It is cle ar th at th e orient ation of a degener ate extension seg ment
ca nnot be in consistent as it is defined glob al ly r ather th an loca lly Consider
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G ode s ic Vo o oi d i
e
r
n
a g ra m
18
B A o o
.
r
n
v
( a)
F ig
u re
10
f to
D e ini i
.
(b)
n of
o
t tio
ri e n a
n
.
other type o f edge g Le t us st art b y restricting o u r attention to a tri an gl e
A entered through its side f The orient ati on is defined in terms o f where the
t angent to g i ntersects f As the t angent v ari es co ntinuously al ong g ( lemm a
it continues to m e et f on the
and continues to inters ect f ( lemm a
so the orient ation o f g is consistent on a
s a me s i de of the point o f t angency
single co nnected porti on of g in the interior o f A Th e ambiguous ca s e o f the
Thus it is suffici ent to
t an gent co i nci ding with f is excluded by lemm a
show th at the orient ation st ays consistent when crossing chords sep ar ating tri
an gles
And indee d it is fo r suppose g cross es the ed ge f sep ar ating A fro m
A i e A is enter e d though f By lemm a
g intersects f tr an svers al ly
By s m oo thn es s o f b isectors ( lemm a
the t an gent to g at g n f still
m eets f Acco rding to o ur definition of orient ation g will be oriented out
w ard fro m A in the interi or of A ne ar f it will be oriented fro m A to A at
its inters ection point with f an d it will be oriented inw ar d A in the interi or
o f A ne ar f
thus m aking the orient ation consistent everywhere al ong g D
an y
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So
the orient ation of an edge is well defined Le t us proceed to cl as sify
the vertices of the forest F 0 P 2 an d to prove th at the ori ent ation indeed
yields a root directed forest We observe th at the le av e s of the dir e cted
for e st i e vertice s with zero i h degree are ex actly the points of intersection
o f F an d e together with vertice s o f 6 P 2 th at h ave extension seg m ents of F
eman ating fro m them Th ese vertices o f F 0 P 2 indeed h ave degree o n e an d
their inci dent edges ar e directed aw ay fro m them An intern al vertex o f
F 0 P2
neither a root nor a leaf ) in the interi or o f A h as o u t degree o f
ex actly o n e as shown by the following lemm a The cas e of an intern al ver
tex l andin g o n a chord o f the tri angul ation will be h an dled sep arately ( cf
note
below )
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Le mm a
degree one
Every
vertex
v
of F
HP;
in the interior of A
h as
out
.
G e o de s i c Vo o no i d ia g
r
ra
m
l9
B A o o
.
r
n
v
Suppose there are tw o or m ore edges em an ating fro m v N ote
th at thes e edges are Vo ro n o i e dg e s an d thus v is necess arily a Voronoi vertex
for an ex tension seg ment ca nnot e ma na t e fro m a point in the interior of P
So suppose there are two or m ore Voronoi ed ges em an ating fro m v Choose
As the ed ges are adj acent there m ust b e a sing l e
a p ai r o f a dja ce n t edges
V oronoi ce ll VP ( t) ( loca l ly ) wedged b etw een t h em an d the s ai d e dg es m ust
for so m e sites s an d r Le t t b e the
b e portions o f b isectors b (s t ) an d b
Note th at by definition o f orient ation the
anchor of v with respect to t
t angents 11 to b (s i ) an d 12 to b ( t r ) at v intersect f when ex tended b ack
through v By co nstruction VP ( t ) is ( local l y ) wed ged b etw ee n the h al f lines
of 11 and 12 ex tending a w ay fro m f ( s ee Fi gure 1 1 a)
In p articu l ar by st ar s hapednes s o f VP ( t ) t must lie in the wed ge W co ntrad
icting the fact th at al l an chors are co nt ai n e d in the polygon over whi ch the
di agr am h as already b een constru cted and t hus m ust lie o n f or on the side o f
—
v
u st li e
opposite
A
d
b
e
visi
b
l
e
fro
m
through
i
n p ar t i cu l ar
m
n
f
f
f
a
between v an d t
Suppose there is no edge eman ating fro m v If v is a bre akp oint
a
m ee ting point o f a bisector an d an extension seg ment) two arcs of the bi s ec
tor join s m oothly at v thus gu ar anteeing the presence of at le as t one outgo
ing edge Hence v must be a Voronoi vertex As G 0 P 2 h as no le aves in the
interi or o f A there must b e tw o o r more Voronoi edges incident to v Co n
sider the t angents to thes e edges at v and choose the two edges whose
t angents intersect f closest to either o f its tw o endpoints ( cf Fi gure 11b)
Si mi l arly to the above ar gument the l ar ger w edge at v bounded by the tw o
edg e s must ( local ly) belong to a single cell VP ( t ) fo r so me t ES and the s ai d
ed ges mus t be porti ons of b (s t ) an d
resp ectively for so m e s r ES
Le t [ 1 ( 12 ) be the li ne t angent to b (s t ) ( b
respectively ) at v Le t W be
th e s mal ler wedge at v bounded by the h alf lin es of 11 an d 12 th at do not
in tersect f By st ar shapedness of V; ( t) ii (v t) must point into W ag ai n con
tradi ctin g the f act th at the p ath h ( v t) exits A thr ough f an d i ts portion
Pr o o f
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b( s
F ig
o
Ge de s
ic Vo o oi d i
r
n
a g ra
m
u re
20
11
,
t) I
.
B A o o
.
r
n
v
cont ained in f is a str ai ght line D
Th e verti ces o f F 0 P 2 o n 6 P 2 th at are neither an chor points nor lie on e
ar e roots as shown by the following l errun a
Lemm a
Let A be a tri an gl e entered thr ough f an d let its remai ning
No vertex v o f F H P ; in the interi or of f or f h as
tw o si des be f an d f
an outgoing ed ge g th at em an ates fro m v into the interior o f A
Pr o o f Th e ed ge g in qu estion must intersect ( say) f tr ansvers ally at v
an d so co i n c i des with f by
otherwise
it
is
degener
ate ex tension seg m ent
a
(
an d s m oothness o f arcs
lemm a
the
b y lemm a
an d
an d note
t an gent to g at v must intersect f m aking g an i n coming edge at v by defini
tion o f orient ati on on g CI
No t e
There can ex ist intern al vertices of F 0 P 2 th a t lie on a chord
f sep ar ating two tri angles A an d A o f P 2 with A entered t hrough f an d A
shows th at there can be no edge le avi ng
through f I n thi s cas e lemm a
v into A an d re as oni ng si mil ar to lemm a
shows th at there is e xa c t ly o ne
ed ge eman ating fro m v into A thereby showing th at v h as ind e ed out degree
one an d thus is a v al id intern al vertex o f the root di rect e d forest
C o r ol l ar y
F O P ; is a root direct ed forest with le aves o n e and at
anchor points roots at non an chor vertice s of F on 6 P 2
e
al l intern al non
root nodes being Voronoi vertices and bre ak points and h aving i n degree o f at
least two
D e fi n i t i o n
Le t A be a tri ang l e entered through f an d let f an d f
be its tw o remai ning sides Define the s w e ep c urve s to be the broken line
co nsis ting o f a seg ment p aral lel to f at dist an ce 1 fro m it exten ding fro m f to
f together with tw o portions o f f an d f fro m the endpoints o f f to the
endpo i nts of the s ai d seg ment Le t r be the dist ance fro m f to the vertex o f A
incident to f and f
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C oro l l ary
In this not ation s (fo r O$ r $ r) sep ar ates the verti ces
o f F 0 P 2 into tw o sets in such a w ay th at al l
directed
edges
betw
een
two
(
)
vertice s o f di fferent sets eman ate fro m below s
fro m verti ce s lying o n
the side o f s th at cont ai ns f) and termi n ate above
Moreover if T is such
th at s p as ses through a vertex v of F 0 P 2 in the interior o f A o r f there is
p re c i s e ly o n e edge incident fro m v an d the s aid e dge extends fro m v into the
are a a bove s
( Figures 12 3 &b)
0 P 2 ) 0 A) can be construct ed b y swee ping A
Le mm a
F GA
with s
Pr o o f All th a t is nee ded to build F in A is to system atically locate a
p air o f chi ldren with a co mm on p ar ent in F n A an d repl ace the children b y
the p arent thereby adv ancing the sweep curve through the fores t one step
Th is i n turn reduces to mai nt ai ni ng the inters ection of F with current swee p
curve and repeatedly locating the p ai r o f adj ace nt edg e s whose inters ection
D
point li es o n s with le as t
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r
,
T
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T
G e o de s ic Voro no i
dia g ra m
21
B A o o
.
r
n
v
( b)
( a)
Fig
u re
12
.
Be
h
av i
o
t
r at a v e r e x o
fF
a
.
We are now ready to descri b e the al gorithm for ex tending Vo r; 1 (S to
for the current polygon P m m is
P 2 Th roughout the algo rith m F fl P m
grows fro m P I to P More pr ecisely maint ain the
maint ai ned w hile P m
Vo rEW m ( S
pl an ar m ap indu ce d by
together with F fl P w m ,
For e ach bounded face R of t his m ap (we will refer to it as a re g ion) m ain
t ain the following i nf orm ation : the site s to whose Voronoi ce ll R belongs
the an chor Efo r the shortes t p ath fro m s to an arb itr ary po i nt of R an d the
v al ue
Th erefore given an arb itr ar y f ace R o f the pl an ar m ap an d a
po i nt x in R the site s closest to x an d the dist an ce d (s x ) can be determi ned
in cons t ant tim e
We will assum e th at the pl an ar m ap for P I is av ai l able ini ti al ly Since
possessing Vor; 1 (S ) i mpli e s knowled ge of the sorted order in which ed ges of
F m ee t the edge e sep ar ating P 2 fro m P 1 we can build in line ar ti me a
se arch tree co nt aini ng this inform ation
More precisely the tr ee will
represent those regions o f the induced pl an ar m ap on P 1 th at are adj acent to
e
Nam ely it will co nt ai n for e ach such region R the own e r o f R ( the site to
whose Voronoi cell R belongs ) the a n cho r of R ( the l as t vert ex on the shor
test p ath fro m the owner to an arbitr ary point in R) an d th e w e ig ht o f th e
an chor ( the dist ance betw ee n the owner an d the an chor )
Th e tree m us t sup
po rt ins ert de l e te an d sp l i t oper ations in ti me log ar i thm ic in its m axim u m size
( see for ex ample red bl ack tree s o f
Th e al gori thm for constructing F 0 P 2 fro m Va r } l (S ) ( an d thus F 0 P 1 )
and thereb y o b t aining F will proceed b y perfor mi ng so m e ini ti al setup at the
ed ge e an d then tr aversing the tri angul ation tree o f P 2 st arting fro m the tri
an gle adj ace nt to e :
Al g o rith m B
Inp u t: A tri angul ated polygon P sep ar ated by a chord e o f the t ri an gu l ati on
in to su bpo l ygo ns P 1 an d P 2 and the pl an ar map induced on P I by Var} 1 (S )
cu
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G o de s ic Vo o n oi d i
e
r
a g ra m
22
.
B A o o
.
r
n
v
m ust corres pond to the d i s appe ar ance o f th e reg i on b etw een th em
fro m T an d the repl ace ment of the two b isector ar cs ( or a b is ect or arc
At thi s point the
an d an ex tension seg m ent ) by a new bisect or arc
region is deleted from T If p does n o t represent a v alid intersection
it is simply discarded The two newly cre ated p ai rs of curves adj acent
o n s T h ave their intersections co m puted an d located in the tri an gul a
.
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,
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tion If they l and in A they are added to Q ; otherwise they ar e
Th e process repe ats until the
added to the appropri ate buckets
queue is empty
m
iv
hen
the
construction
inside
A
is
co
pleted
the
two
tr
ees
W
( )
co rrespondi ng to f an d f ar e produce d b y splitti ng the cu rrent ver
sion o f T ( co nce ptu al ly ) cutting s at v Note th at s coincides with
a
t
a
e
this
point
so
split
ing
corre
ly
cre
t
s
the
two
ini
ti
l
T
c
t
a
t
U
f f
trees repres enting s o fo r the tw o unvisited tri an gles adj acent to A In
cas e f ( respectively f ) is actu ally an edge o f P the co rresponding
search tree is co nverted to adj ace ncy inform ation fo r F al ong it
We proceed to an al yze the ti me co mplexity o f Algo rithm B Steps l ( i )
an d 1 ( ii ) cle arly t ak e line ar ti m e As there is only a line ar nu mber of ini ti al
can did ate p airs of adj ace nt curves on s step 1 ( iii ) requir e s
ti me
One can e asily see th at the ti m e co mplex ity of p art 2 is 0 ( l o gn ) per
ca n did ate i nters ection for point loca tion
for queue m ai nte
n ance oper ati ons ( as no can did ate is ever in tw o queues ) 0 ( l o g (n + k )) for
tree deletion for e ach v a l id inters ection
a V oronoi vert ex o r a b re ak
point ) an d
for tree insertion and split per vertex of P 2 Th e
l ast thr ee bounds follow as soon as it is shown th at the size o f al l queues an d
trees is at all ti m es
In f act e ach candid ate inters ection is ei ther
cre ated ini ti all y ( there ar e only 0 ( IL I) = 0 (n + k ) of such ) or add e d during a
deletion fro m o r insertion into a tree ( at most two new ca ndid ates per i nser
tion o ne new ca ndi d ate per deletion )
Th ere ar e at most n inserti ons an d
o (n + k ) deletions note th at the u niverse does not neces s arily decre as e al l
the m e ; in f act it can grow by as much as n due to introducti on of new
extension segments whi ch can occur at most once per vertex o f P 2 Th us the
to ta l size of al l queues and trees is at every point of the al gorithm bounded
as desired
by
In other words p art 2 spends log arithmi c ti me per
can did ate intersection which there being 0 (n + k ) candi d ate inters ections
bo u nds the executi on ti me of p ar t 2 by 0 (( n + k ) l o g (n
Therefore the extension o f the aug mented V oronoi di agr am of a set o f k
points fro m a porti on o f an n gon to the whole polygon is acco mplished in
ti me by Algorithm B providing the d es ired b ound for
p art 2 (v ) of Algori thm A Observe th at it is i mpossible to i mprove Algo
ri thm B so th at it runs in ti me o ( n + k
without
odifying
the
re
in
ng
a
i
m
m
)
p arts o f Algori thm A for such an i mprovement wo u l d red uce the tim e co m
contr adicting the lower
pl exi ty o f the entire al gorit hm to
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G e o de s ic Vo o oi di
r
n
a g ra m
24
B A o o
.
r
n
v
boun d discuss e d in Section 6
.
Merg in g th e tw o d i a gr a ms
In thi s section we des cri b e a l ine ar ti m e al gorithm for m erg i ng the V oro
noi di agr ams o f two sets o f sites in a polygon assu mi ng th at a chord o f the
polygon sep ar ates the tw o sets o f sites Ag ai n before det ai ling the al gori thm
we will ex am ine so m e properti es of Voronoi di agr ams o f such sets o f sites
Le t A be a set of sites in P an d x be a point in P Th en
De fi n i t i o n
Let a s horte s t
th e dis ta n c e fro m x to A is defined by d (x A)
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,
S
th fro m x to A be
Note th at
d (x s )
d
where s EA is such th at
h (x A) is not uniquely defined for x o n an edge
h (x , A)
pa
,
,
of
h (x , s )
Vorp
be disjoint non empty sets of sites Th e
b i s e c t o r o f A an d B is b (A B )
{ x EP Id (x A)
It is e as ily ch eck ed th at b (A B ) is the uni on o f the V oronoi
No te
edges of Vo rp (A U B ) th at sep ar ate the cell o f an A site fro m th at o f a B site
an d the V oronoi verti ces th at lie o n the bound ary of at le as t o n e ce ll o f a site
in A and at least o n e ce ll o f a site in B In p articul ar b (A B ) h as co mplexity
line ar in the su m o f sizes of A B an d P since it is ( the union o f) a co llection
Moreover fo r any Voronoi e dge e th at
o f Voronoi edges an d vertices
b (A B ) cont ai ns it also co nt ains the two Voronoi verti ces in ci dent to e
Moreover
Hence b (A B ) is a subgr aph o f G ( as defined in Section
b (A B ) can not term i n ate at Voronoi vertices in the interi or o f P o r co nt ain
isol ated Voronoi verti ces as the fact th at a Voronoi vertex v belongs to
b (A B ) al re ady i mpli es th at v is in Voronoi cells of at leas t one A site an d at
le as t o ne B site Th i s in turn i mplies the existence o f at le as t two ( and at leas t
o n e for v E6 P ) Voronoi edg e s eman ating fro m v th at sep arate cells o f sites
fro m di fferent sets In f act thi s ar gument shows th at there is al w ays an even
number o f edges belonging to b (A B ) an d eman ating fro m an arbitr ary Voro
noi vertex i n the in terior o f P Th us b (A H) can be deco mposed in to edge
disjoint si m ple p aths an d cycles Paths connect tw o points o f (BF an d cycl es
ca n be considered n o t to touch 6 P
for a cycle th at does touch it can be
reg arded as a p ath connecting a point o f 6 P to its elf
De fin i t i o n
H (A B )
In other words
{ x EP Id (x A)
H (A B ) is the set o f points in P lying closer to so m e site of A th an to an y site
Le t A, B C P
De fin i t i o n
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of
B
.
Notice th at trivi ally
is a p artition o f P an d
by a s imple continuity argu ment b (A B ) actu al ly sep ar ates H (A B ) fro m
H (B A) in the sens e th at they cann ot be conn ected by a p ath th at does not
m eet b
L emm a
Suppose s EA Le t VP (s ) denote the Voronoi ce ll of s in
Vo rP (A) an d V ; (s ) denote its ce ll in Va rp (A UB ) Th en
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G ode s ic Vo onoi d i a g
e
r
ra m
25
B A o o
.
r
n
v
( )
’
Vp
Pr o o f
'
V
p
Vp
s
(s )
fl
.
{ x Id ( s
(s )
,
x
)
r
{ x Id (s
m)
s
,
fl
x
{
EA U
)
{x
F]
Id (x , A)
x
d (r
,
x
)
m )n
s
u
s
flH
In
i n te ri o r ( VP
Si mil arly o ne shows th at
p arti cu l ar since inte ri or ( V p ( s ) ) is connect ed it is si mply th e conn ect ed co m
ponent o f the set in t e rior ( VP ( s ) ) b (A B ) cont aining s
Le m m a
Su ppose th at Vo r; (A) an d Vo r; (B ) are gi ven Th en
b ot h b (A B ) an d
g iven a po i nt o n every connected co m ponent of
Vo r; (A U B ) can b e co nstru cted in ti m e 0 ( IA | + IB I+ n ) where n is th e
n u mber of vertices o f P
Pro o f Fro m e ach point given on b (A B ) b uild conn ect ed co m ponents of
b (A B ) by the usu al Sh am os Hoey sca n of the two Voronoi di agr am s ( see
for ex ample [L] o r
Note th at o n e ca n follow a bisector of two sites as
long as the two an chor points st ay fixed Th e an chor points ch ange precisely
when the bisector crosses an ex tension seg ment thereb y movin g fro m o ne
region o f the shortes t p ath p artition around a site to an other At thi s point
o n e can recover the new anchor poin t as it is si mply th e anchor as soci ated
wi th the sho rtest p ath p artition region being entered Th e fine s tructure of
th e Voronoi cells ( n am ely their shortest p ath p artition) is so mewh at si mil ar
to the spokes o f Kirkp atrick [K] A co mplication aris es when vertices of
degree gre ater th an tw o are encountered on b
At such a Voronoi ver
tex v the l ist o f owners an d anchors of adj acent Voronoi ce lls ( and shortest
p ath p arti tion regions ) is kn own allowing o ne to determine a l l the ed ges
le avi ng v th at sep arate a cell of an A site fro m th at of a B site thus en ab li ng
the Sh amos Hoey sca n to tr ace a l l br anches of b
Th e additi on al effort
req ui red is propo rtion al to degree of v in Varp (AUB ) thus it is linear over
the cons truction o f
Note th at thi s co mplica tion could be avoided
co mpletely by t aking gener al position to mean in additi on th at no four
poin ts are o n a geodesic circle thus excluding Voronoi verti ces of degree
above three an d forcing degree o f at most two for b
No w split both aug mented Voronoi di agr ams al ong b (A B ) an d co llect
the po rtions of Voronoi ce lls re ach able fro m their owners ; the resulting co l
lection is ( by lemm a
the aug mented Voronoi di agr am o f A U B Note
th at o ne need not be concerned with upd ating the shortes t p ath p art ition o f
the cells as the effect of the merge o n it is li mi ted to t runcation of extension
seg ments down to new cell bound ar i es which is done auto matical ly du ring
the Sh amos Hoey scan Cl
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G e o d s ic Vo o oi d i
e
r
n
a g ra m
26
e
B A o o
.
r
n
v
Observe
th at the tw o steps d es cri b ed ab ove need not be done sep ar ately
It is m ore co nvenient to perform truncation of Voronoi ce lls by b (A B ) at the
ti me it is bein g constructed concurrently upd ating the tw o Voronoi di agr am s
an d m erging them al ong b
No t e 4 2 0 To repe atedly locate the intersection of a ( si mple) arc of a
bis e ctor with the bound ary of cu rrent regions o f Var; (S 1 ) an d Var; (S 2 ) in
we fo l low Kirk p atrick [K] an d introduce a further
an efficient m ann er
refinem ent of o ur m ap th at will gu ar antee th at e ach individu al region h as
b ounded co mplex ity thereb y al l owing us to tr ace an arc o f the bisector
wi thi n each such subregion in const ant ti m e In thi s extr a refinem ent in addi
tion to the extens ion seg ments every vertex o f BP an d every Voronoi vertex
lyi ng in a Voronoi ce ll are conn ect ed by a shortes t p ath to the owner o f the
cel l ( actu al ly o ne nee ds onl y connect each such point to i ts anchor by a
str aight segment so the resulting structure is still li ne ar ) The structure
des cri bed coinci des with the Voronoi tri an gul ation o f [AA] ( Figure
Moreover since e ach seg m ent introduced is cont ained in th e sho rtest p ath to
the owner it will not inters ect the contour b (A B ) more th an once It is
al so e as y to see th at every f ace o f the resulting m ap is bound e d by two seg
ments and ( in gener al ) an ar c o f a hyperbol a or a polygon bound ar y seg ment
an d thus h as co mplexity bounded by a co nst ant It is e asy to veri fy th at this
finer s tructu re al lows Kirkp atrick s tr acing procedure to run in linear ti m e
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V
G o d ic Vo o oi d i
e
es
r
n
o o oi t
r
n
a g ra m
l tio
ri a n g u a
Fig
n of
u re
13
P fo r
27
.
th
t
re e s i e s o f
F i g u re 1
.
B A o o
.
r
n
v
Co r o l l a r y
If al l co mponents o f b (A B ) m eet 6 P Va r; (A U B ) can
b e produced fro m Va r ; (A) an d Va r; (B ) in ti m e 0 ( IA I+ IB I+ n )
It is sufficient to loca te al l points o f
Since b oth
Pr o o f
Vo r; (A) an d Vo r; (B ) are line ar in si z e it is possible to tr ace 6 P i n line ar
ti me p arti tionin g it into 0 (n + k ) str aight seg ments e ach of whi ch lies in
ex actly o ne region o f e ach p artition E ach seg ment co nt ai ns points not only
ne arest to a uni que s EA and a unique t EB but al so h aving a uni que p ai r o f
with
an d d ( t t ) k nown
Th us o n e ach seg m ent the an a
an chors
lyt ic expressions for
d
s
an d
d (x A)
d (x s )
(
)
Ix
j
s
d (x B )
a (x t )
I x t I d (t t)
Since the tw o functions are well b eh aved ( o ne would li k e to s ay
are known
polyn o mi al o f degree two bu t they in fact involve r adica ls ) the points
where they coincide are bounded in nu mber by a co nst ant ( actu ally by two )
an d ca n b e co m puted in const ant ti m e
E ach such point co rresponds to an
intersection o f b (A B ) and 6 P thereby en abli ng us to co mpute a l l such i nter
sections in li near ti me an d per mitting the applica tion of lemm a
N ote th at the cas e when the fun ctions d (x A) an d d (x B ) co inci de is
e as ily shown to vi ol ate the gener al position as su mption as in such a cas e i = t
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Le mm a
If e is an intern al chord o f P an d A an d B are sets of sites
of P on Opposite sides of e b (A B ) co nt ains no cycles In p articul ar the co n
di ti o ns of co roll ary
are s atisfi e d
Pro o f Denote th e p art of P th at cont ai ns A by P 1 and the co mpl em en
t ary p ar t by P 2 ( refer to Figure
Notice th at H (A B ) is the uni on o f interiors o f Voronoi ce lls of A sites ( in
,
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.
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,
F ig
G o d s ic Vo o n oi d i
e
e
r
a g ra m
28
u re
14
.
B A o o
.
r
n
v
together with th e Voronoi edges sep ar ating these cel l s Hence
if there is a cycle C in
it must co mp l etel y enclose a cell o f s ay A
site s i e there is no p ath in H (A B ) connecting s to 6 P Choose x to b e the
point o f e closest to s Observe th at either x lies in the interi or o f e and
u(x s ) is perp endicul ar to e or x is one o f the endpoints of e an d the an gle
between u (x s ) an d the vector pointing al ong e aw ay fro m x h as m easure
2
Ext end h (x s ) p as t s until it intersects 6 P at point y ( Figure
contr ary to our as su mption th at s is enclos ed
We cl ai m th at h (s y ) C H
by a cycle
Consider z Eh (s y ) an d t EB C P z If
d (z t)
( note th at s t since A an d B are disjoint) so
let us assu me
By the above observ ation the angle betw een ii (x z )
It w as shown by [PS R] th at the
an d ii (x t ) h as m e as ure a rr/ 2
side of a geod e sic tri angle lying ag ai nst such an an gle is stri ctly the longest
In p ar ticul ar
Va rp (A U B ))
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,
d (z , t)
H ence d ( z B )
d (z , x )
$
D
therefore 2 E
as asserted
Co r o l l a r y
The di agr am s Var; (SL ) an d Va r; (SR ) i n step 2 ( vi )
Algo ri thm A ca n be m erged in ti m e 0 (n + k )
To s u mm arize the procedure for merging the two di agr ams consists
loca ting all points o f b (A B ) on 6 P by a line ar sca n followed by tr acing
every co nn e cted co mponent of b (A B ) a l a Kirkp atrick [K]
an d
,
.
of
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,
,
.
A sp ec i a l ca s e
of
.
,
5
of
.
It is i nter e sting to discuss
v ari ation of our al gorithm th at co mputes the
geod e sic Voronoi di agr am fo r a set o f k sites among which are a l l refl ex ver
tice s of the enclosing n gon in 0 ( (n + k ) l o g ( n + k ) ) ti m e Th is v ari an t
a
-
.
F ig
G e o de s ic Vo o n o i d ia g
r
ra
m
29
u re
15
.
B A o o
.
r
n
v
ex ploits the fol l owing property o f the geodes i c V orono i d i agr am of s u ch a set
—
the Voronoi cel ls are st ar sh aped in the Eucl idean m etric as a b end
o f sites
o n a p ath fro m a point in a cell to the owner of the ce ll would produce a
reflex vertex i e a closer site Th e gener al str ategy is still divide an d
conquer with the polygon recursively S plit into two roughly equ al p arts and
the res ul ting di agr ams merged a la Sh am os an d Hoey [SH] However the
ex tens ion step is avoided r ather a structure si mi l ar to the ex tension o f Voro
site less regions is bui l t at the b otto m o f recursion an d
n o i di agr am into
m ai nt ained together with the V oronoi di agr am proper through the m erging
steps Th e r ation al e behind t his appro ach is th at the onl y sit e s th at can o w n a
given point o f th e polygon are those vi sible fro m it Consequently during
the merge step the onl y points o f one h alf of the polygon th at can be t ak en
over by sites o f the other h al f are those vi sible fro m the sites through the
window o f the divi ding chord In p ar ticul ar it is eno ugh to co nstru ct the
V oronoi di agr am o f the set of sites o f one h al f of the polygon as visi b le
thr ough this window Thi s notion is identi cal to the notion of the pee per s
V oronoi di agr am o f [B S] b u t fortun ately in our cas e it is e as ily see n to h ave
line ar si ze Moreover we can show th at the geodesic Voronoi di agr am o f a
set of sites in a polygon with peeper s Voronoi di agr ams att ach ed to so m e
o f the polygon ed ges ( in f act to those which are chords of tr i angul ation of
the origin al polygon ) is a st ructure l ine ar in the si ze o f the polygon and the
num ber o f sites an d two such st ructures can b e m erged in line ar ti m e Th i s
al gorithm for constructing the geodesic V oro
yi elds an
noi di agr am in this speci al cas e Th e det ails o f thi s f as ter al gori thm w i ll be
repo rted sep ar ately
Note th at the al gorithm s k etched above al lows in p articul ar to co m pute
the geodesic Voronoi di agr am o f the ve r t i c e s of a si mple n gon in ti me
0 ( n l o gn ) An al tern ative 0 ( n l o gn ) al gorith m is provided by work of Lee
an d Lin
which involves the cal cul ation of
( see al so [Chew ] an d
the g e n e ra l ized D e l a u nay tri a ng ul a t ion of a si mple polygon P It is a tri angul a
tion o f the polygon with the property th at the circle circu ms cribed around
e ach face o f it does not co nt ain ( in its interior ) an y vertex of P v i s i bl e si mul
t an eously fro m all thr ee vertices o f the s ai d tri an gul ar f ace In the cas e o f a
convex polygon this definition giv e s precisely the convention al Del aun ay tri
an g u l ati on o f P which is th e du al of the co nvention al
oronoi
i
gr
V
a
d
am o f
(
)
the vertices o f P which in turn coincides ( inside o f P ) with the geo desic
Voronoi di agr am of the vertices of P I n f act it is not difficul t to observe
th at for a gener al si mple polygon P the gener alized Del aun ay tri angul ation
is ( essenti al ly ) the du al o f the geodesic Voronoi di agr am ( in f act it ca n be
shown to h ave more ed ges th an the du al ) [LL] ( and al so [ Chew ] [WS] ) pro
vide an 0 (n l o gn ) algori th m for constru cting this tri angul ation an d the
di agr am can be reco nstru cted fro m it in line ar ti me by visiting the neighbors
of e ach vertex in cyclical order and thereby reco nstructing its Voronoi ce ll
Th u s we obt ai n an al tern ative 0 ( n 10 gn ) algorith m for co mputing the geodesic
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G o d s ic Vo o oi d i
e
e
r
n
a g ra m
30
B A o o
.
r
n
v
Ack n o w l e dg e m e nts
I wish to th ank m y Ph D advisor Mich a Sh ar i t for suggesting thi s prob
l e m fo r v al u ab l e discussions on i ts solution an d for h aving the p atience to
read thr ough num erous vers i ons o f thi s p aper I wo ul d especi al ly l i k e to
th ank the anonym ous referee for hi s/he r constructive co mm ents an d su gges
ti ons
7
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8
.
B i b l i o gr a p h y
.
F Au re nh amm e r Power di agr ams : properties al gorithms an d
78 96
applica tions
S IAM J Co mp u t ing 16
Voronoi di agr am for po i nts in a si m ple
T As an o an d T As ano
polygon
D i s c re t e Alg o ri t hms a nd Co mp l ex i ty : P ro c J ap a n US
Aca demi c
Jo i n t S e mi na r ( Persp e ctives in Co m puti ng Vo l
Pres s 19 8 7 5 1 6 4
F Au re nh arn m e r and H Ed el s bru nn e r An Opti m al al gori thm
for construct ing the weighted V oronoi di agr am in the p l ane P a t
[ A]
$
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[AF W]
2 5 1 2 57
17
te rn re cog n i i o n ,
-
.
Aronov S Fortune an d G Wilfong The furthest site g eo
desic Voronoi di agr am P ro c 4 i h ACM Sy mp o n Co mp u t a t i o n a l
Ge ome try 1 9 8 8 pp 22 9 24 0
A B al ts an an d M S harir On the shortest p aths b etw e en tw o
convex polyhedr a J ACM 3 5
2 67 2 8 7
B Ch az elle A theorem o n polygon cutting with applica tions
P roc 23 rd Sy mp o n The o ry of Co mp u t i n g 1 98 2 pp 33 9 34 9
L P Chew
Constr ained Del aun ay tri angu l ations Alg o ri thmi ca
to appe ar
B
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[ Ch ]
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[ Chew ]
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[ CD]
Chew and K L Drysd ale III
Voronoi di agr am s b as ed on
convex dist ance functions Pro c of the ACM Sy mp o n Co mp u ta
ti ona l G e o m e try
1 98 5 pp 2 35 2 44
H Edel s brunn er and R Seidel
Voronoi di agr am s and arr an ge
ments D i s cre t e a n d Co mp u t Geo m 1
2 5 44
Fortune
A sw ee pl ine algorith m fo r V oronoi di agr am s
LP
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[ES ]
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Alg ori thmi ca 2
15 3 174
-
,
[ GHLST] L
.
J Hershberger D Le ven M Shari r an d R E Tarj an
Line ar ti me al gorithm s for visibility an d sho rtest p ath problems
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