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5
R2
n !#"#$%'&
v1 , v 2 , . . . , v n
vi
(*),+,-/./0
Vi = {x ∈ R2 | d(x, vi ) ≤ d(x, vj ), j 6= i, j = 1, . . . , n}
(5.1)
132465 & v 37/89;:<>=?A@>B (Voronoi region) $>CED& v F /GIH>J LK/ $'M
MN 83O#P /GIHJ V , V , . . . , V C 9 LQGR :
i
i
1
n
2
V n = {Vi | i = 1, . . . , n}
:<S=?3T
(Voronoi diagram)
1 83O
UWV6XZYZ[\]_^
`" KG'a>b {v | i = 1, . . . , n} c v $ v 0ed D,&#f 0;g,hLiSQ#0ejklLm>Q
89 i>Q60Sn Mo/pN 5_q>r O > F P ki Cstouf!v H>J QGRxw _N 8 y&>z L{|
0L}S~G (half plane) $>CEDxN 83O K v S€xN 9t‚ ƒ 0 H 1S„…G† &/‡6 c ˆ
A x ∈ H ‰ v CZŠ v Œ‹Ž > 5 N 83O‘/’ C r K v $ v (j = 3, 4, . . . , n) 0
gyhj/k/lmQ>i v ‚ 3ƒ G 0 H 1„…G† & x ∈ H F v C#Š;‰ v ‹ M P $E“ c
ŠtN 83O v#N#Š”& H ∩ H ∩ · · · ∩ H F&• ˆ LK CZŠ'‰ K/ v ‹ M –Aa>b0 „5
oG—uŠ˜& v (™),+/-. >š cG› ŠeN …Gœ O 5 "A_stoG& K/ v $ v 0;g,hiGQyAjk/lmSQ
i
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12
2
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1
1
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1j
13
j
1
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1j
1
1n
1
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1
H15
H12
v5
j
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v2
v4
v1
H13
v3
i 1 Q 9
37
vi
‚ ƒ S 0
Hij
13„ 8 P $ 38 † & v 6, >H J F
i
(5.1)
“WŠ Vi = Hi1 ∩ · · · ∩ Hi,i−1 ∩ Hi,i+1 ∩ · · · ∩ Hin
$  P $ ‰ 1 %GN 83O#P c t& _SL ‰Av
“ c ŠtN 8 :
5.1. HJ V F R 1 9ŒO
P Gc _>tHJ FG& x> ( ) $S‰C † &z / &FzŒ :<
=? ! & :6<G=? " $>C † N 83O
StHGJ V /G F& ƒ G H (j 6= i, j = 1, . . . , n) #$–{&%,c N 83OP #$ F ' % 1 n − 1 ( ŠtN 8 y&) F /> $ › › M #$ ‰ ŠtN 8O
$ › 9 #$6*0 ,Š + 5 " K,–0 & :#<G=? "-&.0/13254! $;MGMyN 83O
5.2. ZFG&ze 0 67 8#9 K,–0;g œ 8 j,k/lmGQGi–0{ % 1 9ŒO
9: FG<
& ;”v S a N 9 S = >@? 5 N 5xqr O
5.3. 1 FG& ABDC (– S a N 9LO
E † 5& F&GH K & I œG1 M ,9 J b F ,IH>J F K | › Š & L ,3 N
F M
_
G
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r
(Z
> a N 9GP $3‰ ŠIN 8OP C M (O a N 9
& P ›W $Q
‰ ; v R9 J b &,z Œ F STVU*WX 2 (degenerate) $'MSM–N 83O
i
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2
ij
c t & /> >F &Gz S 06&7 89 Ef v LKGc mYZ Š
N 38 O E † &
>A 1 & _, e F v , v c m YZ ”Š & 6/>
5.2
12
1
2
38
v1
p
e12
v2
v3
F v , v c mYZ Š'N 83O 5 "Lseo& S e $ e S 1 9
6/> p F v , v , v c mYZ ŠtN 83OP E c “ c 9 C r & { Š;N 8 :
5.4. HJ V p 0 ) ƒ d(p, v ) S0
 $e& P FP ›
 $‰ C v K#0 A Š &z ‚Z • LK F€xN › M O
P C r #0  *% ;”v–‰ K#0 €N › M y P $ 0 $;MGMyN 83O
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1
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3
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5.1
13
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V
Ef v _,SIH>J V , V ; v S 0 5 o/M 9 $%Œ& V $ V F! U5WX 2
$'MGMN 83O ;”v >tHGJ V 0'd Dy& P "# 5 oM 9 x>tHJ V K, v $
1 g$ † & ,GtHJ V "%#'&( 0 %) 8–9P $ 1 %GN 8O#P +*0 V 8
v 0;iGQ
1 ›W & 8 $ o SIH>J — P › seo, ! 9 - <.0/1 (Delaunay net) 1 8O
,>eHJ 203 5 oM › † &54 6879 F K,ZLab {v | i = 1, . . . , n} &:;
&QGR $ › Š &z ; v ;˜v 8 ;0#0 - <%.</=8>8? (Delaunay triangle) $>CEDxN 83O
i
j
i
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1
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39
V2
v3
V3
v4
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V4
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40
UWV6XZYZ[ V \]_^
67 ;# LF/& "#y8#9 x//eHJ f v K#0Ag œ8 i/Q 1 8 c ;& 4
0 M + …>† &zt ŒK/ 6&7 89 /> Œj/k/lmGQSi 1 9>P $!G“ c Š;N 83O
5.2 z C r › /S yF C ( ŠtN 8 y& 5.3 CZŠ”&<;”v S a N#Š;N 83O M
† & 4 % 67+9 — 9 z' 4 %687 ;& F ; v 6,3 7 5 o/M 9 “
1 8O w & 5.4 $ " S
0 M + …G† & _,S >c 789 4 67 ;&
L / N 1 YZ F m 5 M P $EG“ c ŠtN 83O ;0y C`v * / N 1 YZ m 5 M F>&>z
;&#
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o C ,–0 A 9 F # 1 83O
5.5. FG&54 8 67 ;0 1 9LO
T2
T1
T4
T3
687%9 —/MGo‰'& xyS3 $ ‘– 4 867 ;– "#
4
%687 &; T $ 08 8–9 T , T , T F "+# 5 o,M 9 $;MSM>& T
%) 89GP $ 5 N 5_qSr O
x
$ 0;iSQ 1 g œG1 P "+#&( 0
67 ; & v>M3o_sIo,u 9 F 6,SŒ >š ›>› ŠtN
4
( „ { $ › s;o,M 9 0 T8? $;MGMyN 8O
5.6. _/SIH>J "%# &%( 0 ) 5 "‰ 4 %67 ;& "#&( 0 ) 5 "Z‰ S 1 9AO
1
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&( 0 $ T (i = 2, 3, 4)
FS& P * #0 8 $ o …Gœ O,P C r › "#&
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41
w o,& PP 1
E 1.3 "' ‹60 P 5 So,pN 5qGr O _E–" n A v , v , . . . , v ) 1 v x‰ ‹ M F ˆ › 9 1–5qr c O zŒ F v , v , . . . , v
0ŒK, $ 8#9 GA 07 5 o,p † “ c ŠŒN 83O v6N6Š”& v 0ŒK, ‰Av x,>;HJ
_StH>J ŒK,y ) & v ‰ ‹ M 5 N 83O
V "#y89
5.7. L K x‰ ‹ M K/ FG&Gz LKy 6/>IH>J 8"#,8–9 _>IH>J AKGy
) 9LO
P >c e& v x‰ ‹ M #0 v 8 8G F>& V "#,8–9 />IH>J AK $ YZ#0 89 8 1 C  &;• $ Y&Z F 89 › M 1 & 6,S' 7 5
o † !0 "# $  P $E 1 %SN 83O
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U V6XZY–[%'& \'(')_\+*,.-0/2143
W
V 056!7 8:99 ‰;< ›G| F& 8 1 _SL y $ P= 1
$ N 5 " O E † & i=1› † &
>?A@B 1: V := R $t—–%A& O:C
DFE.GIH k 0 k = 2, 3, . . . , n N 1KJ ŠML 83O
>?A@B k: v $ v 0;gZhiQ–Ljk/l3m>Qi l 0Sn %A&ze 05#$ $ 8#9 v ‚ ƒ G
H 0 8:9 & V := V ∩ H $I—  O
P C r ›G| 8 $t0& L DENGIH 1 ki l $ k − 2 (–3kSi l (j = 2, 3, . . . , k − 1) $ 0 89Z9 Š &Gz FO F 1 + 2 + 3 + · · · + (n − 2) = (n − 3n + 2)/2 E 5 N 83O
,GL *0 7 8#9 FG& P P0 8 $ o v (i = 1, . . . , n) & 5 o /“ › † ›
› M 1 & g Q O(n ) ‰ cSc so 5 N r P $ › ŠtN 83O >' S0 RT C  7
8–9 FG& : $ ‘ & U
KVW X QR YZ 0 [ M 9GP $E 1 %GN 83O
5.2
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1
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1k
1
1k
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1j
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VW C 9 ,S' 7 v>M3o`y‡
_/GA V *0 Š & P K/ v 0 VW
v ,v ,v c › 9
9 Œ V *0 Š;N 83O
1
2
3
3
4
4
5 N 5 q>r O Nba>& C`v L K
5 o K/ v , v , v , v c ›
1
2
3
4
42
C!v 'K/Sc › 9 4 % 687%9
9GP $E 1 %N 83O 5.5 –C Š
† Œ V 1 % Š 1
F>&Sz ŒK0 4 % 687 ;&ZG $ 8–9GP $ 1 k F89
& P 4 867 ;0– c ŒzŒ jGi60 C = …
8O
K/ v VW w N 83OZP v ‰ ‹ M K–0 & 8 1 ,SŒ V 0 2>465 oM
9 K, v , v , v ) c d D&ze 0 w (= v ) $SC h P $ 5 N 83O K, v $ w 0'gyh
i>Q#Lj/k,lmQ>i0Sn %'& P $ w 37/8–9 _StHGJ V (w ) (= V ) 0 89 N 83O f v S r & ˆ cG{>|Z0 p $ 5 N 83OP p sIo,M 9 G
1 HJ V (w ) "#y89 /GIHGJ LK/#0 w (= v ) $ 5 N 83O
3
4
1
3
4
2
3
1
2
4
1
1
2
1
1
1
2
v1
1
1
v1
V1
V1
v4
v2
w1 = v2
v3
V2
v3
V(w1) = V2
V3
V3
;0 gZhi>Q–AjklmQ>i0Sn %L& P $ w _SHJ V (w ) (= V )
x $ %G–0 89 & p 1 › M |,S#0 p $ 5 N 83O z 5 o p s;o/M 9
1 HJ V (w ) "#,8#9 /GIHGJ LK/#0 w (= v ) $ 5 N 83O
D–& xy, v $ w 0Ag–hSi,Q63jZkylSm,Q,i0/n MGo w y,;H,J V (w )
_S $ S#0 8:9 & p 1 › M |Z0 p $ 5 N 83O
(= V ) / #" K, M'v O J b F ‘’ *0yw J Š L 5 N 8 Z& P>P 1 F p
seoyM 9 , 1 ,tHJ V (w ) "#869 x,GtH/J LK, w {
5 & VW w #" K/ v 0 Š p , p , p c › 9 ,u yo,MN 83OP < *% 9 _,S 0 5 9GP $ 1 , ! 9 & K v , v , v , v c › 9 _
V 1 83O
P rI5 o , –"NM;v AKc › 9 /S V v 9 LK v 0 VW 5 & V 0 7
9: F
v4
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w2
2
1
2
2
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2
2
3
3
3
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3
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43
w2 = v1 V(w2) = V1
V1
v1
p1
p1
p2
p2
v4
v4
v2
v2
v3
V2
w3 = v3
p3
V2
V3
V(w3) = V3
V1
v1
p1
V4
p2
v4
v2
v3
V2
p3
V3
5 "–$%/$ >‘ ’ P C sEo 6; V 0 7
0 VW 5 o OC 1 Œ V 0 7
5
k+1
VW
Vk
K
vk+1
(vk+1 )
begin
K, v x‰
K/ v $ w
|y–0 p $
p s;o,M
k+1
k+1
1
1
1
5 N S8 O %{ FS& 6/A
5 N 8 :
‹ M K#0 v , . . . , v c v &,zŒ 0 w $t—  ;
0;gyhi>QAjk/lm>QGi $ _GHGJ V (w ) %S–0 8 9 & ˆ cG{
89 ;
9 1 & V (w ) $ "%#89 _StHGJ AK/–0 w $ #8 9 ;
i := 2;
while w1 6= wi do begin
1
k
1
1
1
2
44
$ w $ 0tg/hAiQŒjGk>lAmQi $ #>IHJ V (w ) +3y0 89 & 3 p
O#%G–0 p $ 8#9 ;
p s;oM 9 x/G 1 & V (w ) $ "# 5 Zo M 9 _Z>tHGJ LK/#0 w $ 8
9;
K>
vk+1
i
i
i−1
i
i
i
i+1
i := i + 1;
end;
&
p1 , . . . , pi−1
<*% 9 _,SA
Vk
Li>Q60 5 &, !–" 60
V k+1
$ 89
end;
] \
w o V:W (v ) DFE G2H "!#$&%('()* v
!+-, )* w . v , . . . , v /021(354(6578(9 7:; . 1<=>?A@B 7 CD5E
!
F ; . 1"GIH J ?IK ML N(OQP O R 7TS L
UV .XW 0ZY"G .\[^]`_(a :
< !(F("]Zbc(d )*.Iefg < . w h ] 7:ji !\kmlno2pq V (w) !r\s 7G
kTlMnjo`p
q )* u tu
k+1
1
1
k+1
k
d(u, vk+1 ) < d(w, vk+1 )
(5.2)
ui
w
vk+1
kTlMnjo ;
45
w
vk+1
ui
vk+1
w
. 7 )* u G`/ / . H ] 7: + + u S L )* u L 4<= u )* w v
!+-, )* 7:
(5.2) . 7 )* u <= u < . w h ] 7: u f(kmlMn5o2p(q V (w) !(rs 7G klno-pq )* t\ (5.2) . 7 )* G
/ / . H ] 7:
k+1
kTlMnjo ;
,*
46
(vk+1 )
begin
)*
v1 , . . . , v k
/0 cd ! 3 .Ie(f u < .
w
h 7"G
;
nearest := false;
while nearest = false do begin
kTlMnjo`p
q V (w) !(r\s 7 G klno-pq ) *^. u h 7^G ;
V (w) h V (u ) !(r\s 7"G kmM
l n5oAp(q u M 02/(> U )*^.
1
1
u2
h 7G
;
i := 2;
while ui 6= u1 and d(ui , vk+1 ) ≥ d(w, vk+1 ) do begin
kln
oZpq
h 7^G ;
u
V (w)
h
V (ui )
!rs 7^G klMn
oApq V (ui−1 )
h F
L G gU ) *.
i+1
i := i + 1;
end;
if u1 6= ui then w := ui else nearest := true
end
end;
!"#$
%'&)( 3 N .] h <= u kmln5o ;
] 7
Vn
. 8+*7"G-,(i %(' V F i S !/.+0 ^
:
132546879 ,i %' (
begin
)*
v 1 , v2 , v3
/0 L G klno ;
V3
for k = 3, . . . , n − 1 do
,* (v
N ; V . 8G
k+1
end
k+1 )
h %
'
(vk+1 )
. 8mG
;
.;: \kml(njo ;
Vk
!)(*
vk+1
.`%
' u kml(n
o
kTlMnjo ;
47
end;
CD^. H^G ! u ]-b5N (,(* (v ) . 1 G h2u while
F\)* v h w hX 7"G j!#" k %$&')( ] 7: j ?
$&u*+ while FkTln5o-p
q V (w) !
rs 7"G m , kTln
o-p(q V (u )
0 : < 0 )* /0 d(u , v ) < d(w, v )
(i = 1, . . . , m ) O -/. < G + ^h ] /
. 7 &* 1 L w h je =5< ] 7 u kmlMn(o2p(q h rs 7^G ( 3 kTlno2p
q.!2 Xu m − 3 , kmlMnjo`p(q V (u ) F %3 O -4. < G gh PZ]Z_a !: 5 !! 6[ 0 <
I)* 78 F k / PA]Z_a /0 u i * :P 9 ] 7 :
k+1
k+1
j
j
i
i
j
k+1
k+1
i
k
X
(mj − 3) ≤ k.
k
X
mj ≤ 4k
j=1
< S P u O -4. < G kmln(o2pq 78
j=1
hAL P u O(k) CD v !+-, )* w 1
3j/ PZ] 7:
N %' (v ) F u %'()* v !!; 7^G ,)* w .`[ 135/<= u w h k
lnoAp(q V (w ) !(r\s 7^G klMn
o-pq )M* hX< !%=>T.?A@CB 8g64 ')(M ] 7 u D FEG + 5 !!6[ 0 < )* F78 k !IHJ 7"G N <M /
// P`]Z_a :
%+& KLh )* 8 M 3%N3%O [ G h .P[ GQ _ <= uR
,i %(' (
DCjD k+1
k+1
k+1
1
O
n
X
k=3
k
!
= O(n2 )
S .Z[ 0 <G Mh Q / PA] 7:
1
1
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