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36
5
R2
n !#"#$%'&
v1 , v 2 , . . . , v n
vi
(*),+,-/./0
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(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 :
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2
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(Voronoi diagram)
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ŠtN 83O v#N#Š”& H ∩ H ∩ · · · ∩ H F&• ˆ LK CZŠ'‰ K/ v ‹ M –Aa>b0 „5
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=? ! & :6<G=? " $>C † N 83O
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$ › 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<
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5.3. 1 FG& ABDC (– S a N 9LO
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>?A@B 1: V := R $t—–%A& O:C
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>?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
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v4
v2
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v3
V2
v3
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V3
V3
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w2 = v1 V(w2) = V1
V1
v1
p1
p1
p2
p2
v4
v4
v2
v2
v3
V2
w3 = v3
p3
V2
V3
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V1
v1
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V4
p2
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v2
v3
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0 VW 5 o OC 1 ΠV 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
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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
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!+-, )* 7:
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/ / . 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;
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begin
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v 1 , v2 , v3
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V3
for k = 3, . . . , n − 1 do
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N ; V . 8G
k+1
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k+1 )
h %
'
(vk+1 )
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;
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Vk
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vk+1
.`%
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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
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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
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