36 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 SxN 9t 0 H 1S G &/6 c A x ∈ H v CZ v > 5 N 83O/ C r K v $ v (j = 3, 4, . . . , n) 0 gyhj/k/lmQ>i v 3 G 0 H 1 G & x ∈ H F v C#; v M P $E c tN 83O v#N#& H ∩ H ∩ · · · ∩ H F& LK CZ' K/ v M Aa>b0 5 oGu& v (),+/-. > cG eN G O 5 "A_stoG& K/ v $ v 0;g,hiGQyAjk/lmSQ i 1 2 1 12 2 12 1 1 12 1j 13 j 1 j 1j 1 1n 1 i 1 H15 H12 v5 j H14 v2 v4 v1 H13 v3 i 1 Q 9 37 vi S 0 Hij 13 8 P $ 38 & v 6, >H J F i (5.1) W Vi = Hi1 ∩ · · · ∩ Hi,i−1 ∩ Hi,i+1 ∩ · · · ∩ Hin $ P $ 1 %GN 83O#P c t& _SL Av c tN 8 : 5.1. HJ V F R 1 9O P Gc _>tHJ FG& x> ( ) $SC &z / &Fz :< =? ! & :6<G=? " $>C N 83O StHGJ V /G F& G H (j 6= i, j = 1, . . . , n) #${&%,c N 83OP #$ F ' % 1 n − 1 ( tN 8 y&) F /> $ M #$ tN 8O $ 9 #$6*0 , + 5 " K,0 & :#<G=? "-&.0/13254! $;MGMyN 83O 5.2. ZFG&ze 0 67 8#9 K,0;g 8 j,k/lmGQGi0{ % 1 9O 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 G1 M ,9 J b F ,IH>J F K | & L ,3 N F M _ G > > / > r (Z > a N 9GP $3 IN 8OP C M (O a N 9 & P W $Q ; v R9 J b &,z F STVU*WX 2 (degenerate) $'MSMN 83O i i 2 ij c t & /> >F &Gz S 06&7 89 Ef v LKGc mYZ N 38 O E & >A 1 & _, e F v , v c m YZ & 6/> 5.2 12 1 2 38 v1 p e12 v2 v3 F v , v c mYZ 'N 83O 5 "Lseo& S e $ e S 1 9 6/> p F v , v , v c mYZ tN 83OP E c c 9 C r & { ;N 8 : 5.4. HJ V p 0 ) d(p, v ) S0 $e& P FP $ C v K#0 A &z Z LK FxN M O P C r #0 *% ;v K#0 N M y P $ 0 $;MGMyN 83O e13 1 1 3 12 2 3 i 5.1 13 i V Ef v _,SIH>J V , V ; v S 0 5 o/M 9 $%& 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 %) 89P $ 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 $ &z ; v ;v 8 ;0#0 - <%.</=8>8? (Delaunay triangle) $>CEDxN 83O i j i i 1 1 j 1 i 1 i 39 V2 v3 V3 v4 v2 v7 V4 v1 v5 V1 V5 V7 v6 V6 40 UWV6XZYZ[ V \]_^ 67 ;# LF/& "#y8#9 x//eHJ f v K#0Ag 8 i/Q 1 8 c ;& 4 0 M + > &zt K/ 6&7 89 /> j/k/lmGQSi 1 9>P $!G c ;N 83O 5.2 z C r /S yF C ( tN 8 y& 5.3 CZ&<;v S a N#;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 tN 83O ;0y C`v * / N 1 YZ m 5 M F>&>z ;&# ,& 0 )% $ 5 D 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 89 T , T , T F "+# 5 o,M 9 $;MSM>& T %) 89GP $ 5 N 5_qSr O x $ 0;iSQ 1 g G1 P "+#&( 0 67 ; & v>M3o_sIo,u 9 F 6,S > > tN 4 ( { $ s;o,M 9 0 T8? $;MGMyN 8O 5.6. _/SIH>J "%# &%( 0 ) 5 " 4 %67 ;& "#&( 0 ) 5 "Z S 1 9AO 1 2 3 4 1 ) 1 %/N 8O 4 &( 0 $ T (i = 2, 3, 4) FS& P * #0 8 $ o G O,P C r "#& i 1 &'4 687 ;& 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 9 15qr c O z F v , v , . . . , v 0K, $ 8#9 GA 07 5 o,p c N 83O v6N6& v 0K, Av x,>;HJ _StH>J K,y ) & v M 5 N 83O V "#y89 5.7. L K x M K/ FG&Gz LKy 6/>IH>J 8"#,89 _>IH>J AKGy ) 9LO P >c e& v x M #0 v 8 8G F>& V "#,89 />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 1 1.1 n 2 1 1 2 n 1 1 1 1 1 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 ML 83O >?A@B k: v $ v 0;gZhiQLjk/l3m>Qi l 0Sn %A&ze 05#$ $ 8#9 v 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 &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 so 5 N r P $ tN 83O >' S0 RT C 7 89 FG& : $ & U KVW X QR YZ 0 [ M 9GP $E 1 %GN 83O 5.2 i 2 1 1 k 1 1k 1 1k 1 ik 1j 1k 2 i 3 \'] \^+_ P>P 1 F>&U VW C 9 ,S' 7 v>M3o`y _/GA V *0 & P K/ v 0 VW v ,v ,v c 9 9 V *0 ;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 V 1 % 1 F>&Sz K0 4 % 687 ;&ZG $ 89GP $ 1 k F89 & P 4 867 ;0 c z jGi60 C = 8O K/ v VW w N 83OZP v M K0 & 8 1 ,S 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 37/89 _StHGJ V (w ) (= V ) 0 89 N 83O f v S r & 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>QAjklmQ>i0Sn %L& P $ w _SHJ V (w ) (= V ) x $ %G0 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 0AghSi,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 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 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 $ w2 2 1 2 2 4 2 2 3 3 3 1 3 3 2 3 3 3 3 4 1 2 1 3 1 2 3 4 4 4 5 4 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 V 0 7 5 k+1 VW Vk K vk+1 (vk+1 ) begin K, v x K/ v $ w |y0 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 &,z 0 w $t ; 0;gyhi>QAjk/lm>QGi $ _GHGJ V (w ) %S0 8 9 & 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/hAiQjGk>lAmQi $ #>IHJ V (w ) +3y0 89 & 3 p O#%G0 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