Exact Nearest Neighbor Search in High Dimensions

0 downloads 0 Views 216KB Size Report
We present an algorithm for solving the nearest neighbor problem with respect to Lo-distance. It requires no pre- processing and storage only for the point set P.
Exact



Nearest Neighbor Search in High Dimensions Helmut Alt



Laura Heinrich-Litan



Institut fur ¨ Informatik Freie Universitat ¨ Berlin D-14195 Berlin, Germany

Institut fur ¨ Informatik Freie Universitat ¨ Berlin D-14195 Berlin, Germany

[email protected]

[email protected]

.  -  & 10 23 4  /5 6   8"!7:#9< ;>=. & +?$%&' 5@B 'A>(F= HM)$QI¦SI -/- §.& ;   'D.  Jm¥—=. ‘FI )$Q'': C3;|D'=.& g«.¦S@ ² D.   5?5o. D  '²'  5oI%D'    5o'I!apef'¦S 5?²­F³G*!alo x ³ ? ?= JD ³J&o¸  ¹'§Qµ])º&¶«. @ { ´L& 3 'I 5?' ¦J& §  W°  ' § { x

1.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. SCG’01, June 3-5, 2001, Medford, Massachusetts, USA. Copyright 2001 ACM 1-58113-357-X/01/0006 ... 5.00.

»

/IA|,G2¼6ef5)g.½yH :'W Um&&l: 5)-'*D.  2*' +   ?x' ? ')0tRI¼6$QefU )2x _= ¾h!N¿ I%lo@ H. / &  (&gM' !­x  ' J L C)D'* 5o H

 F$Qo5 ?MF G 'CD D' `F = & E@‘€tF'=W 5)!aE  x% D 5?'E'  $QS ?&I‘% :D FF=.=I  +=  L ? S$Q o ; C´D'N3 CID'o HW So® / 3 ''I G &*À 6VWgE!f5o GI5. I'/o-  &  $Q,M02&83 W ' ' kI‹5)„o™% †>‚ ‡@ Á/Œ„+Â?‚ISIEo KY'%=WF=W!m Ã+D' +5G =  (  'IR I& 'IMD ' 6 @=.L +€t?'' S56I!:  š% = ,)-%$%5o3 ; -/D S  +3HG  ))C=yD'x'3 ' q=   .G CD'  HI5  I'J x 2¼6 efHJUh g.'l:À!#  JNKsm37­ Ä;|3=.!'' +?)'=.5 m  x ]GÅEA|•Æ Å ?= Ǖ (@  £$ 5IW  vÈ>5D E!M= I F >H { o;

) +d* 5o   % (-/ d§Q.) =s-'!a 8 xÂJ‡|ƒ?fQ@ F@ʄŒÉ +ˆ#I D'ƒ5o†> ‚D  kD !a  E' “I!#5& ,%  '3kH•'~')F*=. ;;  !f''5s  /C% 6D Ì6  H4!a /Í F W)'xP-' IVD.I'/02 ' &À- 38Hz =. @ +? ¯J' '5 d= '!23Ëh  ,  +d-H4  3H£,% I-/  W +),®. 3 ;  +  m  D ®. /'' P%0P & P/= $Q/,F-= H,'´L5L H%RI ''=k°k= ID' G Z3]I^h0202''5? kL' '=  -F  W-'= Ê0/tF=. R)=B&%$Q£ +  &'  )C=VD'3 |H£  !3H 'k ' .6 )&® +RH*) & m  F )^{z H# +Ikipj  ` i %kf-/mpH< ,. !`š85)= 3!GD. 'tWF  N5)$> x'5)' '= )=65D'x-/ k/& ,&'   x

-''ID.d2.= |H®.@M/ P5o' 6)2=sF5?) F$QD'M' ISI3  I 2!6$% +&IH.= i Ijf ` 5 Mi   kf&'5mp5HsGI -'H  `V$®.5> H& x + ?``I(& !#M *5$Q D 

5®.M-/'  I)=V?  q55= ID '    I8CD'&3 r.%5$Q $Q)=B¡%@ s '  ,5)Š   x30! )   2U4Uy!a < Us '-   r%' 5Å   2U‘Å @N¨mo&23 U‘ :x.-/0t:G F'):$Q J-%HE e+ +] «Qlh­+ D'5?` F‡ )  Um @   Œ 9Ž.#e   ‘s l Œ 9Ž.#e   Š  ‘s'l !    [Œ [ Å  !  i Š  2 i Š  2 $ ¼6e+]*l Å U¬] $ ¼6e+]*l ’—5)D `!2e+])«l:0tGo + Œ 9Ž.#e   Š  ‘s'lÐÅ Œ 9Ž #e   ‘s'l !    [ Å Œ !  i Š  2 •¼6e+]*l w p ' ,0t !a dYÅ1Umi¢ ^  _  ¼ ¡ H J   O_ 2   U ¢ @  5t 'ND. ÀI '%=` À'3H‚o„o Žƒ… … ˆ : ƒ†fˆfŒVƒ/Î /Â?‚oh‚

6. REFERENCES

$

'(*)

+-, '.)

+

&%

'

0/

21

3

54

1

&+-, '.)76

(

81

9

+

81

;:

ƒ¤F†O¢.ƒ=%xmH 'µ§†f[„?[™'F[%Â@=,†f™%¸R„‚o@'…6°kƒF>Î 0P'Ž ‰QfŒ@h„o´ˆ#†a Š M …) ®.  J ; )   +xF '  =.I± %'-/  '+—€q3|)H0t@ A| ? = 2P„+ Œ)ÂI5 * $%*†a'Š