Triangular Mesh Decimation in Parallel Environment

EUROGRAPHICS Workshop on Parallel Graphics and Visualization, Girona, Spain, pp.39-52, ISBN 84-8458-025-3, 2000.

7ULDQJXODU0HVK'HFLPDWLRQLQ3DUDOOHO (QYLURQPHQW 0DUWLQ)UDQF9iFODY6NDOD ^PDUW\VNDOD`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

 0HVKGHFLPDWLRQ 

3RSXODUWHFKQLTXHV

'HFLPDWLRQ PHWKRGV DUH VLPSOLILFDWLRQ DOJRULWKPV WKDW VWDUW ZLWK D SRO\JRQLVDWLRQ W\SLFDOO\ D WULDQJXODWLRQ  DQG VXFFHVVLYHO\VLPSOLI\LWXQWLOWKHGHVLUHGOHYHORIDSSUR[LPDWLRQLVDFKLHYHG0RVWRIGHFLPDWLRQDOJRULWKPVIDOOLQWR RQHRIWKHWKUHHFDWHJRULHVGLVFXVVHGEHORZDFFRUGLQJWRWKHLUGHFLPDWLRQWHFKQLTXH

EUROGRAPHICS Workshop on Parallel Graphics and Visualization, Girona, Spain, pp.39-52, ISBN 84-8458-025-3, 2000.

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

(GJHGHFLPDWLRQ 2WKHUVXEVHWRIGHFLPDWLRQWHFKQLTXHVLVSRLQWHGWRWKHHOLPLQDWLRQRIWKHZKROHHGJH:KHQDQHGJHLVFRQWUDFWHG DVLQJOHYHUWH[UHSODFHVLWVHQGSRLQW7ULDQJOHVZKLFKGHJHQHUDWHWRDQHGJHDUHUHPRYHG+RSSH>@ZDVWKHILUVW ZKRKDVXVHGHGJHFRQWUDFWLRQDVWKHIXQGDPHQWDOPHFKDQLVPDFFRPSOLVKLQJVXUIDFHVLPSOLILFDWLRQ,WLVQHFHVVDU\ WRHYDOXDWHWKHLPSRUWDQFHRIHGJHVEHIRUHWKHFRQWUDFWLRQ2QHRIWKHEHVWNQRZQWHFKQLTXH>@XVHVTXDGULFHUURU PHWULFV IRU WKH HGJH YHUWH[ SDLUV  HYDOXDWLRQ 7KH HGJHV DUH FRQWUDFWHG DFFRUGLQJ WR WKHLU LPSRUWDQFH ± WKH OHVV LPSRUWDQW HGJHV ILUVW VLPLODUO\ WR WKH FDVH RI YHUWH[ UHPRYDO 8QOHVV WKH WRSRORJ\ LV H[SOLFLWO\ SUHVHUYHG HGJH FRQWUDFWLRQDOJRULWKPVPD\LPSOLFLWO\DOWHUWKHWRSRORJ\E\FORVLQJKROHVLQWKHVXUIDFH

3DWFKWULDQJOH GHFLPDWLRQPHWKRGV 7HFKQLTXHVZKLFKHOLPLQDWHHLWKHURQHWULDQJOHRUDQ\ODUJHUDUHDEHORQJWRWKHODVWJURXS7KHVH PHWKRGVGHOHWH VHYHUDO DGMDFHQW WULDQJOHV DQG WULDQJXODWH WKHLU ERXQGDU\ ,Q FDVH RI RQH WULDQJOH WKLV RQH LV GHOHWHG WRJHWKHU ZLWK WKUHHHGJHVDQGWKHQHLJKERXUKRRGLVWULDQJXODWHG7KHHYDOXDWLRQRIWKHUHGXFHGHOHPHQWVUHTXLUHV PRUHFRPSOH[ DOJRULWKPVLQWKHVHPHWKRGV



2XUDSSURDFK±IUDPHZRUNRIRXUDOJRULWKP

(DFKRIWKHDERYHPHQWLRQHGDSSURDFKHVKDYHWKHLUDGYDQWDJHVDQGGLVDGYDQWDJHV>@:HKDYHWULHGWRH[WUDFWWKH DGYDQWDJHVRIDOODSSURDFKHVDVZLOOEHSUHVHQWHGLQWKHIROORZLQJSDUW :H KDYH VWDUWHG ZLWK YHUWH[ GHFLPDWLRQ PHWKRGV DQG XVHG WKH 6FKURHGHU¶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¶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

EUROGRAPHICS Workshop on Parallel Graphics and Visualization, Girona, Spain, pp.39-52, ISBN 84-8458-025-3, 2000.

 0XOWLWKUHDGSURJUDPPLQJ :H KDYH GHYHORSHG WKH DOJRULWKP IRU WKH :LQGRZV 17 SODWIRUP XVLQJ %RUODQG 'HOSKL 7KHUH DUH TXLWH HDV\ DQG HIILFLHQWWRROVIRUPXOWLWKUHDGSURJUDPPLQJ,QWKLVVHFWLRQZHVKRUWO\LQWURGXFHWKHVWDQGDUG'HOSKLFODVV77KUHDG IRUPXOWLWKUHDGSURJUDPPLQJZKLFKKDVEHHQXVHGLQRXULPSOHPHQWDWLRQ



77KUHDGFODVV

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¶VJOREDOYDULDEOHV 1RWHWKDWZHKDYHREWDLQHGYHU\JRRGVSHHGXSRISDUDOOHOSDUWVZLWKWKHVWDQGDUGSURJUDPPLQJWRROV

 6XSHULQGHSHQGHQWVHW 

,QGHSHQGHQWVHW

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



6KDUHGPHPRU\SUREOHPV

7RVWRUHWKHLQIRUPDWLRQDERXWYHUWLFHVZHXVHDGDWDVWUXFWXUHVLPLODUWRDZLQJLQJHGJH:LWKHDFKYHUWH[LVNHSW WKHQXPEHUYHUWH[GHJUHH DQGLQGH[HVRIWULDQJOHVQHLJKERXUV WKDWVKDUHWKHYHUWH[ $IWHUVHYHUDOH[SHULPHQWVZHIRXQGWKDWWKHLQGHSHQGHQWVHWGHVFULEHGDERYHLVQRWIXOO\DGHTXDWHLQRXUFDVH 7KH FULWHULRQ RI QR VKDULQJ HGJH EHWZHHQ WZR LQGHSHQGHQW YHUWLFHV LV QRW VXIILFLHQW IRU HIIHFWLYH SDUDOOHOLVDWLRQ 8VLQJVXFKDQLQGHSHQGHQWVHWRIYHUWLFHVOHDGVWRVLWXDWLRQVZKHQZHKDYHWRXVHDQ\V\QFKURQLVDWLRQPHFKDQLVP WRSURWHFW WKH PHPRU\ VKDUHG E\ WKUHDGV 7ZR HOLPLQDWHG YHUWLFHV IURP WKH LQGHSHQGHQW VHW FDQ VKDUH WKH VDPH QHLJKERXUVVRWZRWKUHDGVFKDQJHDWWULEXWHVRIRQHQHLJKERXULQJYHUWH[DWRQFHGXULQJWKHWULDQJXODWLRQ7RDYRLG VXFK FRQIOLFWV ZH KDYH XVHG PHFKDQLVP FDOOHG FULWLFDO VHFWLRQV 6LQFH WKH FRPSXWDWLRQ LV TXLWH IDVW WKHVH FULWLFDO VHFWLRQVUDSLGO\GHFUHDVHWKHHIIHFWLYLW\RIWKHDOJRULWKP:HIRXQGWKDWWKHVHTXHQWLDODOJRULWKPLVHYHQIDVWHUWKDQ

EUROGRAPHICS Workshop on Parallel Graphics and Visualization, Girona, Spain, pp.39-52, ISBN 84-8458-025-3, 2000.

WKH SDUDOOHO RQH ZLWK FULWLFDO VHFWLRQV RQ WZR SURFHVVRUV PDFKLQHV HDFK SURFHVVRU ZRUNV RQO\ RQ   RI LWV SHUIRUPDQFH 



,QGHSHQGHQWVHW±ZLWKRXWQHHGRIFULWLFDOVHFWLRQV

,QYLHZRISUREOHPV H[SODLQHG LQ SUHYLRXV SDUDJUDSK ZH KDYH GHYHORSHG DQG XVH D VXSHU LQGHSHQGHQW VHW ZKHUH HYHU\WZRWULDQJOHVLQFOXGLQJWZRLQGHSHQGHQWYHUWLFHVFDQQRWVKDUHDQHGJHVHH)LJXUH

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

 $OJRULWKP 

6HTXHQWLDODOJRULWKP

+DYLQJ GHVFULEHG D IUDPHZRUN XSRQ ZKLFK RXU QHZ DOJRULWKP LV EDVHG ZH FDQ ORRN DW LW PRUH LQ GHWDLO IRU WKH SUHVHQWMXVWWKHVHTXHQWLDOYHUVLRQ 

q q q



 

7KH WRSRORJ\ RI HDFK YHUWH[ KDV WR EH HYDOXDWHG 7KH LPSRUWDQFH RI YHUWLFHV LV FRPSXWHG DFFRUGLQJ WRWKHLUWRSRORJ\XVLQJ6FKURHGHU¶VPHWKRG $OOWKHYHUWLFHVDUHVRUWHGLQDVFHQGLQJRUGHU RIWKHLULPSRUWDQFH 7KHYHUWLFHVDUHUHGXFHGLQRUGHUE\WKHLPSRUWDQFH)RUUHGXFWLRQZHQHHGWRVSHFLI\WKHRSWLPDOHGJH IRU FRQWUDFWLRQ ZKLFK JRHV IURP WKH UHGXFHG YHUWH[ 7KHQ WKH FRQVLVWHQF\ LV FKHFNHG ,I DQ\ RI WKH FRQVLVWHQF\ FKHFNV IDLOV RWKHU HGJH LV WR EH WDNHQ RWKHUZLVH WKH YHUWH[ LV UHPRYHG PRYHG WR WKH HQGSRLQWRIWKHFKRVHQHGJHDQGWZRQHLJKERXULQJWULDQJOHVDUHUHPRYHG 

7KHZKROHSURFHVVLVUHSHDWHGXQWLOWKHGHVLUHDSSUR[LPDWLRQLVUHDFKHG



$OJRULWKPDQDO\VLV

2XU RULJLQDO JRDO ZDV WR LPSOHPHQW DQ HIILFLHQW WULDQJOH PHVK VLPSOLILFDWLRQ DOJRULWKP LQ SDUDOOHO HQYLURQPHQW LQ RUGHUWRREWDLQIDVWHUUHVSRQVHIRUYHU\ODUJHGDWDVHWV :HFDQVHHWKDWRXUVHTXHQWLDODOJRULWKPGHVFULEHGLQSUHYLRXVDUWLFOHZRUNVLQWKUHHVWHSV   $FFRUGLQJWRWKHYHUWH[SRVLWLRQLQDPHVKZHUHFRJQL]HW\SHVRIYHUWLFHVVLPSOHERXQGDU\FRUQHULQWHULRUHGJH DQGFRPSOH[

EUROGRAPHICS Workshop on Parallel Graphics and Visualization, Girona, Spain, pp.39-52, ISBN 84-8458-025-3, 2000.

q q q







9HUWH[WRSRORJ\DQGLPSRUWDQFHHYDOXDWLRQ7KLVSDUWFDQEHGRQHLQSDUDOOHOEHFDXVHDOOYHUWLFHVDUH REYLRXVO\LQGHSHQGHQWWRWKHRWKHUV 6RUWLQJ:KHQWKLVSDSHULVZULWWHQWKHUHLVDQLPSOHPHQWDWLRQXVLQJVHTXHQWLDO4XLFN6RUWDOJRULWKP 7KH UHDVRQ LV WKDW WKLV ZRUN LV VWLOO LQ SURJUHVV :H NQRZ WKDW WKHUH DUH ZHOONQRZQ SDUDOOHO VRUWLQJ DOJRULWKPVVRZHKDYHSRLQWHGRXUHIIRUWWRWKHRWKHUSDUWVRIWKHDOJRULWKPIRUHPRVW 'HFLPDWLRQ7KLV SDUW RI WKH DOJRULWKP FRXOG UXQ HIIHFWLYHO\ LQ SDUDOOHO EXW ZH KDYH WR H[SHFW VRPH LPSRUWDQWUHVWULFWLRQV

8VLQJ WKH LQGHSHQGHQW VHW RI YHUWLFHV ZH FDQ VSOLW WKLUG VWHS RI WKH DOJRULWKP WKH GHFLPDWLRQ  LQ WR WZR SDUWV ±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



3DUDOOHOYHUVLRQRIWKHDOJRULWKP

2XUQHZSDUDOOHODOJRULWKPFDQEHGHVFULEHGDV  'LYLGHWKHVHWRIYHUWLFHVLQWR1SDUWVZKHUH1LVHTXDOWRWKHQXPEHURIIUHHSURFHVVRUV q *HWWKHQXPEHURISURFHVVRUV q 'LYLGHWKHVHWRIYHUWLFHVLQWR1SDUWVRIWKHVDPHQXPEHURIYHUWLFHV  5XQ1WKUHDGVWRHYDOXDWHYHUWH[LPSRUWDQFHDFFRUGLQJWRLWVWRSRORJ\(DFKWKUHDGPDNHVDFRPSXWDWLRQRQ LWVRZQVHWRIYHUWLFHV q 'HWHUPLQHDYHUWH[WRSRORJ\ q )RUVLPSOHERXQGDU\RULQWHULRUHGJHYHUWLFHVFRPSXWHWKHLULPSRUWDQFH7KHLPSRUWDQFHIRUDQ\RWKHU W\SHRIYHUWH[LVVHWWRDQ\KLJKYDOXH³LQILQLWH´   $IWHUDOOWKUHDGVILQLVKWKHLUWDVNVRUW4XLFN6RUWDOJRULWKP DOOWKHYHUWLFHVDFFRUGLQJWRWKHLULPSRUWDQFH LQLQFUHDVLQJRUGHU  )LQGDVXSHULQGHSHQGHQWVHWRIYHUWLFHV)RUHDFKYHUWH[GR q ,IWKHYHUWH[LVPDUNDVXQXVHGLQWKHLQGHSHQGHQWVHWLQLWLDOO\DOOYHUWLFHVDUHPDUNHGDVXQXVHG FKHFN LWVQHLJKERXUV,IDOOWKHQHLJKERXUVDUHXQXVHGSXWWKHYHUWH[LQWRWKHVXSHULQGHSHQGHQWVHWDQGPDUN WKHYHUWH[DQGDOOLWVQHLJKERXUYHUWLFHVDVXVHG q 8VHGYHUWLFHVDQGWKHLUQHLJKERXUVDUHVNLSSHG  'LYLGHWKHVXSHULQGHSHQGHQWVHWRIYHUWLFHVLQWR1SDUWV  5XQ1WKUHDGVIRUGHFLPDWLRQ(DFKWKUHDGPDNHVWKHGHFLPDWLRQRQLWVRZQVHWRIYHUWLFHV q )RUHDFKHOLPLQDWHGYHUWH[ILQGWKHRSWLPDOHGJHIRUFRQWUDFWLRQWKDWLQFOXGHVLW q 7HVWWKHFRQVLVWHQF\RIWKHPHVKLIWKLVHGJHLVFRQWUDFWHGUHPRYHG  q ,IWKHFRQVLVWHQF\WHVWLV2.UHPRYHWKHYHUWH[DQGWULDQJXODWHWKHDULVLQJKROHRWKHUZLVHILQGDQRWKHU VKRUWHGJHDQGJRWRWKHSUHYLRXVSRLQW  5HSHDWVWHSV±XQWLOWKHUHTXLUHGGHJUHHRIWKHPHVKUHGXFWLRQLVUHDFKHG

EUROGRAPHICS Workshop on Parallel Graphics and Visualization, Girona, Spain, pp.39-52, ISBN 84-8458-025-3, 2000.

 ([SHULPHQWDOUHVXOWV ,Q WKLV VHFWLRQ ZH SUHVHQW UHVXOWV RI RXU H[SHULPHQWV WKDW FRPSDUH DQ DFFHOHUDWLRQ DQG HIILFLHQF\ REWDLQHG ZLWK GLIIHUHQWGDWDVHWVXVLQJGLIIHUHQWQXPEHURISURFHVVRUVWKUHDGV  :HXVHGLIIHUHQWGDWDVHWVIRUWKHH[SHULPHQWVVHH7DEOH 0RGHOQDPH 1RRIWULDQJOHV 1R2IYHUWLFHV +RUVH   %RQH   %HOO   +DQG   'UDJRQ   +DSS\E   7DEOH'DWDVHWVXVHG 7DEOHVKRZVWLPHFRPSDULVRQDFKLHYHGE\UHGXFLQJWKHPRGHOVXVLQJWR±SURFHVVRUFRPSXWHU'(//3RZHU (GJH±QWLXP,,,FDFKH0%0+]*%5$0UXQQLQJRQWKH:LQGRZV  0RGHO QDPH

7LPH>VHF@REWDLQHGZLWKGLIIHUHQWQXPEHURI SURFHVVRUVWKUHDGV XVHG         +RUVH         %RQH         %HOO         +DQG         'UDJRQ         +DSS\E         7DEOH2EWDLQHGWLPHLQVHFRQGV IRUUHGXFWLRQRQWRSURFHVVRUVDFWLYH :H LQYHVWLJDWHG WKH DFFHOHUDWLRQ DQG WKH HIILFLHQF\ IRU GLIIHUHQW VL]H RI GDWD VHWV DFFRUGLQJ WR WKH QXPEHU RI SURFHVVRUVXVHG



6SHHGXSFRPSDULVRQ

)LJXUHDVKRZVD JUDSKRIWKHVSHHGXSFRPSDULVRQ 7KH VSHHGXS D LV FRPSXWHG IURP WRWDO WLPHV VHTXHQWLDO DQG SDUDOOHOSDUWVRIWKHDOJRULWKPWRJHWKHU XVLQJH[SUHVVLRQ 

D=

WLPH  WLPH1



ZKHUH1 LVDQXPEHURISURFHVVRUVXVHGDQGWLPH1LVWKHWLPHREWDLQHGLI1SURFHVVRUVWKUHDGV DUHXVHG )LJXUH E VKRZV WKH VSHHGXS RI LQGLYLGXDO VHTXHQWLDO DQG SDUDOOHO SDUW RI WKH DOJRULWKP IRU WKH +DSS\E PRGHO 7RWDO WLPH PHDQV UXQ WLPH RI WKH ZKROH DOJRULWKP ,PS HYDO LV WLPH IRU LPSRUWDQFH HYDOXDWLRQ 4VRUW LV VHTXHQWLDOVRUWLQJWLPH,VHWLVWLPHIRUVHOHFWLQJWKHVXSHULQGHSHQGHQWVHWDQG'HFLPLVWLPHRIWKHGHFLPDWLRQSDUW

EUROGRAPHICS Workshop on Parallel Graphics and Visualization, Girona, Spain, pp.39-52, ISBN 84-8458-025-3, 2000.

7RWDOVSHHGXS 



 KRUVH

S X G HH S V

ERQH EHOO



KDQG GUDJRQ KDSS\E



,'($/ &$6( 





















QXPEHURIWKUHDGVSURFHVVRUVXVHG

D

6SHHGXS +DSS\E PRGHO 







7RWDOWLPH ,PSHYDO 4VRUW ,VHW 'HFLP ,'($/&$6(

S X G H H S V





























QXPEHURIWKUHDGV

E )LJXUH  7KH DFFHOHUDWLRQ RI WRWDO FRPSXWDWLRQ WRWDO WLPH  SDUDOOHO DQG VHTXHQWLDO SDUWV WRJHWKHU WKH DFFHOHUDWLRQ LV FRPSXWHG IRU VHYHUDO PRGHOV RI GLIIHUHQW DPRXQW RI WULDQJOHV D  DQG FRPSDULVRQ RI DFFHOHUDWLRQRISDUDOOHODQGVHTXHQWLDOFRGHFRPSXWHGIRUWKH+DSS\EPRGHOE 

EUROGRAPHICS Workshop on Parallel Graphics and Visualization, Girona, Spain, pp.39-52, ISBN 84-8458-025-3, 2000.

,Q)LJXUH\RXFDQVHHWKHDFWXDOUXQWLPHVLQSHUFHQW RIWKHYDULRXVDOJRULWKPVWHSVIRU+DSS\EPRGHO@DQGWKHGLIIHUHQFHDPRQJWKHPLVWKHPHWKRGRIYHUWH[ LPSRUWDQFH HYDOXDWLRQ 7KH IROORZLQJ PHWKRGV KDYH EHHQ XVHG 6FKURHGHUµV PHWKRG $YHUDJH 1RUPDOV +HLJKW GLIIHUHQFH$EVROXWH%LQDU\6FDSH2UGHU5DQGRPDQ\RISUHYLRXVILYH DQG(GJH6ZLWFKLQJ

EUROGRAPHICS Workshop on Parallel Graphics and Visualization, Girona, Spain, pp.39-52, ISBN 84-8458-025-3, 2000.

$OJRULWKPVWDELOLW\    

6FKURHGHU $YHUDJHQRUPDOV +HLJKW'LIIHUHQFH $EVROXWH%LQDU\ 6FDSH2UGHU 5DQGRP (GJH6ZLWFKLQJ

S X HG H S V    

,PSHYDO

4VRUW

,VHW

'HFLP

7RWDOWLPH

,PSHYDO

4VRUW

,VHW

'HFLP

7RWDOWLPH

SDUWLDOFRPSXWDWLRQ

)LJXUH7KHVSHHGXSRISDUWLDOSDUWVRIWKHDOJRULWKPIRUVHYHUDOYHUWH[LPSRUWDQFHHYDOXDWLRQKHXULVWLFV &ULWHULRQRIPLQLPDODUHDWULDQJXODWLRQRQWKHOHIWVLGHWKHVKRUWHVWHGJHFRQWUDFWLRQFULWHULRQRQWKHULJKW VLGHRIWKHJUDSK ,WLVREYLRXVWKDWWKHDOJRULWKPLVLQGHSHQGHQWRIWKHKHXULVWLFXVHGLQPHDQLQJRIWKHHIILFLHQF\DQGWKHVSHHGXS



3URJUDPRXWSXW

)LJXUHDQG)LJXUHVKRZH[DPSOHVRIRULJLQDODQGUHGXFHGPRGHOV

)LJXUH  $ERQH PRGHO FRXUWHV\ &\EHUZDUH  DW GLIIHUHQW UHVROXWLRQV WKH RULJLQDO PRGHO ZLWK  WULDQJOHV D UHGXFHGWRWULDQJOHVE WULDQJOHVF WULDQJOHVG 

EUROGRAPHICS Workshop on Parallel Graphics and Visualization, Girona, Spain, pp.39-52, ISBN 84-8458-025-3, 2000.

)LJXUH7KH+DSS\EPRGHOFRXUWHV\*D7HFK DWGLIIHUHQWUHVROXWLRQVWKHRULJLQDOPRGHOZLWKWULDQJOHV D UHGXFHGWRWULDQJOHVE WULDQJOHVF WULDQJOHVG 

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

$FNQRZOHGJHPHQWV :H ZRXOG OLNH WR H[SUHVV RXU WKDQNV WR '(// &RPSXWHU &]HFK 5HSXEOLF IRU HQDEOLQJ XV WR FDUU\ RXW DOO WKHH[SHULPHQWVRQWKHLUSURFHVVRUFRPSXWHUW\SH'(//3RZHU(GJH±QWLXP,,,FDFKH0%0+] *% 5$0 :H DOVR EHQHILWHG IURP WKH ODUJH PRGHO UHSRVLWRU\ ORFDWHG DW *HRUJLD ,QVWLWXWH RI 7HFKQRORJ\ 85/ KWWSZZZFFJDWHFKHGXSURMHFWVODUJHBPRGHOV

EUROGRAPHICS Workshop on Parallel Graphics and Visualization, Girona, Spain, pp.39-52, ISBN 84-8458-025-3, 2000.

5HIHUHQFHV >@ : 6FKURHGHU - =DUJH : /RUHQVHQ 'HFLPDWLRQ RI 7ULDQJOH 0HVKHV ,Q 6,**5$3+  &RQIHUHQFH 3URFHHGLQJVSDJHV-XO\ >@ 0 *DUODQG 0XOWLUHVROXWLRQ 0RGHOLQJ 6XUYH\  )XWXUH 2SSRUWXQLWLHV ,Q WKH 6,**5$3+  FRXUVH QRWHV  >@ 0*DUODQG3+HFNEHUW6XUIDFH6LPSOLILFDWLRQ8VLQJ4XDGULF(UURU0HWULFV,Q6,**5$3+&RQIHUHQFH 3URFHHGLQJV >@ '.LUNSDWULFN2SWLPDO6HDUFKLQ3ODQDU6XEGLYLVLRQV6,$0-&RPSSDJHV >@ % -XQJHU - 6QRH\LQN 6HOHFWLQJ ,QGHSHQGHQW 9HUWLFHV IRU 7HUUDLQ 6LPSOLILFDWLRQ ,Q :6&*  3URFHHGLQJV 3LOVHQ8QLYHUVLW\RI:HVW%RKHPLDSDJHV)HEUXDU\ >@ + +RSSH 7 'H5RVH 7 'XFKDPS - 0F'RQDOG : 6WXHW]OH 0HVK RSWLPL]DWLRQ ,Q 6,**5$3+  &RQIHUHQFH3URFHHGLQJVSDJHV >@ + 3 6HLGHO 6 &DPSDJQD / .REEHOW 5 6FKQHLGHU - 9RUVDW] 0HVK 5HGXFWLRQ DQG ,QWHUDFWLYH 0XOWLUHVROXWLRQ 0RGHOLQJ RQ $UELWUDU\ 7ULDQJOH 0HVKHV ,Q 6&&*  3URFHHGLQJV &RPHQLXV 8QLYHUVLW\ %UDWLVODYDSDJHV 7KLV ZRUN ZDVVXSSRUWHGE\7KH0LQLVWU\RI(GXFDWLRQ RI 7KH &]HFK 5HSXEOLF ± SURMHFW 96  $FDGHP\ RI 6FLHQFHVRI7KH&]HFK5HSXEOLF±SURMHFW$