Quantum Algorithm for Finding a Maximum Clique in an Undirected ...

-,2692/12 



68%0,77('$&&(37(' 



















2ULJLQDO6FLHQWL¿F3DSHU

4XDQWXP$OJRULWKPIRU)LQGLQJD0D[LPXP&OLTXHLQDQ 8QGLUHFWHG*UDSK $ODQ%RMLü

DODQERMLF#LQKU

,1/WG 0DURKQLFHYD+5=DJUHE&URDWLD

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ܱሺȁܸȁඥʹȁ௏ȁ ሻZRUVWFDVHWLPHFRPSOH[LW\DQGܱሺඥʹȁ௏ȁ ሻEHVW FDVHWLPHFRPSOH[LW\$OJRULWKP VVSDFHFRPSOH[LW\IRUHDFKFDVHLVܱሺȁܸȁሻ .H\ZRUGVTXDQWXPFRPSXWLQJTXDQWXPDOJRULWKPPD[LPXPFOLTXH

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ܱሺඥܰሻ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ܱሺሺŽ‘‰ ܰሻ ଷ ሻZKLFKLVH[SRQHQWLDOO\IDVWHUWKDQDQ\NQRZQDOJRULWKPIRULQWHJHUIDFWRUL]DWLRQ IRU D FODVVLFDO FRPSXWHU 6KRU V DOJRULWKP UHSUHVHQWV RQH RI WKH VWURQJHVW SURRIV WKDW D 


























































%2-,û48$1780$/*25,7+0)25),1',1*$0$;,080&/,48(

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ܱሺȁܸȁଶ ሻ WLPH FRPSOH[LW\ IRU JUDSKV ZKHUH QXPEHU RI YHUWLFHV LV ȁܸȁ ൑ ͳͺ +RZHYHU WKH\ GLGQ WLQWURGXFHWKHWLPHFRPSOH[LW\RIWKHSURSRVHGDOJRULWKPIRUDQDUELWUDU\ODUJHJUDSK :HQJ/RQJ&KDQJ7LQJ7LQJ5HQ0DQJ)HQJ-XQ/XR.DZXX:HLFKHQJ/LQ0LQ\L*XR DQG /DL &KLQ /X SURSRVHG D TXDQWXP DOJRULWKP WKDW VLPXODWHV '1$EDVHG DOJRULWKP IRU ILQGLQJPD[LPXPFOLTXHSUREOHP>@7KHWLPHFRPSOH[LW\RIWKHLUDOJRULWKPLVܱሺȁܸȁඥʹȁ௏ȁ ሻ ,Q WKLV SDSHU , ZRXOG OLNH WR WDNH D GLIIHUHQW DSSURDFK WKDQ DXWKRUV RI H[LVWLQJ TXDQWXP DOJRULWKPVIRUPD[LPXPFOLTXHDQGSURSRVHDQHZTXDQWXPDOJRULWKPIRUPD[LPXPFOLTXHLQ DQ DUELWUDU\ JUDSK ZKLFK LV EDVHG RQ *URYHU V DOJRULWKP &RPSOH[LW\ RI D QHZ TXDQWXP DOJRULWKPZLOOEHDQDO\]HGDQGFRPSDUHGWRWKHFRPSOH[LW\RIH[LVWLQJTXDQWXPDOJRULWKPV IRUWKHPD[LPXPFOLTXHSUREOHP *URYHU VDOJRULWKPLVGHVFULEHGLQVHFWLRQZKHUHDVVHFWLRQFRQWDLQVWKHORJLFRIWKH SURSRVHG DOJRULWKP WLPH DQG VSDFH FRPSOH[LW\ DQDO\VLV RI WKH SURSRVHG DOJRULWKP LV GHVFULEHGLQVHFWLRQDQGVHFWLRQSURYLGHVWKHFRQFOXVLRQDQGUHIHUHQFHWRIXWXUHUHVHDUFK

 *URYHU V$OJRULWKP *URYHU VDOJRULWKPLVDZD\RIILQGLQJDQHOHPHQWLQDQXQVRUWHGOLVWZLWK1HOHPHQWVXVLQJ WKH TXDQWXP FRPSXWHU ,W ZDV GLVFRYHUHG DQG SURSRVHG E\ /RY *URYHU LQ  ZKR GHVFULEHGLWLQKLVIDPRXVZRUN$IDVWTXDQWXPPHFKDQLFDODOJRULWKPIRUGDWDEDVHVHDUFK >@$QDOJRULWKPLVEDVHGRQDQDPSOLWXGHDPSOLILFDWLRQRIWKHEDVHVWDWHZKLFKGHVFULEHVD SRVLWLRQ RI D VHDUFKHG HOHPHQW LQ WKH OLVW 7KH WLPH FRPSOH[LW\ RI *URYHU V DOJRULWKP LV ܰ ܱሺඥܰሻFRPSDUHGZLWKܱሺ ሻWLPHFRPSOH[LW\IRUDQDYHUDJHFDVHRQDFODVVLFDOFRPSXWHU,W ʹ

KDVDOVREHHQVKRZQWKDWWKHWLPHFRPSOH[LW\RIܱሺඥܰሻLVRSWLPDOIRUDTXDQWXPFRPSXWHU IRUWKHXQVRUWHGOLVWVHDUFKLQJSUREOHP>@(DFKLWHUDWLRQRI*URYHU VDOJRULWKPDPSOLILHVWKH SUREDELOLW\DPSOLWXGHRIWKHEDVHVWDWHZKLFKGHVFULEHVDSRVLWLRQRIDQHOHPHQWLQWKHOLVW 4XDQWXPFLUFXLWIRUDVLQJOHLWHUDWLRQZLWKLQ*URYHU VDOJRULWKPLVVKRZQLQ)LJXUH>@ 

 )LJXUH4XDQWXPFLUFXLWIRUDVLQJOHLWHUDWLRQRI*URYHU VDOJRULWKP



,WHUDWLRQRI*URYHU VDOJRULWKPXVHV2UDFOHZKLFKLGHQWLILHVWKHSRVLWLRQRIWKHVHDUFKHG HOHPHQWZKLFKLVVWRUHGLQVLGHDTXDQWXPUHJLVWHURIVL]H݊ ൌ Ž‘‰ ଶ ܰTXELWV2UDFOHXVHVWKH IXQFWLRQI[ ZKLFKUHWXUQVLIVWDWH[LVWKHRQHVHDUFKHGRULIVWDWH[LVQ WWKHRQHVHDUFKHG 2UDFOH DOVR UHTXLUHV LWV RZQ TXDQWXP UHJLVWHU ZRUNVSDFH  IRU LWV RSHUDWLRQV 2YHUDOO VSDFH 

 



!

"

#

$

!

%

&

'

(

#

)

!

&

*

+

*

,

-

*

.

#

/

/

&

0

-

1

0

2

-2851$/2),1)250$7,21$1'25*$1,=$7,21$/6&,(1&(6

FRPSOH[LW\RIDVLQJOH*URYHULWHUDWLRQLVܱሺ݊ሻ$IWHU2UDFOHKDVILQLVKHG+DGDPDUG VJDWH RQ Q TXELWV 3KDVH VKLIW JDWH DQG WKHQ DJDLQ +DGDPDUG V JDWH RQ Q TXELWV ZLOO DPSOLI\ WKH ʹ SUREDELOLW\ RI VHDUFKHG EDVH VWDWH IRU ܱሺ ሻ $IWHU ܱሺඥܰሻ LWHUDWLRQV SUREDELOLW\ RI WKH ξܰ GHVLUDEOHEDVHVWDWHZLOOEHDQGZHFRQGXFWDPHDVXUHPHQWRIWKHTXDQWXPUHJLVWHUILUVWQ TXELWV )LJXUH>@VKRZVWKHTXDQWXPFLUFXLWIRU*URYHU VDOJRULWKP 

 )LJXUH4XDQWXPFLUFXLWIRU*URYHU VDOJRULWKP

 7KH LQLWLDO VWDWH IRU *URYHU V DOJRULWKP LV ȁ߰଴ ‫ ۄ‬ൌ ȁͲ‫ ۄ‬%HIRUH WKH H[HFXWLRQ RI WKH ILUVW *URYHU VLWHUDWLRQ+DGDPDUG VJDWHSXWVWKHV\VWHPLQDXQLIRUPVXSHUSRVLWLRQRIDOOSRVVLEOH EDVHVWDWHVDVVKRZQLQWKHIROORZLQJHTXDWLRQ ȁ߰‫ ۄ‬ൌ

ଶ೙ ିଵ

ͳ

෍ ȁ‫ۄݔ‬Ǥ

ξʹ௡

௫ୀ଴

 ,WFDQDOVREHVKRZHGWKDWLIWKHHOHPHQWZKLFKZHDUHVHDUFKLQJIRULQDQXQVRUWHGOLVW DSSHDUVWWLPHVWKHQDOJRULWKPFDQSURJUHVVLYHO\LQFUHDVHWKHVHWRIQXPEHUVIURPZKLFKZH XQLIRUPO\FKRRVHDQXPEHURILWHUDWLRQVIRU*URYHU VDOJRULWKP7KHWLPHFRPSOH[LW\RIVXFK ܰ

D YHUVLRQ RI *URYHU V DOJRULWKP LV ܱሺට ሻ 7KH IROORZLQJ VWHSV GHVFULEH WKH YHUVLRQ RI ‫ݐ‬

*URYHU V DOJRULWKP IRU DQ XQNQRZQ QXPEHU RI VROXWLRQV SURSRVHG E\ 0LFKHO %R\HU *LOOHV %UDVVDUG3HWHU+¡\HUDQG$ODLQ7DSSLQWKHLUZHOONQRZQSDSHU7LJKWERXQGVRQTXDQWXP VHDUFKLQJ>@   /HW7EHWKHXQVRUWHGOLVWDQG[WKHVHDUFKHGYDOXH,QLWLDOL]HP DQGVHWȜ  $Q\YDOXHRIȜVWULFWO\EHWZHHQDQGZRXOGGR   &KRRVHMXQLIRUPO\DWUDQGRPDPRQJWKHQRQQHJDWLYHLQWHJHUVVPDOOHUWKDQP  ,QLWLDOL]HTXDQWXPUHJLVWHURIWKHVL]H݊ ൌ Ž‘‰ ଶ ܰLQVWDWHȁ߰଴ ‫ ۄ‬ൌ ȁͲ‫ۄ‬DQGWKHQXVH +DGDPDUG V JDWH RQ HYHU\ TXELW WR SXW WKH TXDQWXP UHJLVWHU LQ WKH XQLIRUP ೙ ିଵ ଵ VXSHUSRVLWLRQRIEDVHVWDWHVȁ߰‫ ۄ‬ൌ ೙ σଶ௫ୀ଴ ȁ‫ۄݔ‬ ξଶ  $SSO\MLWHUDWLRQVRI*URYHU¶VDOJRULWKP  2EVHUYHWKHUHJLVWHUOHWLEHWKHRXWFRPH  ,I7>L@ [WKHSUREOHPLVVROYHG([LW  2WKHUZLVHVHWPWR݉݅݊ሺߣ݉ǡ ξܰሻDQGJREDFNWRVWHS

 3URSRVHG$OJRULWKP

7KHPD[LPXPFOLTXHLQDQXQGLUHFWHGJUDSK* 9( LVWKHODUJHVWVXEVHWRIDVHWRIJUDSK V YHUWLFHV ZKHUH HDFK SDLU RI HOHPHQWV LQ WKH VXEVHW LV FRQQHFWHG >@ 3URSRVHG TXDQWXP DOJRULWKP IRU ILQGLQJ D PD[LPXP FOLTXH LQ DQ DUELWUDU\ XQGLUHFWHG JUDSK LV EDVHG RQ WKH YHUVLRQ RI *URYHU V DOJRULWKP ZKLFK ILQGV WKH VHDUFKHG HOHPHQW LQ DQ XQVRUWHG OLVW LQ WLPH ܰ

ܱሺට ሻZKHUHWLVWKHQXPEHURIHOHPHQWVZKLFKVDWLVI\WKHVHDUFKFULWHULD%HIRUHGHVFULELQJ ‫ݐ‬





























































3

%2-,û48$1780$/*25,7+0)25),1',1*$0$;,080&/,48(

WKH VWHSV RI WKH SURSRVHG DOJRULWKP LW LV QHFHVVDU\ WR GHILQH WZR LPSRUWDQW WKLQJV IRU WKH DOJRULWKPWKHLQIRUPDWLRQWKDWZLOOEHVWRUHGLQWKHTXDQWXPUHJLVWHUDQGWKHSURFHVVLQJORJLF RIWKHDOJRULWKP  ,QIRUPDWLRQ6WRUHG,QVLGHWKH4XDQWXP5HJLVWHU $OO SRVVLEOH VXEVHWV RI D VHW RI JUDSK V YHUWLFHV VKRXOG EH VWRUHG LQ D TXDQWXP UHJLVWHU ZLWKWKHVDPHSUREDELOLW\DPSOLWXGH7KHUHDUHʹȁ௏ȁ SRVVLEOHVXEVHWVVRWKHUHIRUH_9_TXDQWXP ELWV ZLOO EH QHHGHG WR VWRUH VXFK LQIRUPDWLRQ 7KH DOJRULWKP ZLOO DFKLHYH WKLV TXDQWXP UHJLVWHU VVWDWHE\DSSO\LQJ+DGDPDUG VRSHUDWRURQHYHU\TXELWLQWKHTXDQWXPUHJLVWHUZKLFK LVLQLWLDOL]HGLQWKHVWDWHȁ߰଴ ‫ ۄ‬ൌ ȁͲ‫ۄ‬,W VLPSRUWDQWWRHPSKDVL]HWKDWWKHTXDQWXPUHJLVWHUZLOO KDYHʹȁ௏ȁ SRVVLEOHEDVHVWDWHVDQGZKHQZHPHDVXUHWKHTXDQWXPUHJLVWHUZHZLOOUHFHLYHRQH RIWKHʹȁ௏ȁ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ȁ‫ۄݎ‬DQGUXQVDJDLQ,QLWLDOO\ WKHYDOXHLQWKHYHFWRUȁ‫ۄݎ‬ZLOOEHȁͲ‫ۄ‬7KHDOJRULWKPLWHUDWHVWKURXJKWKHVHVWHSVXQWLOWKHVL]H RI WKH UHWULHYHG FOLTXH LV QRW ODUJHU WKDQ WKH YDOXH VWRUHG LQ WKH YDULDEOH N RU WKH UHVXOWLQJ VXEVHWLVQRWDJUDSK VFOLTXH)LQDOO\WKHDOJRULWKPUHWXUQVWKHYHFWRUȁ‫ۄݎ‬DQGLILWLVGLIIHUHQW WKDQȁͲ‫ۄ‬RQHRIWKHPD[LPXPFOLTXHVRIDJLYHQJUDSKLVIRXQG  7KHVWHSVRIWKHDOJRULWKPDUHDVIROORZV  ,QLWLDOL]HYDULDEOHN DQGYHFWRUȁ‫ ۄݎ‬ȁͲ‫ۄ‬  'HILQH IXQFWLRQ I[  IRU DQ RUDFOH ZKLFK ZLOO EH XVHG LQVLGH *URYHU V DOJRULWKP )XQFWLRQ I[  UHWXUQV LI EDVH VWDWH [ UHSUHVHQW DFOLTXH RI WKH JUDSK DQG LWV VL]H LV ODUJHUWKDQYDOXHVWRUHGLQWKHYDULDEOHNRWKHUZLVHIXQFWLRQUHWXUQV  ,QLWLDOL]H TXDQWXP UHJLVWHU RI VL]H ݊ ൌ Ž‘‰ ଶ ܰ LQ VWDWH ȁ߰଴ ‫ ۄ‬ൌ ȁͲ‫ ۄ‬DQG WKHQ XVH +DGDPDUG V JDWH RQ HYHU\ TXELW WR SXW WKH TXDQWXP UHJLVWHU LQ WKH XQLIRUP ೙ ିଵ ଵ VXSHUSRVLWLRQRIEDVHVWDWHVȁ߰‫ ۄ‬ൌ ೙ σଶ௫ୀ଴ ȁ‫ۄݔ‬ ξଶ  5XQ*URYHU VDOJRULWKPIRUDQXQNQRZQQXPEHURIVROXWLRQV  0HDVXUHTXDQWXPUHJLVWHU,IRXWFRPHLVDFOLTXHDQGLWVVL]HLVODUJHUWKDQYDOXHLQ WKHYDULDEOHNWKHQSXWRXWFRPHLQWKHYHFWRUȁ‫ۄݎ‬DQGFOLTXHVL]HLQYDULDEOHNDQGJR WRVWHSRWKHUZLVHJRWRVWHS  5HWXUQYHFWRUȁ‫ۄݎ‬

 &RPSOH[LW\$QDO\VLV  6SDFH&RPSOH[LW\ $Q DOJRULWKP QHHGV D _9_ TXELW UHJLVWHU LQ WKH XQLIRUP VXSHUSRVLWLRQ VWDWH WR VWRUH DOO VXEVHWV RI D VHW RI JUDSK V YHUWLFHV 7KH DOJRULWKP DOVR QHHGV VSDFH WR VWRUH WKH VL]H RI FXUUHQWO\ IRXQG FOLTXH ZKLFK LV UHWXUQHG E\ DOJRULWKP V LWHUDWLRQ %HFDXVH WKH VL]H RI WKH PD[LPXPFOLTXHRIJUDSKLVPD[LPDOO\_9_DOJRULWKPQHHGVFODVVLFDOŽ‘‰ ଶ ȁܸȁELWVIRUYDULDEOH N WR VWRUH VXFK LQIRUPDWLRQ 7KH DOJRULWKP ZLOO DOVR QHHG DGGLWLRQDO FODVVLFDO _9_ ELWV IRU YHFWRUȁ‫ۄݎ‬WRVWRUHWKHODVWVXFFHVVIXOO\IRXQGFOLTXHRIDJUDSK7RVXPPDUL]HDQDOJRULWKP QHHGVRQH_9_TXELWUHJLVWHURQH_9_ELWFODVVLFUHJLVWHUDQGRQHŽ‘‰ ଶ ȁܸȁFODVVLFDOUHJLVWHUWKDW LVWKHRYHUDOOFRPSOH[LW\RIDOJRULWKPLV2_9_ ,WLVLPSRUWDQWWRHPSKDVL]HWKDWWKHVSDFH 



6

4

5

7

8

9

:

;

8




?

:

@

8

=

A

B

A

C

D

A

E

:

F

F

=

G

D

H

G

I

-2851$/2),1)250$7,21$1'25*$1,=$7,21$/6&,(1&(6

FRPSOH[LW\RIDOJRULWKPVWHSLVQRWUHOHYDQWEHFDXVHZHFDQGHWHUPLQHWKHVL]HRIWKHVXEVHW DQGFKHFNLIVRPHVXEVHWRIDVHWRIJUDSK VYHUWLFHVLVFOLTXHLQSRO\QRPLDOVSDFHRI_9_HYHQ RQDFODVVLFDOFRPSXWHU  7LPH&RPSOH[LW\ )RUWKHSXUSRVHVRIDWLPHFRPSOH[LW\DQDO\VLVSOHDVHFRQVLGHUWKHIROORZLQJXQGLUHFWHG JUDSK 

 )LJXUH*UDSK*ZKHUHȦ*  _9_

,WVPD[LPXPFOLTXHLQFOXGHVDOOJUDSK VYHUWLFHVDQGLWVFOLTXHQXPEHULVȦ* Ȧ* LV WKHFOLTXHQXPEHURIJUDSK* 9( WKDWLVWKHQXPEHURIYHUWLFHVLQDPD[LPXPFOLTXHLQ JUDSK*>@ ,Q DQ DUELWUDU\ JUDSK * 9(  WKHUH DUH DW OHDVW ʹఠሺீሻ െ ሺ߱ሺ‫ܩ‬ሻ ൅ ͳሻ FOLTXHV 7KLV LV EHFDXVH HYHU\ VXEVHW RI D PD[LPXP FOLTXH ZKLFK KDV PRUH WKDQ RQH HOHPHQW LV DOVR D JUDSK VFOLTXH)RUWKHJUDSKVKRZQLQ)LJXUHȦ*  WKHUHDUHDWOHDVWʹସ െ ሺͶ ൅ ͳሻ  FOLTXHVLQWKHJUDSK 6LQFH WKH ORJLF RI WKH DOJRULWKP LV WR LWHUDWLYHO\ LQFUHDVH WKH VL]H RI D VHDUFKHG JUDSK V FOLTXH WKHUH FRXOG EH PD[LPDOO\ Ȧ*  LWHUDWLRQV RI DQ DOJRULWKP $OJRULWKP V LWHUDWLRQV LQ ZRUVWFDVHVFHQDULRDUH   ,WHUDWLRQ  DOJRULWKP ILQGV D JUDSK V FOLTXH RI VL]H  ,Q JHQHUDO WKH ZRUVW FDVH FRPSOH[LW\ RI WKLV LWHUDWLRQ LVܱ ቌඨ

ቀ

ʹȁܸȁ

ഘሺಸሻ ቁ൅ǤǤ൅ቀഘሺಸሻቁ ʹ

ഘሺಸሻ

ʹȁܸȁ

ቍ WKDW LV ܱ ቆට ʹȁܸȁ

VKRZQLQ)LJXUHWKHFRPSOH[LW\RIWKLVLWHUDWLRQLVܱ ቆට

ଵଵ

ଶഘሺಸሻ ିሺఠሺீሻାଵሻ

ቇ

ቇ )RU JUDSK

 ,WHUDWLRQDOJRULWKPILQGVDJUDSK VFOLTXHRIVL]H,QJHQHUDOZRUVWFDVHFRPSOH[LW\

RI WKLV LWHUDWLRQ LV ܱ ቌඨ

ቀ

ʹȁܸȁ

ഘሺಸሻ ቁ൅ǤǤ൅ቀഘሺಸሻቁ ͵

ഘሺಸሻ

ʹȁܸȁ

ቍ WKDW LV ܱ ቌඨ

ଶഘሺಸሻ ିሺఠሺீሻାଵሻିቀഘሺಸሻቁ ʹȁܸȁ

VKRZQLQ)LJXUHWKHFRPSOH[LW\RIWKLVLWHUDWLRQLVܱ ቆට

ସ

ʹ

ቇ

ቍ )RU WKH JUDSK

  Ȧ* ,WHUDWLRQDOJRULWKPILQGVRQO\RQHRIDJUDSK VFOLTXHRIVL]HȦ* ,QJHQHUDO

WKHFRPSOH[LW\RIWKLVLWHUDWLRQIRUWKHZRUVWFDVHVFHQDULRLVܱ ቌඨ

ʹȁܸȁ

ഘሺಸሻ ቀഘሺಸሻቁ

ቍWKDWLVܱሺඥʹȁܸȁ ሻ

 Ȧ* ,WHUDWLRQDOJRULWKPFRXOGQRWILQGDFOLTXHWKHVL]HRIZKLFKLVODUJHUWKDQȦ*  ,QJHQHUDOWKHFRPSOH[LW\RIWKLVLWHUDWLRQLVܱሺඥʹȁܸȁ ሻ 



K

L

M

N

O

P

M

Q

R

S

T

O

U

M

R

V

W

V

X

Y

V

Z

O

[

[

R

\

Y

]

\

^

4

J

%2-,û48$1780$/*25,7+0)25),1',1*$0$;,080&/,48(

 :H FDQ VHH WKDW WKH ZRUVW FDVH VFHQDULR LV ZKHQ WKH YDOXH RI D VHDUFKHG FOLTXH V VL]H ZKLFK LV VWRUHG LQ WKH YDULDEOH N LV LQFUHPHQWHG E\ RQO\  DIWHU HDFK DOJRULWKP V LWHUDWLRQ 7KHPRVWFRPSOH[LWHUDWLRQLVWKHODVWRQHEHFDXVHWKHUHLVQ WDJUDSK VFOLTXHZKLFKLVODUJHU WKDQFOLTXHQXPEHUȦ* DQGFRPSOH[LW\RIODVWLWHUDWLRQLVܱሺඥʹȁܸȁ ሻ,IWKHUHLVDPD[LPXP RI Ȧ*  LWHUDWLRQV DQG Ȧ*  FDQ PD[LPDOO\ EH _9_ DQG LI WKH PD[LPXP FRPSOH[LW\ RI RQH DOJRULWKP V LWHUDWLRQ LV ܱሺඥʹȁܸȁ ሻ ZH FDQ FRQFOXGH WKDW WKH FRPSOH[LW\ IRU WKH ZRUVW FDVH VFHQDULRLVܱሺȁܸȁඥʹȁܸȁ ሻ ,WLVDOVRLPSRUWDQWWRHPSKDVL]HWKDWWKHSUREDELOLW\RIWKHZRUVWFDVHVFHQDULRLVDWOHDVW  ܲ௪௖

ସ

ఠሺீሻ

ቀఠሺீሻ ቁ ఠሺீሻ

൯ ൫ఠሺீሻ ଶ

ఠሺீሻ

ିଵ

߱ሺ‫ܩ‬ሻ ߱ሺ‫ܩ‬ሻ ൒ ‫כ‬Ǥ Ǥ‫כ‬ ൌ ෍ ൬ ൰ቌ෍ ൬ ൰ቍ  ݅ ݆ ቀఠሺீሻ ቁ ቁ ൫ఠሺீሻ ൯൅Ǥ Ǥ ൅ ቀఠሺீሻ ௝ୀ௜ ௜ୀଶ ଶ ఠሺீሻ ఠሺீሻ

 )RUWKHJUDSKVKRZQLQ)LJXUHWKHSUREDELOLW\RIWKHZRUVWFDVHVFHQDULRLVܲ௪௖ ൒ ଶସ

଺ ‫כ‬ ଵଵ

ହ ‫ ͳ כ‬ൌ  ହହ %HVW FDVH VFHQDULR LV ZKHQ D JUDSK GRHV QRW KDYH D FOLTXH 7KHQ WKH DOJRULWKP ZLOO WHUPLQDWHDIWHURQO\RQHLWHUDWLRQRIWKHFRPSOH[LW\ܱሺඥʹȁܸȁ ሻEXWLIDJUDSKKDVDWOHDVWRQH FOLTXHWKHQWKHSUREDELOLW\IRUWKHEHVWFDVHVFHQDULRLV  ܲ௕௖

ିଵ

ఠሺீ

ͳ

߱ሺ‫ܩ‬ሻ ൰ቍ  ൒ ൌ  ቌ෍ ൬ ఠሺீሻ ఠሺீሻ ݅ ቁ ൯൅Ǥ Ǥ ൅ ቀ ൫ ଶ

௜ୀଶ

ఠሺீሻ

 )RU WKH JUDSK VKRZQ LQ )LJXUH  WKH SUREDELOLW\ IRU WKH EHVW FDVH VFHQDULR LV ܲ௕௖ ൒ ଵ ଵ ൌ   ଺ାସାଵ ଵଵ ,WVKRXOGEHHPSKDVL]HGWKDWWKHWLPHFRPSOH[LW\RIDOJRULWKPVWHSLVQRWUHOHYDQWIRU WKHZRUVWDQGEHVWFDVHVFHQDULRDQDO\VLVEHFDXVHZHFDQGHWHUPLQHWKHVL]HRIWKHVXEVHWDQG YHULI\ZKHWKHUDVXEVHWRIDVHWRIJUDSK VYHUWLFHVLVDFOLTXHLQSRO\QRPLDOWLPHRI_9_HYHQ RQDFODVVLFDOFRPSXWHU

 &RQFOXVLRQDQG)XWXUH:RUN 7KH QHZ TXDQWXP DOJRULWKP IRU ILQGLQJ D PD[LPXP FOLTXH LQ DQ DUELWUDU\ XQGLUHFWHG JUDSK KDV EHHQ LQWURGXFHG ,W LV EDVHG RQ D YHUVLRQ RI *URYHU V DOJRULWKP IRU ILQGLQJ DQ HOHPHQW LQ XQVRUWHG OLVW ZKHQ WKH H[DFW QXPEHU RI SRVVLEOH VROXWLRQV LV XQNQRZQ 7KH SURSRVHG TXDQWXP DOJRULWKP KDV WKH VDPH WLPH FRPSOH[LW\ LQ ZRUVW FDVH DV TXDQWXP DOJRULWKPSURSRVHGE\/RQJ&KDQJ7LQJ7LQJ5HQ0DQJ)HQJ-XQ/XR.DZXX:HLFKHQJ /LQ0LQ\L*XRDQG/DL&KLQ/XWKDWLVܱሺȁܸȁඥʹȁ௏ȁ ሻ3URSRVHGTXDQWXPDOJRULWKPKDVZRUVH WLPHFRPSOH[LW\LQZRUVWFDVHWKDQDOJRULWKPSURSRVHGE\$QGUHZ0&KLOGV(GZDUG)DUKL -HIIUH\*ROGVWRQHDQG6DP*XWPDQQEXWRQO\IRUWKHFDVHZKHUHQ ,WLVREYLRXVWKDWD SURSRVHG DOJRULWKP FDQQRW EH LPSURYHG VLJQLILFDQWO\ LH WR DFKLHYH WKH SRO\QRPLDO WLPH FRPSOH[LW\LQZRUVWFDVH EHFDXVHWKHJHQHUDOLGHDRIDQDOJRULWKPLVWRILQGWKHVXEVHWRID VHWRIJUDSK VYHUWLFHVXVLQJ*URYHU VDOJRULWKPZKLFKKDVܱሺඥܰሻWLPHFRPSOH[LW\ZKHUH1LV WKHQXPEHURIWKHVXEVHWRIDVHWRIJUDSKYHUWLFHVWKDWLVܰ ൌ ʹȁ௏ȁ 'HVSLWHWKLVIDFWWKHUHDUH VRPH SRVVLEOH LPSURYHPHQWV 2QH RI WKH LGHDV IRU DOJRULWKP LPSURYHPHQW LV WR JXHVV WKH VHDUFKHGFOLTXHVL]HLQDOJRULWKPLWHUDWLRQE\XVLQJELQDU\VHDUFKORJLF7KHPHQWLRQHGWULFN FRXOGLPSURYHWKHWLPHFRPSOH[LW\RIWKHZRUVWFDVHVFHQDULRWRܱ൫ሺŽ‘‰ ଶ ȁܸȁሻ ξʹȁ௏ȁ ൯$OVR RQHRIWKHWRSLFVIRUIXWXUHUHVHDUFKLVWKHFDOFXOXVIRUWKHDYHUDJHFDVHDQGWLPHDQGVSDFH DQDO\VLVEDVHGRQVLPXODWLRQRIWKHSURSRVHGDOJRULWKPXVLQJWKHTXDQWXPFRPSXWHUVLPXODWRU 

 a

_

`

b

c

d

e

f

c

g

h

i

j

e

k

c

h

l

m

l

n

o

l

p

e

q

q

h

r

o

s

r

t

-2851$/2),1)250$7,21$1'25*$1,=$7,21$/6&,(1&(6

RQ WKH FODVVLFDO FRPSXWHU IRU UDQGRPO\ JHQHUDWHG JUDSKV )LQDOO\ , FRQFOXGH WKDW WKH SURSRVHGDOJRULWKPLVQ WWKHEHVWSRVVLEOHZD\WRVROYHWKHPD[LPXPFOLTXHSUREOHPEXWLW GHVFULEHVDQHZDSSURDFKDQGXVHRI*URYHU VDOJRULWKPLQILQGLQJWKHPD[LPXPFOLTXHLQDQ DUELWUDU\JUDSK

5HIHUHQFHV >@ %HQQHWW &KDUOHV + %HUQVWHLQ ( %UDVVDUG * 9D]LUDQL 8 6WUHQJWKV DQG :HDNQHVVHV RI 4XDQWXP &RPSXWLQJ 6,$0 -RXUQDO RQ &RPSXWLQJ YROXPH   >@ %R\HU0%UDVVDUG*+¡\HU37DSS$7LJKWERXQGVRQTXDQWXPVHDUFKLQJ )RUWVFKULWWHGHU3K\VLNYROXPH  >@ &KDQJ:/5HQ77)HQJ0/XR-:HLFKHQJ/LQ.*XR0&KLQ/X/ 4XDQWXP $OJRULWKPV RI %LRPROHFXODU 6ROXWLRQV IRU WKH &OLTXH 3UREOHP RQ D 4XDQWXP &RPSXWHU  85/ KWWSDU[LYRUJIWSDU[LYSDSHUVSGI >@ &KLOGV $ 0 )DUKL ( *ROGVWRQH - *XWPDQQ 6 )LQGLQJ FOLTXHV E\ TXDQWXP DGLDEDWLF HYROXWLRQ 4XDQWXP ,QIRUPDWLRQ DQG &RPSXWDWLRQ  SDJHV  ±   >@ 'LYMDN%/RYUHQFLF$'LVNUHWQDPDWHPDWLNDVWHRULMRPJUDIRYD7,9$7LVNDUD 9DUDåGLQ >@ )UDQFR 5 *URYHU V DOJRULWKP KWWSDU[LYRUJSGIYSGI

DQG

KXPDQ

PHPRU\



85/

>@ *URYHU /. $ IDVW TXDQWXP PHFKDQLFDO DOJRULWKP IRU GDWDEDVH VHDUFK ,Q WK $QQXDO $&0 6\PSRVLXP RQ WKH 7KHRU\ RI &RPSXWLQJ SDJHV  ±  3KLODGHOSKLD86$ >@ .D\H 3 /DIODPPH 5 0RVFD 0 $Q ,QWURGXFWLRQ WR 4XDQWXP &RPSXWLQJ 2[IRUG8QLYHUVLW\3UHVV86$ >@ /DYRU & /LEHUWL / 0DFXODQ 1 *URYHU¶V DOJRULWKP DSSOLHG WR WKH PROHFXODU GLVWDQFH JHRPHWU\ SUREOHP ,Q 3URFHHGLQJV RI 9,, %UD]LOLDQ &RQJUHVV RI 1HXUDO 1HWZRUNV >@ 0DWHXV32PDU@ 1LHOVHQ 0LFKDHO $ &KXDQJ ,VDDF / 4XDQWXP &RPSXWDWLRQ DQG 4XDQWXP ,QIRUPDWLRQ WK $QQLYHUVDU\ (GLWLRQ &DPEULGJH 8QLYHUVLW\ 3UHVV 1HZ @ 3LWWHQJHU $UWKXU 2 $Q ,QWURGXFWLRQ WR 4XDQWXP &RPSXWLQJ $OJRULWKPV %LUNKlXVHU%RVWRQ >@ 6KRU33RO\QRPLDOWLPHDOJRULWKPVIRUSULPHIDFWRUL]DWLRQDQGGLVFUHWHORJDULWKPV RQDTXDQWXPFRPSXWHU6,$0-RXUQDORQ&RPSXWLQJ9ROXPH SDJHV  >@ 8GUHVFX03URGDQ/9ODGXWLX0,PSOHPHQWLQJ4XDQWXP*HQHWLF$OJRULWKPV $6ROXWLRQ%DVHGRQ*URYHU V$OJRULWKP,Q3URFHHGLQJVRIWKHUG&RQIHUHQFHRQ &RPSXWLQJ)URQWLHUVSDJHV± >@ 9HQWXUD ' 0DUWLQH] 7 $ TXDQWXP DVVRFLDWLYH PHPRU\ EDVHG RQ *URYHU¶V DOJRULWKP ,Q 3URFHHGLQJV RI WKH ,QWHUQDWLRQDO &RQIHUHQFH RQ $UWLILFLDO 1HXUDO 1HWZRUNVDQG*HQHWLF$OJRULWKPVSDJHV± 

 v

w

x

y

z

{

x

|

}

~



z

€

x

}



‚



ƒ

„



…

z

†

†

}

‡

„

ˆ

‡

‰

_

u

%2-,û48$1780$/*25,7+0)25),1',1*$0$;,080&/,48(

>@ =DOND&*URYHU¶VTXDQWXPVHDUFKLQJDOJRULWKPLVRSWLPDO3K\V5HY$YROXPH  ±



 a

_

Š

b

c

d

e

f

c

g

h

i

j

e

k

c

h

l

m

l

n

o

l

p

e

q

q

h

r

o

s

r

t