+ All Categories
Home > Documents > Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk...

Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk...

Date post: 06-Mar-2021
Category:
Upload: others
View: 1 times
Download: 0 times
Share this document with a friend
44
1 p 2 p ?
Transcript
Page 1: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Fkdswhu 4

LQWURGXFWLRQ

�Wkh ixqgdphqwdo sureohp ri frppxqlfdwlrq lv wkdw ri uhsurgxflqj dw rqh

srlqw hlwkhu h{dfwo| ru dssur{lpdwho| d phvvdjh vhohfwhg dw dqrwkhu srlqw1�

�Fodxgh Vkdqqrq

Wklv fkdswhu suhvhqwv d ihz vlpsoh h{dpsohv wr looxvwudwh wkh sulqflsohvlqyroyhg lq vriw lwhudwlyh ghfrglqj +VLG, lq Vhfwlrq 41 D eulhi vxppdu|ri gljlwdo frppxqlfdwlrqv dqg srwhqwldo dssolfdwlrqv lv jlyhq lq Vhfwlrqv5 dqg 6/ iroorzhg e| dq ryhuylhz ri wkh fkdswhuv lq wklv errn lq Vhfwlrq71 D jxlgh wr qrwdwlrq dqg v|perov lv surylghg lq Vhfwlrq 81

41 VRIW LWHUDWLYH GHFRGLQJ

�Vriw� ghfrglqj uhihuv wr wkh xvh ri suredelolvwlf lqirupdwlrq dv lq0sxw wr d ghfrglqj dojrulwkp/ dv rssrvhg wr �kdug� ghfrglqj/ zkhuh wkhlqsxw lv d vhtxhqfh ri v|perov +h1j1/ elwv,1 Vhfwlrq 414 jlyhv vrph sure0ohpv uhodwhg wr frlq suredelolwlhv/ looxvwudwlqj wkh lghd ri vriw ghfrglqj1Lq Vhfwlrq 415/ wkh frxqwlqj ri vroglhuv looxvwudwhv wkh xvh ri lwhudwlyhghfrglqj wr shuirup d frpsxwdwlrq1 Vhfwlrq 416 ghvfulehv wkh phvvdjh0sdvvlqj dojrulwkp +PSD,/ zklfk lv d vriw lwhudwlyh ghfrglqj whfkqltxh1

414 FRLQ SUREDELOLWLHV

Zh frqvlghu d sx}}oh uhodwhg wr frlq suredelolwlhv= Vxssrvh wkhuh duhwkuhh frlqv/ zklfk duh nqrzq wr eh doo khdg ru doo wdlov1 Lq dgglwlrq/vxssrvh wkdw wkh �uvw frlq lv nqrzq wr eh d khdg zlwk suredelolw| �

�/

wkh vhfrqg frlq lv +lqghshqghqwo|, nqrzq wr eh d khdg zlwk suredelolw|�

�/ dqg wkh wklug frlq kdv d xqlirup glvwulexwlrq +zlwk suredelolw| �

2,1

Jlyhq wklv lqirupdwlrq/ zkdw lv wkh suredelolw| wkdw wkh wklug frlq lv dkhdgB +Klqw= Wkh dqvzhu lv qhlwkhu �

�qru �

b1,

4

5 VRIW LWHUDWLYH GHFRGLQJ

1p 2p ?

Iljxuh 4141 Hydoxdwlqj suredelolwlhv rq frlqv

Ohw xv ghvfuleh wklv sureohp pruh suhflvho|1 Ohw {� ghqrwh wkh l0wkfrlq/ iru l @ 4> 5> 6/ zlwk wkh ydoxh {� @ 4 uhsuhvhqwlqj khdgv dqg wkhydoxh {� @ 3 uhsuhvhqwlqj wdlov1 Wkh �uvw frlq lv nqrzq wr eh d khdg zlwksuredelolw| s� @ S +{� @ 4,/ dqg wkh vhfrqg frlq lv nqrzq wr eh d khdgzlwk suredelolw| s2 @ S +{2 @ 4,1 Qrwklqj lv nqrzq derxw wkh wklug frlq/vr lw lv dvvxphg wr kdyh d xqlirup glvwulexwlrq/ s� @ S +{� @ 4, @ �

2 1Wkhvh suredelolwlhv s�/ s2/ dqg s� duh nqrzq dv wkh d sulrul +ru �sulru�/iru vkruw, suredelolwlhv/ dqg fdq frph iurp dq revhuydwlrq ru h{whuqdovrxufh1

Dgglwlrqdo vwuxfwxuh rq wkh frq�jxudwlrq ri wkh frlqv fdq eh ghvfulehge| d frqvwudlqw vhw V @ i+{�> {2> {�,j ri ydolg frq�jxudwlrqv1 Lq wkh fdvhzkhuh wkh frlqv duh doo wdlov ru doo khdgv/ wkh frqvwudlqw vhw lv

V @ i+3> 3> 3, > +4> 4> 4,j =

Xvlqj wklv lqirupdwlrq/ wkh sureohp lv wr �qg wkh suredelolw| wkdw {� @ 4/jlyhq wkdw wkh wkuhh frlqv +{�> {2> {�, vdwlvi| wkh frqvwudlqw V1 Wklvfrqglwlrqdo suredelolw| lv nqrzq dv wkh d srvwhulrul +ru �srvwhulru�/ iruvkruw, suredelolw|/ dqg fdq eh zulwwhq dv

sTLt|� @ S +{� @ 4 m +{�> {2> {�, 5 V, =

+Sulru dqg srvwhulru suredelolwlhv duh glvfxvvhg lq pruh ghwdlo lq Fkdswhu5/ Vhfwlrq 41,

Uhfdoo wkh uxoh iru frqglwlrqdo suredelolwlhv +uhi1 ^Ohr<7`,=

S +D m E, @S +D>E,

S +E,

Lq wklv fdvh/ ohw D uhsuhvhqw wkh hyhqw wkdw wkh wklug frlq lv d khdg+{� @ 4,/ dqg ohw E uhsuhvhqw wkh hyhqw wkdw doo wkuhh frlqv vdwlvi| wkhfrqvwudlqw ++{�> {2> {�, 5 V,1 Wkhq wkh srvwhulru suredelolw| lv

sTLt|� @

SE%� c%2 c%��M7

%�'�

S +{�> {2> {�,

SE%� c%2 c%��M7

S +{�> {2> {�,= +414,

Page 2: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Lqwurgxfwlrq 6

Wkh vxppdwlrq lv ryhu doo frq�jxudwlrqv lq V vxfk wkdw {� @ 4/ qrupdo0l}hg e| vxpplqj ryhu wkh suredelolwlhv ri doo frq�jxudwlrqv lq V1 Lw lvdvvxphg wkdw wkh sulru suredelolwlhv duh lqghshqghqw/ vr wkdw wkh mrlqwsuredelolw| fdq eh idfwruhg dv iroorzv=

S +{�> {2> {�, @ S +{�,S +{2,S +{�,

H{dpsoh 4 +htxdolw|,1 Zkhq wkh frlqv duh hlwkhu doo khdgv ru doowdlov/ wkh vhw ri ydolg frq�jxudwlrqv iru wkh wkuhh frlqv lv ghqrwhgV @ i+3> 3> 3, > +4> 4> 4,j1 Xvlqj +414,/ wkh suredelolw| ri wkh wklug frlqehlqj d khdg lv

sTLt|

� @s�s2s�

s�s2s� . +4� s�, +4� s2, +4� s�,=

Wkhq vxssrvh wkdw wkh sulru suredelolw| wkdw wkh �uvw frlq lv d khdg+{� @ 4, lv s� @

��/ dqg vlploduo| iru wkh vhfrqg frlq/ s2 @

��1 Vlqfh

qrwklqj lv nqrzq derxw wkh wklug frlq/ s� @�2lv dvvxphg1 Wkhq wkh

srvwhulru suredelolw| lv

+4@6,2

+4@6,2 . +5@6,2@ 4@8>

vr wkdw kdylqj wzr suredelolwlhv ri 4@6 ghfuhdvhv wkh fhuwdlqw| wkdwwkh wklug frlq lv d khdg wr 4@8/ dqvzhulqj wkh txhvwlrq srvhg lq wkh�uvw sdudjudsk1

H{dpsoh 5 +sdulw|0fkhfn,1 Qh{w/ vxssrvh wkh vwuxfwxuh lv wkdwwkhuh duh dq hyhq qxpehu ri khdgv dprqjvw wkh wkuhh frlqv1 Lq wklvfdvh/ wkh vhw ri ydolg frq�jxudwlrqv lv

V @ i+3> 3> 3, > +4> 4> 3, > +4> 3> 4, > +3> 4> 4,j =

Li wkh �uvw wzr frlqv kdyh sulru suredelolwlhv s� dqg s2/ wkhq xvlqj+414,/ wkh srvwhulru suredelolw| ri wkh wklug frlq fdq eh ghgxfhg wreh

sTLt|� @ s� +4� s2, . +4� s�, s2=

Lq rwkhu zrugv/ wkh suredelolw| wkdw wkh wklug frlq frphv xs khdgv lvjlyhq e| wkh suredelolw| wkdw h{dfwo| rqh ri wkh rwkhu wzr frlqv lv dkhdg1 Li s� @ s2 @

��/ wkhq qrwklqj kdv ehhq ohduqhg derxw wkh wklug

frlq/ dv sTLt|� @ �21

H{dpsoh 6 +frqvwudlqw,1 Lq jhqhudo/ wkh frqvwudlqw vhw fdq ehdq duelwudu| uhvwulfwlrq rq wkh doorzdeoh sdwwhuqv1 Vxssrvh wkdw wkh

7 VRIW LWHUDWLYH GHFRGLQJ

frqvwudlqw shuplwv h{dfwo| rqh khdg dprqjvw wkh wkuhh frlqv/ vr wkdwwkh frqvwudlqw vhw lv

V @ i+4> 3> 3, > +3> 4> 3, > +3> 3> 4,j =

Jlyhq wkh sulru suredelolwlhv s� dqg s2 iru wkh �uvw dqg vhfrqg frlq/wkh srvwhulru suredelolw| iru wkh wklug frlq fdq eh irxqg dv iroorzv=

sTLt|

� @+4� s�, +4� s2,

s� +4� s2, . +4� s�, s2 . +4� s�, +4� s2,

@+4� s�, +4� s2,

4� s�s2

Lq wklv fdvh/ hyhq zkhq wkh �uvw wzr frlqv kdyh xqlirup suredelolwlhvs� @ s2 @ �

2/ wkh nqrzohgjh wkdw wkh frlqv vdwlvi| wkh frqvwudlqw V

jlyhv d srvwhulru suredelolw| ri sTLt|

� @ ��iru wkh wklug frlq wr eh d

khdg1

Wkh �uvw wzr h{dpsohv/ �htxdolw|� dqg �sdulw|0fkhfn/� fruuhvsrqg wrwkh htxdolw| qrgh dqg sdulw|0fkhfn qrgh glvfxvvhg lq Fkdswhu 51 Wkhwklug h{dpsoh/ �frqvwudlqw/� jlyhv dq h{dpsoh ri d qrq0olqhdu frqvwudlqwvlplodu wr wkh frqvwudlqhg frghv glvfxvvhg lq Fkdswhuv 8 dqg 91 Wkhphvvdjh0sdvvlqj dojrulwkp lv edvhg rq vlpsoh suredelolvwlf fdofxodwlrqvvxfk dv wkhvh/ zklfk duh frpelqhg wr vroyh frpsolfdwhg ghfrglqj sure0ohpv1

415 FRXQWLQJ VROGLHUV

Dq h{dpsoh gxh wr Shduo ^Shd;;` +dovr suhvhqwhg lq ^Pdf34`, vhuyhvwr looxvwudwh krz wkh sdvvlqj ri phvvdjhv lq d judsk fdq vroyh d sureohpdv d glvwulexwhg frpsxwdwlrq1 Iluvw/ vxssrvh wkhuh lv d wurrs ri vroglhuvvwdqglqj lq d urz/ dv slfwxuhg lq Iljxuh 4151 Hdfk vroglhu fdq rqo|frppxqlfdwh zlwk wkh qhljkerulqj vroglhuv gluhfwo| dkhdg dqg ehklqg1Wkh sureohp lv wr frxqw wkh wrwdo qxpehu ri vroglhuv dqg glvwulexwh wklvlqirupdwlrq wr hyhu| vroglhu1

Dq lqwhuhvwlqj vroxwlrq lv jlyhq e| wkh iroorzlqj uxoh= Zkhqhyhu |rx

uhfhlyh d qxpehu iurp rqh qhljkeru/ dgg 4 dqg sdvv wklv qxpehu wr wkh

rwkhu qhljkeru1 Wkh dojrulwkp vwduwv zlwk wkh vroglhuv vwdqglqj dw wkhhqgv ri wkh olqh/ zkr hdfk sdvv d 4 wr wkhlu lpphgldwh qhljkeruv1 Wkhqxpehuv duh lqfuhphqwhg e| 4 hdfk wlph wkh| jhw sdvvhg dorqj1 Wkhlqwhusuhwdwlrq ri wkh phvvdjhv fdq eh dv iroorzv= Li d vroglhu sdvvhvd phvvdjh q wr dqrwkhu vroglhu/ lw phdqv/ �L eholhyh wkdw wkhuh duh q

vroglhuv/ lqfoxglqj p|vhoi/ xsvwuhdp ri wklv phvvdjh1� Zkhq doo phvvdjhvkdyh ehhq sdvvhg/ hdfk vroglhu vxpv wkh lqfrplqj qxpehuv iurp wkhqhljkeruv/ dqg wkhq dggv 4 pruh wr rewdlq wkh wrwdo qxpehu ri vroglhuv1

Page 3: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Lqwurgxfwlrq 8

� � � � �

�����

/ /��

55��

7RWDO� / � 5 � �

Iljxuh 4151 Frxqwlqj vroglhuv xvlqj phvvdjh0sdvvlqj

D judsk fdq eh xvhg wr uhsuhvhqw wkh vroglhuv/ dv vkrzq lq Iljxuh 4161Wkh qrghv uhsuhvhqw wkh vroglhuv/ dqg wkh hgjhv uhsuhvhqw wkh frppxql0fdwlrq olqnv ehwzhhq wkh vroglhuv1 Wkh uhohydqw uxoh iru phvvdjh0sdvvlqjiru judskv lq wklv fdvh lv dv iroorzv= Li d qrgh kdv N . 4 hgjhv/ vxp

wkh lqfrplqj phvvdjhv dorqj hdfk vxevhw ri N hgjhv/ dqg sdvv wkh vxp

dorqj wkh uhpdlqlqj hgjh1 Lq rwkhu zrugv/ li d qrgh kdv N . 4 hgjhv/wkhq dq| wlph lw uhfhlyhv lqfrplqj phvvdjhv dorqj N hgjhv/ lw vxpvwkrvh qxpehuv dqg sdvvhv wkh uhvxow rxw dorqj wkh uhpdlqlqj hgjh1 Li dqrgh uhfhlyhv ohvv wkdq N phvvdjhv/ wkhq lw grhv qrwklqj/ dqg zdlwv irupruh phvvdjhv wr duulyh1 +Rqfh d phvvdjh lv sdvvhg dorqj dq hgjh/ lwlv dvvxphg wr vwd| wkhuh xqwlo lw lv xsgdwhg1, Iru wkh qrghv lq Iljxuh416/ wkhuh lv rqh yhuwlfdo hgjh wkdw lv xvhg iru lqsxwwlqj wkh lqlwldo frxqwri 4/ dqg rqh ru wzr krul}rqwdo hgjhv uhsuhvhqwlqj wkh frqqhfwlrqv wrqhljkerulqj vroglhuv1

Xvlqj wkhvh uxohv/ wkh rughu ri wkh phvvdjh0sdvvlqj lq wkh judsk lvvkrzq lq Iljxuh 4171 Iluvw wkhuh lv dq hgjh wr hdfk qrgh zlwk dq lqfrplqjphvvdjh ri 4/ uhsuhvhqwlqj hdfk vroglhu*v lqlwldo frxqw wkdw wkhuh lv 4vroglhu�qdpho| klpvhoi ru khuvhoi1 Qh{w/ vlqfh wkh wzr qrghv dw wkh hqgvkdyh rqo| 5 hgjhv/ wkh rqh lqfrplqj phvvdjh lv vx!flhqw wr wuljjhu dqrxwjrlqj phvvdjh wr wkh qhljkerulqj vroglhu1 Vxevhtxhqwo|/ lq erwk wkh

9 VRIW LWHUDWLYH GHFRGLQJ

D�E�F

� � � � �

�����

D

F

Prior

Forward-backward

Extrinsic

Message-passing rule E

� � � � �

� � � � � �

� � � � �

�����

� � � � � �� � � � � �

Iljxuh 4161 Judsklfdo uhsuhvhqwdwlrq iru vroglhu0frxqwlqj

iruzdug dqg edfnzdug gluhfwlrqv/ hdfk phvvdjh sdvvhg wr d qrgh wuljjhuvwkdw qrgh wr vhqg d phvvdjh lq wkh vdph gluhfwlrq wr wkh qh{w qrgh/ zlwkwkh ydoxh lqfuhphqwlqj e| 4 hdfk wlph1 Ilqdoo|/ zkhq d qrgh uhfhlyhv dphvvdjh iurp erwk wkh ohiw dqg uljkw vlghv/ lw vxpv wkh qxpehuv wrjhwkhuwr jlyh dq rxwjrlqj phvvdjh dorqj wkh erwwrp hgjh1 +Iru wkh fdvh ri wkhwzr vroglhuv rq wkh hqg/ zklfk rqo| kdyh wzr hgjhv/ dq| phvvdjh wkh|uhfhlyh dorqj rqh hgjh lv gluhfwo| sdvvhg dorqj wkh rwkhu1, Wklv rxwjrlqjphvvdjh lv fdoohg wkh �h{wulqvlf� phvvdjh/ dqg uhsuhvhqwv wkh dgglwlrqdonqrzohgjh wkdw wkh vroglhu kdv ohduqhg iurp wkh phvvdjh0sdvvlqj1 +Wkhodehov lq Iljxuh 416 ri �lqwulqvlf/� �iruzdug0edfnzdug/� dqg �h{wulqvlf�uhihu wr wkh phvvdjh0sdvvlqj dojrulwkp lq Fkdswhu 5 dqg wkh iruzdug0edfnzdug dojrulwkp lq Fkdswhu 61, Lq rwkhu zrugv/ wkh ydoxh ri wkhrxwsxw phvvdjh/ zklfk lv 8 lq wklv fdvh/ vd|v wkdw wkh phvvdjh0sdvvlqjkdv frxqwhg 8 dgglwlrqdo vroglhuv lq wkh judsk1 Sxwwlqj wklv wrjhwkhuzlwk wkh lqlwldo phvvdjh ri 4 jlyhv wkh wrwdo qxpehu ri vroglhuv fruuhfwo|dw 91

Wklv phvvdjh0sdvvlqj uxoh zrunv shuihfwo| zkhqhyhu wkh judsk frq0wdlqv qr f|fohv +l1h1/ sdwkv wkdw ehjlq dqg hqg lq wkh vdph qrgh,/ dv

Page 4: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Lqwurgxfwlrq :

� � � � � �

� �

� �

� �

� � � �

� � � �

� � � � �

� � � � � �

�� � � � � �

� �� � � � � �

� � � � � �� �

� � �

� � � �� � � � � �� ��

� � � � �� � �

� � � �� � � � � �� ��

�� �

Iljxuh 4171 Wkh rughu lq zklfk phvvdjhv duh sdvvhg lq wkh judsk

vkrzq lq Iljxuh 418/ doorzlqj iru wkh frpsxwdwlrq ri wkh wrwdo qxpehuri vroglhuv lq d glvwulexwhg pdqqhu1 Wkh vlwxdwlrq ri d vroglhu zlwk 6qhljkerulqj vroglhuv lv looxvwudwhg lq pruh ghwdlo lq Iljxuh 419/ zkhuh wkhvroglhu lv uhsuhvhqwhg e| d qrgh wkdw kdv 7 hgjhv +6 iru wkh qhljkeruv/dqg 4 iru wkh vroglhu*v rzq frxqw,1 Wkhq wkh rxwsxw wr d qhljkerulqjvroglhu lv htxdo wr wkh vxp ri wkh lqsxwv iurp wkh rwkhu wzr qhljkeruvsoxv 41

Rq wkh rwkhu kdqg/ wklv phwkrg ri vroglhu0frxqwlqj fdq idlo edgo| zkhqwkhuh lv d f|foh lq wkh judsk1 Vxssrvh wkdw d jurxs ri vroglhuv vwdqgvlq d flufoh/ dv vkrzq lq Iljxuh 41:1 Xvlqj wkh phvvdjh0sdvvlqj uxoh wrfrxqw wkh wrwdo qxpehu ri vroglhuv uhvxowv lq wkh qxpehuv lqfuhphqwlqjwr lq�qlw|$

416 PHVVDJH0SDVVLQJ RQ JUDSKV

Wkh nh| lghd ri wkh phvvdjh0sdvvlqj dojrulwkp/ dv looxvwudwhg e| wkhvroglhu0frxqwlqj h{dpsoh/ lv wr vroyh d sureohp xvlqj vlpsoh frpsxwd0wlrq xqlwv wkdw h{fkdqjh phvvdjhv1 Zh jlyh d eulhi ghvfulswlrq ri d judsk

; VRIW LWHUDWLYH GHFRGLQJ

1 1 1

1 1

1 1 1

1

1

8

2

7

1

8

7

5

8

4

2

7

1

8

1

2

Iljxuh 4181 Phvvdjh0sdvvlqj zrunv iru frxqwlqj vroglhuv lq d wuhh irupdwlrq

��

��

Iljxuh 4191 D vroglhu zlwk wkuhh qhljkeruv

vwuxfwxuh +Iruqh|*v qrupdo judskv ^Iru34`, wkdw fdq eh xvhg wr shuirupphvvdjh0sdvvlqj1 Dv vkrzq lq Iljxuh 41;/ wkh judsk lv dq xqgluhfwhgjudsk/ lq zklfk d yduldeoh lv dvvrfldwhg zlwk hdfk hgjh/ dqg hdfk qrghuhsuhvhqwv d orfdo frqvwudlqw rq wkh yduldeohv dvvrfldwhg zlwk wkh hgjhvfrqqhfwhg wr wkdw qrgh1 Phvvdjhv fruuhvsrqglqj wr wkh suredelolw| glv0wulexwlrqv iru wkh yduldeoh duh sdvvhg lq erwk gluhfwlrqv dorqj doo hgjhvri wkh xqgluhfwhg judsk1

Page 5: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Lqwurgxfwlrq <

� �

� �

��

Iljxuh 41:1 Phvvdjh0sdvvlqj idlov iru vroglhuv lq d flufoh irupdwlrq

N1

N3

N4

Internal edgeExternal edge

N2 N5

*

N6

N8

N7

Node

Module

Iljxuh 41;1 Judsk frqvlvwlqj ri qrghv dqg hgjhv

Iru wkh judsk/ wkhuh pd| dovr eh hgjhv wkdw duh rqo| frqqhfwhg wrrqh qrgh1 Wkhvh duh wkh h{whuqdo hgjhv +dovr nqrzq dv �kdoi0hgjhv�,dv rssrvhg wr wkh lqwhuqdo hgjhv wkdw frqqhfw wzr qrghv1 Wkhvh h{0whuqdo hgjhv duh xvhg iru sdvvlqj suredelolvwlf lqirupdwlrq lq dqg rxwri wklv vwuxfwxuh1 Vlqfh erwk wkh lqsxw dqg rxwsxw duh phvvdjh yhf0

43 VRIW LWHUDWLYH GHFRGLQJ

wruv fruuhvsrqglqj wr suredelolwlhv/ wklv ghfrglqj surfhvv lv riwhq fdoohg�vriw0lq/ vriw0rxw� +VLVR,1 Wkh ryhudoo ghfrglqj v|vwhp frqvlvwv ri gli0ihuhqw qrghv wkdw h{fkdqjh suredelolvwlf lqirupdwlrq1 Lq dgglwlrq/ lw lvsrvvleoh wr jurxs wkhvh qrghv dqg hgjhv lqwr prgxohv/ zklfk fdq eh frq0qhfwhg wrjhwkhu wr irup odujhu ghfrglqj v|vwhpv1 Qrghv/ prgxohv dqgwkh phvvdjh0sdvvlqj dojrulwkp duh glvfxvvhg lq ghwdlo lq Fkdswhu 51

51 GLJLWDO FRPPXQLFDWLRQV

Zh ghvfuleh d iudphzrun iru gljlwdo frppxqlfdwlrqv/ vwduwlqj �uvwzlwk wkh fkdqqho dqg wkhq prylqj rq wr frglqj dqg prgxodwlrq1

514 FKDQQHO

Gljlwdo frppxqlfdwlrqv lv wkh duw ri jhwwlqj elwv uholdeo| iurp rqhsodfh wr dqrwkhu +ru/ lq wkh fdvh ri gljlwdo vwrudjh/ iurp rqh wlph wrdqrwkhu, dfurvv dq xquholdeoh fkdqqho1 Ohw wkh vhtxhqfh { uhsuhvhqw dfroohfwlrq ri vljqdov {�> wklv pd| eh d �qlwh eorfn { @ +{�> {2> ===> {� ,ru d el0lq�qlwh vhtxhqfh { @ +===> {�3�> {�> {�n�> ===,1 Dv vkrzq lq Iljxuh41</ wkh fkdqqho pd| lqwurgxfh qrlvh dqg glvwruwlrq wr wkh wudqvplw0whg vhtxhqfh {/ zklfk lv uhfhlyhg dv wkh vhtxhqfh |1 Wkh jrdo ri wkhfkdqqho ghwhfwru lv wr xvh wkh uhfhlyhg vhtxhqfh | dqg nqrzohgjh ri wkhfkdqqho fkdudfwhulvwlfv wr surgxfh d uhsolfd e{ ri wkh ruljlqdo vhtxhqfh{1 Pd{lpxp0olnholkrrg vhtxhqfh ghwhfwlrq +POVG, ^Iru:5` frqvlvwv rifkrrvlqj wkh vhtxhqfh wkdw pd{lpl}hv wkh suredelolw| wkdw wkh uhfhlyhgyhfwru lv | li wkh wudqvplwwhg vhtxhqfh { zdv vhqw1

e{�u @ dujpd{

%Su +| m {, =

Lq rxu whuplqrorj|/ wkh whup �ghfrghu� lv xvhg jhqhulfdoo| wr ghvfulehdq| surfhvv wkdw lqyroyhv wkh uhfryhu| ri d gdwd vhtxhqfh wkdw kdv ehhq�hqfrghg/� zkhuh hqfrglqj lv eurdgo| gh�qhg wr lqfoxgh wkh wudqvirupd0wlrq lqwurgxfhg e| wkh fkdqqho1 Dv d uhvxow/ wkh whup �fkdqqho ghfrghu�zloo eh xvhg lqwhufkdqjdeo| zlwk �fkdqqho ghwhfwru� wr ghvfuleh wkh prg0xoh wkdw shuirupv frpsxwdwlrqv wr rewdlq lqirupdwlrq rq wudqvplwwhgvhtxhqfh1

Vrph nh| h{dpsohv ri fkdqqhov duh wkh elqdu| v|pphwulf fkdqqho

+EVF,/ wkh elqdu| hudvxuh fkdqqho +EHF,/ dqg wkh dgglwlyh zklwh Jdxv0

vldq qrlvh +DZJQ, fkdqqho/ dv vkrzq lq Iljxuh 41431 Iru wkh EVFdqg EHF/ wkh wudqvplwwhg vljqdov i{�j duh elqdu|/ dqg wkh uhfhlyhg vlj0qdov i|�j duh glvfuhwh0ydoxhg dv zhoo1 Rq wkh rwkhu kdqg/ iru wkh DZJQfkdqqho/ wkh vljqdov duh uhdo0ydoxhg +ru frpsoh{0ydoxhg,/ dqg wkh uhfhlyhgvljqdo |� lv jlyhq e|

|� @ {� . q�>

Page 6: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Fkdswhu 5

PHVVDJH0SDVVLQJ DOJRULWKP

H soxulexv xqxp +rxw ri pdq|/ rqh,

iurp wkh Juhdw Vhdo ri wkh Xqlwhg Vwdwhv ri Dphulfd

Wklv fkdswhu lqwurgxfhv d iudphzrun iru xqghuvwdqglqj suredelolv0wlf ghfrglqj/ dqg dssolhv lw wr wkh ghfrglqj ri orz0ghqvlw| sdulw| fkhfn+OGSF, frghv1 Vhfwlrq 41 vhwv xs wkh suholplqdulhv iru vriw lwhudwlyh gh0frglqj1 Vhfwlrq 51 h{sodlqv wkh phvvdjh0sdvvlqj dojrulwkp xvlqj qrupdojudskv1 Vhfwlrq 61 suhvhqwv vrph edvlf qrghv dqg prgxohv1 Vhfwlrq 71lqwurgxfhv OSGF frghv dqg glvfxvvhv wkhlu ghfrglqj dojrulwkp1

41 SUHOLPLQDULHV

414 SUREDELOLWLHV

Wklv vhfwlrq uhylhzv vrph edvlf gh�qlwlrqv dqg frqfhswv lq suredelo0lw|1 ^Ohr<7`

41414 EDVLF FRQFHSWV

Iru dq hyhqw H/ zh fdq dvvljq d suredelolw| S +H,/ zklfk lv d uhdoqxpehu ehwzhhq 3 dqg 4/ lqfoxvlyh1 Lw lv xvhixo wr frqvlghu wklqn ri H dvd vhw ri rxwfrphv/ vr wkdw S +H, lv wkh suredelolw| ri fkrrvlqj dq rxwfrphiurp wkdw vhw1 Iru wkh qxoo vhw >/ dqg wkh vdpsoh vsdfh D frqvlvwlqj ridoo rxwfrphv/ zh kdyh S +>, @ 3 dqg S +D, @ 41 Vxssrvh wkdw d vhw rihyhqwv iI&j/ iru n @ 4> 5> ===>N/ irupv d sduwlwlrq ri wkh frpsohwh vhw

ri hyhqwv/ vr wkdw wzr hyhqwv duh qrw vlpxowdqhrxvo| wuxh/ dqg wkh vhw

fryhuv wkh frpsohwh vsdfh ri hyhqwv1 Lq vhw qrwdwlrq/ I& _ I&� @ > li

n 9@ n�/ dqgV

&I& @ D/ zkhuh S +D, @ 41 Wkhq wkh wrwdo suredelolw| lv

56

57 VRIW LWHUDWLYH GHFRGLQJ

htxdo wr 41g[

&'�

S +I&, @ 4= +514,

Iru wzr hyhqwv H dqg I / wkh mrlqw suredelolw| S +H>I, lv wkh sured0elolw| wkdw erwk hyhqwv duh wuxh1 Lq vhw qrwdwlrq/ wklv fruuhvsrqgv wr wkhsuredelolw| ri fkrrvlqj dq rxwfrph iurp wkh lqwhuvhfwlrq H _ I 14 Irudq| hyhqw H/ wkh suredelolw| S +H, fdq eh ghfrpsrvhg dv wkh vxp riwkh mrlqw suredelolwlhv ri H dqg I& 1

S +H, @g[

�'�

S +I& >H, = +515,

Wzr hyhqwv H dqg I duh vdlg wr eh lqghshqghqw h{dfwo| zkhq wkhmrlqw suredelolw| idfwruv lqwr wkh surgxfw ri wkh lqglylgxdo suredelolwlhv=

S +H>I , @ S +H,S +I , +516,

Iru wzr hyhqwv H dqg I / wkh frqglwlrqdo suredelolw| S +I m H, lv wkhsuredelolw| wkdw dq hyhqw I lv wuxh jlyhq wkh hyhqw H lv wuxh/ zklfk fdqeh h{suhvvhg dv iroorzv/

S +I m H, @S +H>I ,

S +H,+517,

iru S +H, A 31 Lq wkh fdvh wkdw wkh hyhqwv H dqg I duh lqghshqghqw/ lwfdq eh vhhq iurp +516, dqg +517, wkdw

S +I m H, @ S +I , +li H>I lqghshqghqw,/ +518,

vr wkdw frqglwlrqlqj grhv qrw fkdqjh wkh suredelolw| li wkh hyhqwv duhlqghshqghqw1 Dsso|lqj wkh frqglwlrqdo suredelolw| h{suhvvlrq rqfh djdlqwr wkh qxphudwru ri +517, |lhogv Ed|hv* wkhruhp +dovr nqrzq dv Ed|hv*uxoh,=

S +I m H, @S +H m I ,S +I,

S +H,+519,

Lq dgglwlrq/ iru suredelolwlhv frqglwlrqhg rq wzr hyhqwv/ d vlplodu uh0vxow dv +517, krogv/

S +H>J m I , @ S +H m I>J,S +J m I, > +51:,

4Wklv mrlqw suredelolw| vdwlvlihv wkh iroorzlqj xsshu dqg orzhu erxqgv=

� E.� n � E8 �3 � $ � E.c 8 � $ 4�? t� E.� c � E8 ��

Page 7: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvd

jh0sdvvlq

jdojrulwk

p58

zklfk

lvnqrz

qdv

wkhfkdlq

uxoh

irufrq

glwlrq

dosure

delolwlhv1

Dvshfldo

fdvhri

+51:,rffx

uvzkhq

S+H>J

mI,@S+H

mI,S+J

mI,/

+51;,

lqzklfk

fdvhwkhwkuhh

hyhqwv

H/I

dqgJ

+lqwkdw

rughu,

duhvdlg

wrvdwlvi|

wkhPdunrysurshuw|1

Wkhvh

hyhqwv

wkhq

irupdPdunryfkdlq/

zklfk

lvghqrwhg

e|H'

I'

J1Wzrfrq

glwlrq

viru

dPdunry

fkdlq

wkdw

duhhtxlydohq

wwr

+51;,duh

S+H

mI>J

,@S+H

mI,

S+J

mH>I

,@S+J

mI,=

LqwklvPdunry

fkdlq

/frq

glwlrq

lqje|wkhhyhq

wIpdnhv

wkhrwk

huhyhq

wvH

dqgJ

lqghshqghqw/dv

lq+516,1

Frqfhs

wvvxfk

dvlqghshqghqfh/

Ed|hv*

wkhruhp

dqgPdunry

fkdlq

vsury

lghwkhedvlf

wrrov

iruwkhgh�qlwlrq

riwkhghfr

glqjdojrulwk

pvlq

Vhfwlrq

51

41415

DOSKDEHWVDQGSUREDELOLW\YHFWRUV

DqdoskdehwDuhs

uhvhqwv

dvhw

riydox

hvwkdw

dydulde

ohfdq

wdnh1Wkh

doskde

hwfdq

ehdglvfuhwh

vhw/vxfk

dvD

@i3>4j/

rudfrq

wlqxrx

vvhw/

vxfk

dvD

@U1+Iru

wklvglvfx

vvlrq/zhuhvwulfw

rxudwwhq

wlrqwr

glvfuhwh

vhwv1,Wkhdos

kde

hwpd|

kdyh

vrphdg

glwlrq

dovwux

fwxuh/

vxfk

dvehlq

jdjurx

sxqghu

dgglwlrq

+h1j1/wkhdg

glwlyh

jurxsrip

hohphqwv/

]6,/ru

d�hog

+h1j1/wkhJdorlv

�hog

I^@JI+t,,1

Iru

dqhyhq

wH/lw

lvsrvvle

ohwr

dvvljqdsure

delolw|

S+H

,/zklfk

lvduhdo

qxpehu

ehwz

hhq3dqg4/

lqfox

vlyh1Wkhq

irududq

grp

ydul0de

oh{ehorq

jlqjwr

dqdos

kde

hwD/lw

lvsrvvle

ohwr

frqvlg

huwkhhyhq

wi{@djwkdw

{htxdov

vrphsduwlfx

oduhohp

hqwd5D1Wkhsure

delolw|

riwklv

hyhqwlvuhs

uhvhqwhg

e|S+{

@d,1

Dvdixqfwlrq

rid5D/wklv

sure

delolw|

S+{

@d,jlyhv

dsuredelolw|

pdvv

ixqfwlr

qiru

wkhydulde

oh{1Vlqfh

wkhvhw

rihyhq

wvi{@dj/

irudoo

d5D/irup

vdsduwlwlrq

riwkh

frpsohwh

vhwri

hyhqwv/

wkhsure

delolwlhv

S+{

@d,ryhu

dooydox

hvlq

wkh

doskde

hwD

vxp

wr41

[@M�

S+{

@d,@

4

59VRIWLWHUDWLYHGHFRGLQJ

Wkhvh

sure

delolwlhv

fdqehuhs

uhvhqwhg

e|dyhfwru

dvzhoo/

zklfk

lvfdoohg

dyhfwru

risure

delolwlhv/

rusuredelolw|

yhfwru/

�$S+{,@

59997

S+{

@d� ,

S+{

@d2 ,

111S�{

@d

��� �

6:::8>

+51<,

zkhuh

wkhdos

kde

hwD

frqvlvwv

ri �d� >d

2 >===>d��

� �1Wkhqrwdwlrq

�$S+{,

wkhq

uhsuhvhq

wvwkhhqwluh

yhfwruri

sure

delolwlhv

vkrz

qlq

+51<,1Qrwh

wkdw

wkhqrwdwlrq

S+{,/

l1h1/zlwk

rxwwkhduurz

/lv

gh�qhg

lqVhfwlrq

51416/wr

uhsuhvhq

wS+{

@d,iru

vrphydox

hd5D/zklfk

fdqehghwhu0

plqhg

iurpwkhfrq

wh{w1

Rqwkhrwk

hukdq

g/dq

duelwudu|

yhfwruri

qrq

0qhjdwlyh

uhdoqxpehuv

+wkdw

grqrw

qhfhvvdulo|

vxp

wr4,

lvfdoohg

dphvvd

jhyhfwru

dqglv

ghqrwhg

e|wkhv|pero�$�1Liwkhydulde

oh{ehorq

jvwr

wkhdos

kde

hwD/

wkhq

wkhphvvdjh

yhfwruzrx

ogkdyh

ohqjwk

mDm/

�$�+{,@

59997

p�

p2

111p

��� 6:::8

>

zkhuh

p&�

3/dq

glw

lvdvvx

phg

wkdw

wkhhqwulhv

riwkhphvvdjh

yhf0wru

duhqrw

doovlp

xowdq

hrxvo|

}hur1Wkhhqwulhv

riwkhyhfwru

fdqdovr

ehzulwwhq

dv�+{

@d& ,

@p& /

lqdq

dorj|wr

wkhsure

delolw|

qrwdwlrq

S+{

@d& ,1

Lwlvsrvvle

ohwr

vhwxsdq

htxlydohq

fhuhodwlrq

+�,rq

phvvdjh

yhfwruvvxfk

wkdw

hdfkphvvdjh

yhfwrulv

htxlydohq

wwr

dsure

delolw|

yhfwrue|

qrup

dol}lqjlwe|wkhvxp

ridoo

hohphqwv1

59997

p�

p2

111p

��� 6:::8

�4

S��

�&'�p& 59997

p�

p2

111p

��� 6:::8

Dvduhvx

owri

wklv

htxlydohq

fhuhodwlrq

/wkhuh

duhmDm�

4ghjuhhv

riiuhhg

rplq

wkhphvvdjh

yhfwru15

5Qrwh

wkdwvlq

fhwk

hphvvd

jhyhfwru

�6

�6

2uuu

6��

�A/vdwlv�

hvdq

htxlydohqfh

uhodwlr

qvx

fkwk

dwpxowls

olfdwlr

qe|dq|srvlwly

hfrqvwd

qwS:

f|lhogvdqhtxlydohqwphvvd

jh

Page 8: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvd

jh0sdvvlq

jdojrulwk

p5:

Zhzloo

dovrfrq

vlghu

wkhvhw

gh�qhg

e|wkhsurgxfw

ripxowls

ohdo0

skde

hwvD%f�D%������

D%g/fruuhvs

rqglqjwr

wkhdoo

srvvle

ohydox

hvri

wkhydulde

ohv/{f >{

� >===>{g1Hdfk

hohphqw

+�f >�

� >===>�g,5D%f�D%�����

�D%g

riwklvvhw

lvfdoohg

dfrq�jxudwlr

q/zkhuh

hdfkydox

h��lvvhohfwhg

iurpwkhds

surs

uldwhdos

kde

hwD� /dq

gfruuhvs

rqgvwr

dsrvvle

ohydox

hiru

wkh

yduldeoh{� 1

41416

LQWULQVLF/H[WULQVLF

DQGSRVWHULRU

Iru

dydulde

oh{/wkhuh

duhvhyhudo

w|shv

risure

delolwlhv

wkdw

fdqeh

dvvrfldwhg

zlwk

wkhydulde

oh1Iru

wkhhyhq

wi{@d& j/

vxssrvh

wkdw

H

lvdq

hyhqwzkrvh

h�hfw

rqwkhydulde

oh{lv

xqghu

txhvwlrq

1Wkhsulr

u

+rud

sulr

ul,sure

delolw|

uhihuvwr

wkhsure

delolw|

S+{

@d& ,

wkdw

wkh

yduldeoh{wdnhv

wkhydox

hd� 1Lq

wkhlwhudwlyh

dojrulwkpvwkdw

zhfrq

vlghu/

krz

hyhu/wkhsulru

sure

delolw|

wkdw

lvxvhg

irudydulde

oh{pd|

ghs

hqg

rqwkhfkrlfh

riwkhhyhq

wH/vr

wkdw

lwlvpruh

dssurs

uldwhwr

vshdn

riwkhsulru

sure

delolw|

iru{

zlwk

uhvshfwwr

H1Wrdyrlg

frqixvlrq

zlwk

wkhwux

hsulru

sure

delolw|/

zhvxevwlwx

whwkhwhup

�lqwulq

vlf�iru

�sulru�

wrghvfule

hwklvsure

delolw|1

Wkhlqwulq

vlfsuredelolw|

iru{zlwk

uhvshfw

wrH

lvghqrwhg

e|

S�?|

.+{@d,@S+{@d,=

Rqwkhrwk

hukdq

g/wkhsrvwhulr

u+ru

dsrvwhulr

ul,sure

delolw|

lvwkh

frqglwlrq

dosure

delolw|

edvhg

rqnqrz

ohgjh

riwkhhyhq

wH1Wkhsrvwh

ulru

sured

elolw|

iru{

zlwk

uhvshfwwr

Hlv

ghqrwhg

STLt|

.+{@d,@S+{@d

mH,=

Wkhlqwulq

vlfdq

gsrvwhulru

sure

delolwlhv

uhsuhvhq

wwkhsure

delolw|

ehiruh

dqgdiwhu

wdnlqjlqwr

dffrxqwwkhhyhq

wH1

Wkhsrvwhulru

sure

delolw|

fdqehzulwwhq

xvlq

jEd|hv*

wkhruhp

+519,dv

TLt|ih�Lh

}��

~

S+{@d

mH,@

4

S+H,

i |h�?t�U

}��

~

S+H

m{@d,

�?|h�?t�U

}��

~

S+{@d,=

+5143,

yhfwr

u/lw

fdq

eh

lqwhus

uhwh

gdvd

srlqwlq

wkhsur

mhfwly

hvs

dfhS

���EU�1

Lqsduwlfx

odu/

wkh

vhwriphvvd

jhyhfwr

uvfr

uuhvsrqg

wrwk

hsrlqwv

lqsur

mhfwlyhvs

dfh

vxfk

wkdwdoo

hohphq

wvri

wkhyhfwr

ukdy

hwk

hvd

phvlj

q+r

uduh

}hur,1

Wkhvh

phvvd

jhyhfwr

uvrffx

s|�*2��

�riyroxph

riwk

hsur

mhfwlyh

vsdfh1

Lqwk

hfd

vhri

���'

2/wk

hsrlqwv

lqwk

hsur

mhfwlyh

sodqhS2EU�

fruuhvs

rqg

wrolq

hvwk

urxjk

wkhrulj

lq/dqg

wkhphvvd

jhyhfwr

uvfr

uuhvsrqg

wrolq

hv+'

6%

zlwk

vorsh6M

dfc"

o�

5;VRIWLWHUDWLYHGHFRGLQJ

Wkhwhup

S+{

@d,wkdw

dsshduv

lqwkhuljk

w0kdq

gvlg

hri

+5143,lvwkhlq0

wulqvlf

sure

delolw|1

Wkhfrp

sohp

hqwdu|

whupS+H

m{@

d,fruuhvs

rqgv

wrwkh�h{

wulqvlf�

sure

delolw|/

zklfk

lvdsure

delolw|

wkdw

ghvfule

hvwkh

qhz

lqirup

dwlrqiru

{wkdw

kdv

ehhq

rewdlq

hgiurp

wkhhyhq

wH1Wkh

h{wulq

vlfsuredelolw|

iru{

zlwk

uhvshfw

wrH

lvgh�

qhg

e|

Si |

.+{

@d,@

f�%S+H

m{@

d,>

+5144,

zkhuh

wkhqrup

dol}dwlrqfrq

vwdqw

f�%@

4S

@M�S+H

m{@

d,

lvqhfhvvdu|

vrwkdw

wkhhqwulhv

lqwkhh{wulq

vlfsure

delolw|

yhfwruS

i |

.+{,

vxp

wr4+l1h1/

S@M�S

i |

.+{

@d,@

4,1Wkhq

wkhuhodwlrq

vkls

ehwz

hhqwkhlqwulq

vlf/h{

wulqvlf

dqg

srvwhulru

sure

delolwlhv

lq+5143,

fdqehuhz

ulwwhqdv

STLt|

.+{

@d,@hf%

S�?|

.+{

@d,S

i |

.+{

@d,>

+5145,

zkhuhhf%

lvdqrup

dol}dwlrqfrq

vwdqwwkdw

wdnhvwkhirup

hf%@ #[@

M�

S�?|

.+{

@d,S

i |

.+{

@d, $

3�

=

Wklv

frqvwdq

wlv

qhfhvvdu|

lqrug

huwr

pdnh

wkhsrvwhulru

sure

delolwlhv

vxp

wr4+l1h1/ S

@M�STLt|

.+{

@d,@

4,1+F

rpsdulq

j+5143,/

+5144,dq

g

+5145,/lwfdq

ehvhhq

wkdw

wkhfrq

vwdqwv

duhuhodwhg

e|hf%

@+f�%S+H

,,3�,1

Lqwhup

vri

sure

delolw|

yhfwruv/wkhsrvwhulru

sure

delolw|

yhfwrulv

surs

ruwlrqdo

wrwkhphvvdjh

yhfwruwkdw

lvjhq

hudwhgiurp

wkhsrlq

wzlvh

pxowls

olfdwlrqri

wkhvh

wzryhfwruv/

6

�$STLt|

.+{,@hf

% ��$S

�?|

.+{,��$S

i |

.+{, �

@hf% 59997

S�?|

.+{@d� ,Si |

.+{@d� ,

S�?|

.+{@d2 ,Si |

.+{@d2 ,

111

S�?|

.

�{@d

��� �Si |

.

�{@d

��� �

6:::8

6Lw

vkrxog

eh

qrwhg

vrph

riwk

hvhtxdqwlwlhv

pd|

qrweh

zhoo0g

h�qhg

liwk

huhduh

}hurv1

Iruh{

dpsoh/

liiruvr

ph&/��?|

.E%'@&�dqg�TLt|

.E%'@&�duh

erwk

f/wk

hqwk

hh{

wulqvlf

suredelolw|

�i |

.E%'@&�lv

qrwzhoo0g

h�qhg

1Lq

sudfwlfd

olp

sohp

hqwdwlr

qv/

wklv

phd

qvwk

dw

wkhsur

edelolwlhv

duh

xvx

doo|

erxqghg

dzd|

iurp

3dqg4+h1j

1/e|"dqg�3"/

uhvshfwly

ho|,1

Page 9: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp 5<

zkhuh wkh hohphqwv ri wkh doskdehw duh�d�> d2> ===> d���

�dqg � uhsuhvhqwv

wkh srlqwzlvh pxowlsolfdwlrq ri wkh wzr yhfwruv1Wr uhfds/ wkh whupv �lqwulqvlf/� �h{wulqvlf/� dqg �srvwhulru� ghvfuleh

suredelolwlhv iru d yduldeoh lq uhodwlrq wr dq hyhqw1 +Lq wkh frxuvh riwkh ghfrglqj dojrulwkp/ wkh hvwlpdwhg suredelolwlhv pd| gl�hu iurp wkhwuxh suredelolwlhv/ vr wkdw lw pd| eh pruh dffxudwh wr xvh whupv vxfkdv svhxgr0lqwulqvlf/ svhxgr0h{wulqvlf dqg svhxgr0srvwhulru wr lqglfdwhwkdw wkhvh duh qrw wkh h{dfw ydoxhv1 Iru frqyhqlhqfh/ krzhyhu/ zh zloofrqwlqxh wr xvh wkh whupv lqvwulqvlf/ h{wulqvlf dqg srvwhulru/ zlwk wkhxqghuvwdqglqj wkdw wkhvh pd| uhsuhvhqw zrunlqj hvwlpdwhv1, Iru fodulw|/wkh lqwulqvlf/ h{wulqvlf ru srvwhulru suredelolwlhv iru { vkrxog eh odehohg�zlwk uhvshfw wr H� +zklfk fdq eh deeuhyldwhg dv �z1u1w1 H�,/ dqgghqrwhg e| S �?|

.+{ @ d,/ S i |

.+{ @ d,/ dqg S

TLt|

.+{ @ d,1 Wklv h{solflw

uhihuhqfh wr wkh hyhqw H pd| eh rplwwhg zkhq lw lv xqghuvwrrg iurpfrqwh{w1

41417 ELQDU\ YDULDEOHV

Iru wkh fdvh ri elqdu| yduldeohv/ zklfk wdnh rq ydoxhv iurp d elqdu|doskdehw D @ i3> 4j/ wkhuh duh vhyhudo htxlydohqw uhsuhvhqwdwlrqv ri wkhsuredelolw|=

41 Wkh suredelolw| s @ S +{ @ 4,/ zklfk olhv lq wkh udqjh ^3> 4`1

51 Wkh suredelolw| yhfwru�$S +{,/ zklfk frqvlvwv ri wzr suredelolwlhv wkdw

vxp wr 41�$S +{, @

�S +{ @ 3,S +{ @ 4,

�@

�4� s

s

61 Wkh orj0olnholkrrg udwlr/

OOU+s, @ orjS +{ @ 4,

S +{ @ 3,@ orj

s

4� s>

zklfk lv d uhdo qxpehu17

71 Wkh vriw elw "+s, @ 5s� 4/ zklfk olhv lq wkh udqjh ^�4> 4`1

E| frqyhqwlrq/ wkh orj0olnholkrrg udwlr xvhv wkh qdwxudo orjdulwkp1Wkh uhodwlrqvkls ehwzhhq wkh vriw elw +uhi1 ^KRS<9`, dqg wkh orj0olnholkrrg

7 Lw lv srvvleoh wr fkrrvh wkh rssrvlwh frqyhqwlrq *L}� E%'f�� E%'��

iru wkh gh�qlwlrq ri wkh orj0

olnholkrrg udwlr/ zklfk wxuqv rxw wr jlyh vlpsohu h{suhvvlrqv lq wkh ghfrglqj ri orz0ghqvlw|

sdulw|0fkhfn frghv1 Zh vwlfn zlwk wkh frqyhqwlrq *L}� E%'��� E%'f�

vr wkdw d srvlwlyh OOU fruuh0

vsrqgv wr % ' �1

63 VRIW LWHUDWLYH GHFRGLQJ

udwlr fdq wkhq eh h{suhvvhg xvlqj wkh k|shuerolf wdqjhqw ixqfwlrq

"+s, @ 5s� 4 @ wdqk

�4

5OOU+s,

�> +5146,

zkhuh wdqk+{, @ e%3e3%

e%ne3%1 Iru d elqdu| yduldeoh/ wkh orj0olnholkrrg udwlrv

fruuhvsrqglqj wr wkh lqwulqvlf dqg h{wulqvlf suredelolwlhv vxp wr jlyh wkhOOU iru wkh srvwhulru suredelolw|/

OOUTLt|

.+{, @ OOU�?|

.+{, . OOUi |

.+{, > +5147,

zkhuh wkh vxevfulsw H lqglfdwhv wkdw lw lv zlwk uhvshfw wr dq hyhqw H1Wklv fdq eh ghulyhg iurp +5145,/ zklfk jlyhv

STLt|

.+{ @ 4, @

sh

sh . +4� s, +4� h,>

zkhuh s @ S �?|.

+{ @ 4,/ dqg h @ S i |.

+{ @ 4,1 Lq wklv orj0olnholkrrg uhs0uhvhqwdwlrq/ lw fdq fohduo| eh vhhq wkdw wkh h{wulqvlf lqirupdwlrq uh hfwvwkh lqfuhphqwdo jdlq lq nqrzohgjh ri wkh srvwhulru ryhu wkh lqwulqvlflqirupdwlrq1 Qrwh wkdw li wkh srvwhulru dqg lqwulqvlf suredelolwlhv duhhtxdo/ wkhq OOUi |

.+{, @ 31

415 QRUPDO JUDSK

D qrupdo judsk J lv dq xqgluhfwhg judsk/ frqvlvwlqj ri qrghv dqghgjhv/ dv lqwurgxfhg e| Iruqh| lq ^Iru34`1 Hdfk qrgh fdq eh frqqhfwhgwr dq| qxpehu ri hgjhv/ dqg wkh hgjhv duh frqqhfwhg wr rqh ru wzrqrghv1 Wkh suhvhqfh ri hgjhv wkdw duh frqqhfwhg wr rqo| rqh qrgh lvd vshfldo surshuw| ri qrupdo judskv1 Dq hgjh frqqhfwhg wr wzr qrghvlv dq lqwhuqdo hgjh/ zkloh dq hgjh frqqhfwhg wr rqo| rqh qrgh lv dqh{whuqdo hgjh1 +Wkhvh h{whuqdo hgjhv duh dovr nqrzq dv �kdoi0hgjhv/��ohdi0hgjhv� ru �grqjohv1�, Wr hdfk hgjh lv dvvrfldwhg d yduldeoh {�

wkdw wdnhv ydoxhv ryhu d �qlwh/ glvfuhwh doskdehw ghqrwhg e| D%�1 Zkhq

uhihuulqj wr wkh frpelqdwlrq ri wkh hgjh dqg lwv dvvrfldwhg yduldeoh/ zhxvh wkh frpsrxqg whup �hgjh0yduldeoh1� Wkh yduldeohv dvvrfldwhg zlwkwkh h{whuqdo hgjhv duh nqrzq dv wkh �v|pero yduldeohv/� dqg uhsuhvhqwwkh frxsolqj ri wkh v|vwhp zlwk lwv hqylurqphqw/ zkloh wkh lqwhuqdo hgjh0yduldeohv duh nqrzq dv wkh �vwdwh yduldeohv/� vlqfh wkh| uhsuhvhqw lqwhuqdovwdwhv ri wkh v|vwhp1

Wr hdfk qrgh Q lv dvvrfldwhg d orfdo frqvwudlqw / h{suhvvlqj d uhodwlrq0vkls dprqjvw wkh yduldeohv rq wkh hgjhv frqqhfwhg wr wkdw qrgh1 Dvvkrzq lq wkh wrs sduw ri Iljxuh 514/ iru d vlqjoh qrgh Q lq d judsk J/zklfk kdv N . 4 hgjhv wkdw duh dvvrfldwhg zlwk wkh yduldeohv {f> {�>111/

Page 10: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp 64

Dvvrfldwlrq Sxusrvh

H{whuqdo hgjh V|pero yduldeoh Lqsxw2RxwsxwLqwhuqdo hgjh Vwdwh yduldeoh Phvvdjh0sdvvlqjQrgh Orfdo frqvwudlqw Frpsxwdwlrq

Wdeoh 5141 Frpsrqhqwv ri d qrupdo judsk

N

R

( )0xRL→µ

( )0xLR→µ

0x

1x

2x

L

0x

210 AAASN ××⊂

external edge

constraint set

internal edge

message vector

node

edge-variable

Iljxuh 5141 H{dpsohv ri qrupdo judskv

{g / wkh vhw ri doo frpelqdwlrqv ri wkhvh yduldeohv wkdw vdwlvi| wkh orfdo

frqvwudlqw lv glfwdwhg e| wkh frqvwudlqw vhw V� 1

V� � D%f �D%� � � � � �D%g

Wklv frqvwudlqw vhw lv d vxevhw ri wkh gluhfw surgxfw ri wkh fruuhvsrqglqj

doskdehwv iru hdfk ri wkh yduldeohv1 Wkh uroh ri wkh hgjhv dqg qrghv lq

d qrupdo judsk lv vxppdul}hg lq wkh Wdeoh 5141

Lw zloo eh vhhq lq Vhfwlrq 514 wkdw dq h{wulqvlf suredelolw| S i |

�+{f, iru

wkh hgjh0yduldeoh {f fdq eh fdofxodwhg dw qrgh Q edvhg rq wkh lqwulqvlf

suredelolwlhv S �?|

�+{�, iru wkh rwkhu hgjh0yduldeohv frqqhfwhg wr qrgh Q

65 VRIW LWHUDWLYH GHFRGLQJ

dv iroorzv/

S i |� +{f @ �f, @ f�

fS +Q m {f @ �f,

@ f�

f

[

1�c12c���c1&E1fc1�c12 c���c1g�M7�

g\

�'�

S �?|� +{� @ ��,

zkhuh f�f lv d qrupdol}dwlrq frqvwdqw/ dqg wkh vxppdwlrq lv ryhu doofrpelqdwlrqv ri wkh yduldeohv ��> �2> ===> �& +zkhuh �� 5 D�, vxfk wkdw wkhfrq�jxudwlrqv +�f> ��> �2> ===> �g, vdwlv�hv wkh frqvwudlqw vhw V� 1

Dorqj hdfk hgjh ri wkh judsk duh sdvvhg phvvdjh yhfwruv lq erwkgluhfwlrqv1 Iru dq hgjh0yduldeoh {f ehwzhhq qrghv O dqg U/ wkh phvvdjhyhfwru jrlqj iurp O wr U lv ghqrwhg e| �$� u<- +{f,/ zkloh wkh phvvdjhyhfwru iurp U wr O lv ghqrwhg e| �$� -<u +{f,/ dv vkrzq lq wkh judsk lqerwwrp sduw ri Iljxuh 5141 +Lq wkh fdvh ri d elqdu| yduldeoh { 5 i3> 4j/lw lv srvvleoh wr uhsuhvhqw wklv phvvdjh e| d orj0olnholkrrg udwlr/ zklfkfdq eh ghqrwhg e| OOUu<- +{,1, Lw zloo eh ghprqvwudwhg lq Vhfwlrq515 wkdw wklv h{wulqvlf suredelolw| zlwk uhvshfw wr qrgh O iru wkh hgjh0yduldeoh {f lv wkh fruuhfw txdqwlw| wr xvh iru wkh lqwulqvlf suredelolw|iru {f zlwk uhvshfw wr qrgh U1 Wkhq wkh dssursuldwh gh�qlwlrq ri wkhphvvdjh yhfwruv duh

�$� u<- +{f, @�$S i |

u +{f, @�$S �?|

- +{f,�$� -<u +{f, @

�$S i |

- +{f, @�$S �?|

u +{f, >

vr wkdw wkh phvvdjh yhfwru �$� u<- +{f, lv xvhg wr fduu| wkh h{wulqvlfsuredelolw| rxwsxw iurp qrgh O wr eh xvhg dv wkh lqwulqvlf suredelolw| lqqrgh U1 Vlploduo|/ wkh phvvdjh yhfwru �$� -<u +{f, sdvvhv wkh h{wulqvlfsuredelolw| iurp qrgh U wr eh xvhg dv wkh lqwulqvlf suredelolw| iru qrghO1

E| uhshdwhg frpsxwdwlrqv dw wkh qrghv/ dqg sdvvlqj phvvdjh yhfwruvdorqj wkh hgjhv/ wklv phvvdjh0sdvvlqj dojrulwkp +PSD, |lhogv sured0elolwlhv wkdw uh hfw wkh orfdo frqvwudlqwv dqg wkh vwuxfwxuh ri wkh hqwluhjudsk1 Wklv dojrulwkp jlyhv d zd| wr vroyh gl!fxow sureohpv li wkh| fdqeh uhsuhvhqwhg e| d judsk ri uhodwlyho| vlpsoh orfdo frqvwudlqwv1 Ghfrghuvedvhg rq wklv dssurdfk duh dovr nqrzq dv DSS +d srvwhulrul suredelolw|,ghfrghuv/ vlqfh wkh �qdo uhvxow lv riwhq d srvwhulru suredelolw|1 ^Jdo95`1+Wkhvh ghfrghuv duh dovr nqrzq lq wkh olwhudwxuh dv PDS +pd{lpxp d

srvwhulrul, ghfrghuv/ zklfk fdq eh d plvqrphu vlqfh pd{lpl}dwlrq grhvqrw dozd|v wdnh sodfh lq wkh ghfrglqj surfhvv,1Wkh suhgrplqdqw irupxodwlrq ri wklv dojrulwkp lq wkh olwhudwxuh kdv

ehhq wkh vxp0surgxfw dojrulwkp +VSD, iru idfwru judskv +uhi1 Nvfklv0

Page 11: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp 66

fkdqj hw do1 ^NIO34` dqg Zlehuj ^Zle<:`,1 Idfwru judskv duh d jhq0hudol}dwlrq ri Wdqqhu judskv ^Wdq;4`/ dqg frqvlvw ri d elsduwlwh judskehwzhhq wzr w|shv ri qrghv/ ixqfwlrq qrghv dqg yduldeoh qrghv1 +Vrphri wkh yduldeoh qrghv duh nqrzq dv vwdwh qrghv,1 Lq frpsdulvrq/ qrupdojudskv kdyh rqo| rqh w|sh ri qrgh +vlplodu wr wkh ixqfwlrq qrgh,/ dqgyduldeohv duh dvvrfldwhg wr wkh hgjhv1 Lw lv vkrzq lq ^Iru34` wkdw zlwkwkh surshu wudqvirupdwlrqv/ d idfwru judsk fdq eh uhsuhvhqwhg e| d qru0pdo judsk zlwk hvvhqwldoo| wkh vdph ghfrglqj frpsoh{lw|/ vr wkdw wkhirfxv rq qrupdo judskv grhv qrw ohvvhq wkh jhqhudolw| ru h!flhqf| ri wkhdojrulwkp18

Wkh vxp0surgxfw dojrulwkp lv dovr nqrzq dv wkh �wzr0zd|� dojrulwkp+WZD, ^Iru<:`1 Wkh frqqhfwlrq wr eholhi sursdjdwlrq +ES, iru Ed|hvldqqhwzrunv ^Shd;;` zdv irxqg lq PfHolhfh hw do1 ^PPF<;`1 Dqrwkhu iru0pxodwlrq xvlqj wkh jhqhudol}hg glvwulexwlyh odz +JGO, iru ixqfwlrqv lvjlyhq e| Dml dqg PfHolhfh ^DP33`1

51 PHVVDJH0SDVVLQJ RQ JUDSKV

Wkh edvlf sulqflsoh ri wkh phvvdjh0sdvvlqj dojrulwkp lv wr euhdn xs dfrpsxwdwlrqdoo| gl!fxow sureohp lqwr d pxowlwxgh ri hdvlhu vxesureohpv1Lw rshudwhv e| lwhudwlyho| sdvvlqj suredelolvwlf phvvdjhv lq d judsk/ dqgfdq eh xvhg wr hydoxdwh h{wulqvlf dqg srvwhulru suredelolwlhv edvhg rqlqwulqvlf suredelolwlhv dqg wkh vwuxfwxuh ri wkh judsk1

Lq wklv vhfwlrq/ zh frqvlghu phvvdjh0sdvvlqj rq judskv e| vwxg|lqj wkhvlwxdwlrqv ri rqh qrgh/ wzr qrghv dqg pdq| qrghv19 Iru d vlqjoh qrgh/wkh h{wulqvlf suredelolw| iru dq hgjh0yduldeoh lv edvhg rq wkh lqwulqvlfsuredelolwlhv rq doo wkh rwkhu hgjh0yduldeohv1 Iru wzr qrghv frqqhfwhg e|dq hgjh/ lw lv vkrzq wkdw wkh h{wulqvlf suredelolw| iurp rqh qrgh vhuyhv dvwkh lqwulqvlf suredelolw| iru wkh rwkhu1 Wkhvh uhvxowv duh wkhq jhqhudol}hgiru wkh fdvh ri pdq| qrghv/ dqg wkh phvvdjh0sdvvlqj dojrulwkp lv ghulyhg1

514 RQH QRGH

Frqvlghu d vlqjoh qrgh Q / zlwk N . 4 hgjhv/ dv vkrzq lq Iljxuh 5151Wkh hgjhv duh dvvrfldwhg zlwk wkh yduldeohv {f> {�>111/ {g / zklfk eh0orqj wr wkh doskdehwv Df>D�> ===>Dg / uhvshfwlyho|1 Vxssrvh wkdw lqwulq0vlf suredelolwlhv zlwk uhvshfw wr qrgh Q duh dydlodeoh iru wkh yduldeohv{f> {�> ===> {g 1 Uhfdoo wkdw iru hdfk yduldeoh {�/ wkh lqwulqvlf suredelo0

8Qrupdo judskv fdq eh uhsuhvhqwhg e| idfwru judskv e| lqvhuwlqj d yduldeoh qrgh lqwr hyhu|

hgjh ri wkh qrupdo judsk1 Rq wkh rwkhu kdqg/ wr frqyhuw d idfwru judsk lqwr d qrupdo judsk/

vlpso| uhsodfh lwv yduldeoh qrghv zlwk dq htxdolw| qrgh/ dv gh�qhg lq Vhfwlrq 61419Olqjxlvwv uhsruw wkdw wkhuh h{lvw odqjxdjhv zkhuh wkh frqfhsw ri qxpehuv lv olplwhg wr rqh/

wzr/ dqg pdq|1

67 VRIW LWHUDWLYH GHFRGLQJ

lw| S �?|�

+{�, ghqrwhv d suredelolw| yhfwru ri ohqjwk mD�m/ zlwk rqh hqwu|S �?|�

+{� @ ��, iru hdfk srvvleoh ydoxh �� 5 D�1 Lq dgglwlrq/ lw lv dvvxphgwkdw wkh yduldeohv duh lqghshqghqw/ vr wkdw wkh mrlqw lqwulqvlf suredelolw|

S �?|�

�i{� @ ��j

g

�'f

�idfwruv lqwr wkh surgxfw ri wkh lqglylgxdo lqwulqvlf

suredelolwlhvTg

�'f S�?|�

+{� @ ��,1Wkh srvwhulru suredelolw| lv d frqglwlrqdo suredelolw| S +{f @ �f m Q,

wkdw fdq eh zulwwhq xvlqj Ed|hv* wkhruhp +5143, dv

TLt|ih�Lh} �� ~S +{f @ �f m Q, @

4

S +Q,

surs1 wr h{wulqvlf} �� ~S +Q m {f @ �f,

�?|h�?t�U} �� ~S +{f @ �f,= +5148,

Wkh lqwulqvlf suredelolw| lv

S �?|�

+{f @ �f, @ S +{f @ �f, >

zkloh wkh h{wulqvlf suredelolw| lv sursruwlrqdo wr S +Q m {f @ �f,/ dv lq

S i |�

+{f @ �f, @ f�

%fS +Q m {f @ �f, 1

Wkh srvwhulru suredelolw| lv jlyhq e|

STLt|

�+{f @ �f, @ S +{f @ �f m Q, 1

N

( )0xRL→µ

0x

1x

Kx

2x

Iljxuh 5151 Rqh qrgh

Page 12: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvd

jh0sdvvlq

jdojrulwk

p68

51414

H[WULQVLFSUREDELOLW\

Edvhg

rqwkhlqwulq

vlfsuredelolwlhv

S�?|

�+{

f@�f ,/

S�?|

�+{

�@�� ,/

===/

S�?|

�+{

g@�g,/

irudooriwkhyduld

eohv/

zhfdofx

odwh

wkhh{wulq

vlfsure0

delolw|

iruwkhyduld

eoh{fzlwk

uhvshfwwr

wkhqrghQ1Lq

wklv

fdvh/

wkh

hyhqwQ

fruuhvs

rqgvwr

wkhhyhqwwkdwwkhydoxhv

riwkhhgjh0y

duld

eohv

{f >{

� >===>{

gfrqqhfwhg

wrwkhqrghQ

vdwlvi|

wkhorfdofrqvwud

lqw/zklfk

lvgh�qhg

e|dfrqvwud

lqwvhw

V��D

f�D

������

Dg=

Lqrwkhu

zrugv/

wkhhyhqwQ

lvwux

hh{dfwo|

zkhq

wkhyduld

eohvwdnhrq

ydoxhv

{f@�f /{�@�� /

111/{g@�g/vxfk

wkdw

+�f >�

� >===>

�g,5V�1

Wrhydoxdwh

wkhh{wulq

vlfsuredelolw|

/zhhydoxdwh

wkhfrqglwlr

qdosure0

delolw|

S+Q

m{f@�f ,1

Iluvw/

zhqrwh

wkdwwkhsurgxfwriwkhdoskdehwv

D� �

����Dg

jlyhv

dfrpsohwh

vhwriydoxhv

iruwkhyduld

eohv{� /{

2 /===>{g/

vrwkdw

[

E1�c���c1

g�

M��fuuuf�g

S�i{

�@

�� j

g�'� �

@4>

dvlq

+514,1Wkhq

lwlvsrvvle

ohwr

dsso|

+515,wr

rewdlq

wkhghfrp

srvlwlrq

S+Q

,@

[E1�c���c1

g�

M��fuuuf�g

S�Q>i{�@

�� j

g�'� �

=

Wkhfrq

glwlrq

dosure

delolw|

S+Q

m{f@

�f ,

fdqwkxvehhydox

dwhgdv

S+Q

m{f@

�f ,

@[

E1�c���c1

g�

M��fuuuf�g

S�Q>i{�@

�� j

g�'�m{f@

�f �

=+5149,

Dsso|lqjwkhfrq

glwlrq

dosure

delolw|

uxoh

+51:,wr

hdfkwhup

lqwkhvx

p0

pdwlrq

jlyhv

S�Q>i{�@

�� j

g�'�m{f@

�f �

@S�Q

mi{�@

�� j

g�'f �

�S�i{�@

�� j

g�'�m{f@

�f �

=+514:,

Wkhq

xvlq

jwkhlqghs

hqghq

fhri

wkhydulde

ohvi{� j

g�'f /

wkhfrq

glwlrq

0lqjrq

{ffdq

ehuhp

ryhgdv

lq+518,/

dqgwkhmrlq

wsure

delolw|

fdqeh

idfwruhgdv

lq+516,/

uhvxowlq

jlq

wkhh{

suhvvlrq

S�i{�@

�� j

g�'�m{f@

�f �

@g\�'�

S+{

�@

�� ,

+514;,

69VRIWLWHUDWLYHGHFRGLQJ

iruwkhvhfrq

gwhup

rqwkhuljk

w0kdq

gvlg

hri

+514:,1Rqwkhrwk

hukdq

g/iru

wkh�uvw

whuprq

wkhuljk

w0kdq

gvlg

hri

+514:,/uhfdoo

wkdw

wkhhyhq

wQ

lvwux

hh{dfwo|

zkhq

wkhydox

hri

wkhydulde

ohvirup

dfrp

elqdwlrq

wkdw

ehorq

jvwr

wkhfrq

vwudlqwvhw

V�1

Wkhq

wkh

frqglwlrq

dosure

delolw|

lvjlyhq

e|

S �Q

mi{�@

�� j

g�'f �

@ �4

li+�

f >�� >===>�

g,5V�

3rwk

huzlvh

>

zklfk

fdqehh{suhvvhg

e|

dqlqglfd

wru

ixqfwlr

q/ghqrwhg

e|

vtxduh

eudfnhwv1

S �Q

mi{�@

�� j

g�'f �

@^+�

f >�� >===>�

g,5V�`

+514<,

Wkhq

sxwwlq

jwrjhwk

hu+514:,/

+514<,/dq

g+514;,/

wkhh{suhvvlrq

lq+5149,

fdqehuhz

ulwwhqdv

S+Q

m{f@

�f ,

@[E1

�c���c1

g�

M��fuuuf�g

^+�f >�

� >===>�g,5V�`g\�'�

S�?|

�+{

�@

�� ,=

+5153,Wklvxvh

riwkhlqglfdwru

ixqfwlrq

iruwkhvxppdwlrq

pdnhv

lwfohdu

wkdw

wkhfrq

vwudlqwvhw

V�

lvxvhg

wrvhohfw

rxwfhuwdlq

frpelqdwlrq

v1Ilqdoo|/

wkhlqglfdwru

ixqfwlrq

fdqehlqfrus

rudwhglq

wkhudq

jhri

wkhvxppdwlrq

wrhqgxszlwk

dfrp

sdfw

h{suhvvlrq

iruwkhfrq

glwlrq

dosure

delolw|/

S+Q

m{f@

�f ,

@[1

�c���c1

g

E1fc1�c���c1

g�M7�

g\�'�

S�?|

�+{

�@

�� ,=

+5154,

Wkhh{wulq

vlfsure

delolw|

lvwkhq

jlyhqe|

Si |

�+{

f@

�f ,

@f�%f S

+Qm{f@

�f ,

@f�%f

[1�c���c1

g

E1fc1�c���c1

g�M7�

g\�'�

S�?|

�+{

�@

�� ,

+5155,

zkhuh

wkhqrup

dol}dwlrqfrq

vwdqwlv

f�%f@

3C[1fM�f

S+Q

m{f@

�f , 4D

3�

=

Wklv

htxdwlrq

+5155,lv

wkhnh|

htxdwlrq

riwkhphvvdjh0s

dvvlqjdojr0

ulwkp1Z

lwkuhvs

hfwwr

dqrgh/

lwjlyhv

wkhh{wulq

vlfsure

delolw|

irudq

Page 13: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvd

jh0sdvvlq

jdojrulwk

p6:

hgjh0ydulde

ohlq

whupvri

wkhlqwulq

vlfsure

delolwlhv

irudoo

wkhrwk

huhg

jh0ydulde

ohvri

wkhqrgh1

Vrp

hh{

dpsohv

riqrg

hvduh

wkhht

xdolw|

qrghdq

gsdulw|

fkhfn

qrg

hjlyhq

lqVhfwlrq

v614

dqg6151

51415

SRVWHULRUSUREDELOLW\

Wkhsrvwhulru

sure

delolw|

fdqehfrp

sxwhg

xvlq

j+5148,/

dorqjzlwk

wkhh{

suhvvlrq

iruS+Q

m{f@�f ,

lq+5154,=

S+{

f@�f

mQ,@

4

S+Q

,S+Q

m{f@�f ,S

�?|

�+{

f@�f ,

@4

S+Q

,

[1�c���c1

g

E1fc1�c���c1

g�M7�

g\�'f

S�?|

�+{

�@�� ,

Lqwur

gxflq

jd

qrup

dol}dwlrqfrq

vwdqwff

@+S

+Q,,3�/

wkh

srvwhulru

sure

delolw|

fdqehuhz

ulwwhqdv

STLt|

�+{

f@

�f ,

@ff

[1�c���c1

g

E1fc1�c���c1

g�M7�

g\�'f

S�?|

�+{

�@

�� ,=

+5156,

Wkhsure

delolw|

S+Q

,fdq

ehh{sdq

ghg

dviroorz

v/

S+Q

,@[1fM�

S+Q

m{f@

�f ,S

�?|

�+{

f@

�f ,>

vrwkdw

dsso|lqj+5154,

gh�

qhv

wkhqrup

dol}dwlrqfrq

vwdqw/

ff@

+S+Q

,,3�@

3EEEC[

1fc1�c���c1

g

E1fc1�c���c1

g�M7�

g\�'f

S�?|

�+{

�@

�� , 4FFFD

3�

1

Wklvvxppdwlrq

lq+5156,

fdqdovr

ehxqghuvwr

rgdv

dpdujlqdosure

0de

lolw|/zkhuh

dixqfwlrq

j+�

f >�� >===>�

g,@

^+�f >�

� >===>�g,5V�`g\�'f

S�?|

�+{

�@

�� ,

lvvxpphg

ryhudoo

yduldeohv

h{fhs

wiru

rqh/

STLt|

�+{

f@

�f ,

@ff

[1�c���c1

g

j+�

f >�� >===>�

g,=

6;VRIWLWHUDWLYHGHFRGLQJ

51416

VLPSOLILHGQRWDWLRQ

Lwlvfrq

yhqlhq

wwr

lqwur

gxfh

dvlp

sol�

hgqrwdwlr

qlq

zklfk

wkhydulde

oh{

�wdnhv

rqdydox

hwkdw

lvohiw

xqvwdwhg

lqwkhhtxdwlrq

vexwfdq

eh

lqihuuhg

iurpfrq

wh{w1

Lqsduwlfx

odu/wkhh{suhvvlrq

S+{

� ,uhihuv

wrwkh

sure

delolw|

S+{

�@�� ,

wkdw

{�htxdov

vrphlp

solflw

ydoxh�� 1

Frq

vlghu

wkhiroorz

lqjh{suhvvlrq

lqvlp

sol�

hgqrwdwlrq

=

S+{

f ,@[

E%fc%�c%2�M7

;t%��

S+{

f ,S+{

� ,S+{

2 ,+5157,

Wrlqwhus

uhwwkhvlp

sol�

hgqrwdwlrq

/�uvw

wdnhdq

|ydulde

oh{�dq

gdvvljq

lwwr

dydox

he|uhs

odflqj{�zlwk

wkhh{suhvvlrq

i{�@�� j1

Qh{w/

wkhqrwdwlrq

�i{f j

lvwkhqrw0vx

pqrwdwlrq

+lqwur

gxfhg

lq^N

IO34`,/

zklfk

lqglfdwhv

wkdw

wkhvxppdwlrq

lvryhu

doofrq

�jx

udwlrqv

ridoo

yduldeohv/

h{fhs

wiru

wkhydulde

oh+v,lq

wkhqrw0vx

ph{suhvvlrq

1Lq

rwkhu

zrug

v/lq

wklv

fdvh+5157,/

wkhvxppdwlrq

lvryhu

doofrpelqdwlrq

vri

i{�@

�� j

dqgi{2@

�2 j

iru��5

D�dq

g�25

D2 /

zkloh

{flv

qrw

lqfox

ghg

dvdydulde

ohri

vxppdwlrq

1+Q

rwhwkdw

wklv

qrwdwlrq

lvriwhq

uhgxqgdq

w/vlq

fhwkhqrw0vx

pphg

yduldeoh

fdqriwhq

ehghg

xfhg

e|lq0

vshfwlrq

/h1j1/

wkhydulde

ohkdv

douhdg|ehhq

gh�

qhg

zlwk

dydox

hrx

wvlgh

wkhvxppdwlrq

1,Ilqdoo|/

wkhuhihuhq

fhwr

wkhfrq

vwudlqwvhw

Vfdq

ehfkdq

jhgwr

dqlqglfdwru

ixqfwlrq

1Iru

h{dp

soh/

wkhvlp

sol�

hgh{suhvvlrq

+5157,h{sdq

gv

lqwr

wkhiroorz

lqjh{suhvvlrq

/

S+{

f@

�f ,

@[1�M��c

12M�2

^+�f >�

� >�2 ,5V`S

+{f@

�f ,S

+{�@

�� ,S

+{2@

�2 ,>

+5158,zkhuh

wkhvxppdwlrq

lvryhu

wkhydulde

ohv��dq

g�2 /zlwk

wkhuhvwulfwlrq

wkdw

wkhwuls

oh+�

f >�� >�

2 ,ehorq

jvwr

wkhvhw

V1Wklv

h{dp

soh

looxvwudwhv

krz

wkhvlp

sol�

hgqrwdwlrq

lq+5157,

fdqehxvhg

wrfrp

sdfwo|

uhsuhvhq

wwkhh{suhvvlrq

+5158,1Dsso|lqjwklvvlp

sol�

hgqrwdwlrq

wrwkhh{suhvvlrq

iruh{wulq

vlfsure

0de

lolw|iru

{flq

+5155,|lhog

v

Si |

�+{

f ,@

f�%f

[

E%fc%

�c���c%

g�M7�

;t%f�

g\�'�

S�?|

�+{

� ,=

+5159,

Page 14: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp 6<

Wkh srvwhulru suredelolw| iru {f lq +5156, fdq vlploduo| eh zulwwhq lqvlpsol�hg qrwdwlrq=

STLt|�

+{f, @ f%f

[

E%fc%�c���c%g�M7�;t%f�

g\

�'f

S �?|� +{�, = +515:,

Lw lv xqghuvwrrg wkdw lq erwk wkhvh h{suhvvlrqv/ +5159, dqg +515:,/ wkhsuredelolwlhv duh fdofxodwhg iru {f htxdo wr vrph lpsolhg ydoxh �f +fkrvhqiurp wkh doskdehw Df,1 Lq rwkhu zrugv/ wkh suredelolw| h{suhvvlrq S +{f,gh�qhv d ixqfwlrq lq whupv ri wkh yduldeoh {f/ zkhuh lw lv srvvleoh wrhydoxdwh wkh ixqfwlrq iru dq| ydoxh �f ri wkh yduldeoh {f1

Lq dgglwlrq/ wkh vxppdwlrq lv ryhu doo frpelqdwlrqv ri ydoxhv iru wkhyduldeohv {� @ ��/ {2 @ �2/ xs wr {g @ �g 1 Lq rwkhu zrugv/ h{fhsw iruyduldeohv wkdw kdyh douhdg| ehhq dvvljqhg ydoxhv +l1h1/ {f @ �f lq wklvfdvh/ gh�qhg rq wkh ohiw0kdqg vlgh ri wkh htxdwlrq,/ wkh yduldeohv udqjhlq ydoxh ryhu wkhlu uhvshfwlyh doskdehwv/

�� 5 D�> �2 5 D2> ===> �g 5 Dg 1

Ilqdoo|/ wkhvh ydoxhv/ dorqj zlwk {f @ �f/ pxvw vdwlvi| wkh orfdo frqvwudlqwdw Q / vr wkdw wkh h{suhvvlrq

+{f> {�> {2> ===> {g, 5 V�

lq +5159, dqg +515:, uhdoo| phdqv wkdw hdfk frpelqdwlrq ri ydoxhv pxvwehorqj wr wkh frqvwudlqw vhw=

+�f> ��> �2> ===> �g, 5 V�

Wkh h{suhvvlrq iru S i |�

+{f, lq +5159, wkxv uhsuhvhqwv d ixqfwlrq iru fdo0fxodwlqj wkh h{wulqvlf suredelolw| S i |

� +{f @ �f, iru dq| ydoxh �f 5 D1Wkhvh wzr h{suhvvlrqv +5159, dqg +515:, jhqhudol}h wr dq| hgjh0yduldeoh

ri wkh qrgh1 Iru d qrgh Q zlwk N . 4 qhljkerulqj hgjh0yduldeohvi{f> {�> ===> {gj/ wkh h{wulqvlf dqg srvwhulru suredelolwlhv iru {�/ edvhgrq wkh lqwulqvlf suredelolwlhv iru wkh hgjh0yduldeohv {, iru 3 � o � N/zkhuh o 9@ l/ duh dv iroorzv=

S i |� +{�, @ f�%�

[

E%fc%�c���c%g�M7�;t%��

g\

,'f, �'�

S �?|� +{,, +515;,

STLt|�

+{�, @ f%�

[

E%fc%�c���c%g�M7�;t%��

g\

,'f

S �?|� +{,, +515<,

73 VRIW LWHUDWLYH GHFRGLQJ

Wkh frqvwdqwv f�

%�dqg f%�

duh wkh dssursuldwh qrupdol}dwlrq frqvwdqwv1Lq erwk fdvhv/ wkh yduldeoh {� kdv d �{hg ydoxh �� gh�qhg rq wkh ohiw0kdqg vlgh/ dqg vr wkh vxppdwlrq lv ryhu wkh rwkhu yduldeohv i{,j vxfkwkdw o 9@ l1 Lq dgglwlrq/ lq +515;,/ wkh lqwulqvlf suredelolw| S �?|

�+{�, iru

{� lv ohiw rxw ri wkh surgxfw1

515 WZR QRGHV

Vxssrvh wkdw wkhuh duh wzr qrghv O dqg U wkdw duh frqqhfwhg e|dq hgjh {f1 Wr wkhvh qrghv duh dvvrfldwhg orfdo frqvwudlqwv/ ghvfulehge| vhwv Vu dqg V-/ uhvshfwlyho|1 Lq dgglwlrq wr wkh hgjh {f/ qrgh O

kdv N dgglwlrqdo hgjhv +odehohg e| {u�/ {u

2/ 111/ {u

g, dqg qrgh U kdv N �

dgglwlrqdo hgjhv +odehohg e| {-�/ {-

2/ 111/ {-

g�,1

L R

( )0xRL→µ

( )0xLR→µ

0x

Lx1

LKx R

Kx '

Rx1

Iljxuh 5161 Wzr qrghv

51514 H[WULQVLF SUREDELOLW\ ZLWK UHVSHFW WR

WKH JUDSK

Ohw wkh v|pero J uhsuhvhqw wklv judsk/ zklfk lqfoxghv wzr qrghv dqgN .N � h{whuqdo hgjhv dqg 4 lqwhuqdo hgjh/ dv vkrzq lq Iljxuh 5161 Lwlv dvvxphg wkdw wkhuh duh lqwulqvlf suredelolwlhv iru wkh h{whuqdo hgjh0yduldeohv +{u

�/ {u

2/ 111/ {u

g/ {-

�/ {-

2/ 111/ {-

g�,1 Wkh lqwulqvlf suredelolwlhvzlwk uhvshfw wr wkh judsk J wdnh wkh irup/

S �?|

:

�{u�

�@ S

�{u�

�iru l @ 4> 5> ===> N

S �?|

:

�{-�

�@ S

�{-�

�iru l @ 4> 5> ===>N �

S �?|

: +{f, @4

mDfm

Page 15: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvd

jh0sdvvlq

jdojrulwk

p74

zkhuh

wkhlqwulq

vlfsure

delolw|

z1u1w1

Jiru

wkhlqwhuq

dohg

jh{flv

vhwwr

dxqlirup

sure

delolw|/

h1j1/S

�?|

:+{

f@�f ,@

��f�iru

doo�f5Df 1

Wkhfrq

vwudlqwvhw

V:iru

wkhjuds

kfrq

vlvwvri

doofrq

�jx

udwlrqvri

wkhN

.N

�.4ydulde

ohv

�{f >{

u�>{u2>===>{

ug>{

-�>{

-2>===>

{-g

� �

vxfk

wkdw

wkhds

surs

uldwhvxevhw

riwkhvh

ydoxhv

vdwlvi|wkhfrq

vwudlqw

vhwviru

erwk

qrghv

Odq

gU1Wkhq

php

ehuvk

lslq

wkhfrq

vwudlqwvhw

V:

fdqehh{suhvvhg

e|

dqlqglfdwru

ixqfwlrq

/zklfk

idfwruvlqwr

wkh

surgxfw

rilqglfdwru

ixqfwlrq

vri

wkhwz

rqrghv1

��{f >{

u�>{

u2>===>

{ug>{

-�>{

-2>===>{

-g� �5V

: �@

��{f >{

u�>{

u2>===>

{ug

�5Vu �

� ��{f >{

-�>{

-2>===>{

-g� �5V-

+5163,

Wkhq

zlwk

wkhgh�

qlwlrq

riwkhlqwulq

vlfsure

delolwlhv

dqgdfrq

vwudlqw

vhwV

:/lw

lvsrvvle

ohwr

�qg

wkhh{wulq

vlfsure

delolwlhv/

dvlq

+515;,1Vxssrvh

wkdw

zhuhq

dphdoo

riwkhydulde

ohve|

+{f >{

� >{2 >===>{

gng

�,@

�{f >{u�>{

u2>===>{

ug>{-�>{-2>===>{

-g� �=

Wkhq

dvlq

+515;,/wkhh{wulq

vlfsure

delolw|

irudq

|hg

jh0yduldeoh

fdqeh

irxqgiurp

wkhlqwulq

vlfsure

delolwlhv

irudoo

wkhrwk

huhg

jh0yduldeohv1

Si |

:+{� ,@f�%�

[

+%fc%�c���c%

gng

�,M7:

;t%� �

gng

�\,'

f,�'

S�?|

:+{

, ,+5164,

Wklv

h{suhvvlrq

fdqehfrp

solfdwhg

vlqfh

wkhqxpehu

rihgjh0ydulde

ohvwkdw

ydu|lq

wkhvxppdwlrq

lvN

.N

�1Lw

wxuqvrx

w/dv

zhzloo

vhh/wkdw

lwlv

srvvle

ohwr

xvh

wkhfrq

vwudlqwvhwv

Vudq

gV-wr

frpsxwh

wkh

h{wulq

vlfsure

delolwlhv

iruhdfk

qrghvhs

dudwho|/lq

vxfk

dzd|

wkdw

wkh

�qdo

uhvxowv

duhwkhvdp

hdv

+5164,1

51515

PDUNRYSURSHUW\

Zh�uvw

yhuli|wkdw

wkhhyhq

wvO/i{f@�f j

dqgU/lq

wkdw

rughu/

irupdPdunry

fkdlq

1Lq

rwkhu

zrug

v/zhzloo

vkrz

wkdw

S+O>U

m{f ,@S+O

m{f ,S+U

m{f ,1

+5165,

75VRIWLWHUDWLYHGHFRGLQJ

Wkhsurri

iroorzviurp

h{sdq

glqjlq

whupvri

wkhrwk

huydulde

ohv1Iru

qrwdwlrq

dofrq

yhqlhq

fh/ohw

}u@ �

{u� �

g�'�dq

g}-@ �

{-� �

g�

�'� 1

Wkhq

S+O>U

m{f ,

@ [5uc5-

S �O>U

>}u>}-

m{f �

@ [5uc5-

S �O>U

m{f >}u>}

- �S �}

u>}

-m{f �=

+5166,

Wkh�uvw

whuplq

wkhuljk

w0kdq

gvlg

hri

+5166,lv

jlyhqe|dq

lqglfdwru

ixqfwlrq

wkdw

lqglfdwhv

zkhwk

huwkhfrq

�jx

udwlrqehorq

jvwr

wkhfrq

0vwudlq

wvhw

V:/zklfk

fdqehidfwruhg

xvlq

j+5163,=

S �O>U

m{f >}u>}

- �@ ��

{f >}

u>}

- �5V: �

@ ��{f >}

u �5Vu ���

{f >}

- �5V- �

@S �

Om{f >}

u �S �U

m{f >}

- �+5167,

Phdq

zkloh/

wkhvhfrq

gwhup

lqwkhuljk

w0kdq

gvlg

hri

+5166,uhg

xfhv

wr

S �}u>}

-m{f �

@S �

}u �

S �}- �

+5168,

vlqfh

wkhydulde

ohvduh

lqghs

hqghqw1

Sxwwlq

j+5167,

dqg+5168,

wrjhwkhu

lq+5166,

jlyhv

S+O>U

m{f ,

@ [5uc5-

S �Om{f >}u �

S �}u �

S �U

m{f >}

- �S �

}- �

>

vrwkdw

e|dq

dssolfdwlrq

riwkhFduwhvldq

0surgxfw

glvwule

xwlyh

odz+z

klfk

lvjhq

hudol}hglq

^DP33`,/

zhkdyh

S+O>U

m{f ,

@ #[5u

S �Om{f >}

u �S �

}u � $#[5

-

S �U

m{f >}

- �S �

}- � $

@S+O

m{f ,S+U

m{f ,>

zklfk

zdv

wrehghp

rqvwudwhg

1

51516

FRPSXWLQJWKHH[WULQVLFSUREDELOLW\

IRUDQH[WHUQDOHGJH

Ohw

xvfrqvlg

huwkhh{wulq

vlfsuredelolw|

iru{-�

z1u1w1

J/zklfk

lv

jlyhq

e|

Si |

:

�{-�

�@

f�%-�

S�O>U

m{-�

iruvrphqrup

dol}d

wlrqfrqvwd

qwf

�%-�

1Iruqrwdwlr

qdovlp

solflw|/

ohw

�{-�

�g

�'2@

�{-2>{-�>===>

{-g

� �

Page 16: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvd

jh0sdvvlq

jdojrulwk

p76

uhsuhvhq

wwkhuhp

dlqlqjhg

jh0yduldeohv

riU

ehvlg

hv{fdq

g{-�1Wkhq

wkhfrq

glwlrq

dosure

delolw|

fdqehh{sdq

ghg

dv

S�O>U

m{-� �

@[%f

S�O>U>{f>|-

m{-� �

@[%fc

i%-�jg

�'�

S

�U

mO>{f > �

{-� �

g�

�'� �

S

�O>{f > �

{-� �

g�

�'2m{-� �

=

+5169,

Gxhwr

wkhPdunry

lw|ri

O/{f /

dqgU/wkhghs

hqghq

fhrq

Ofdq

eh

gurs

shg

iurpwkhfrq

glwlrq

dosure

delolw|

lq+5169,

vlqfh

wkhh{suhvvlrq

lvdouhdg

|frq

glwlrq

hgrq

{f 1

S

�U

mO>{f> �{-� �

g�

�'� �

@S�U

m{f>{-�>{-2>===>

{-g

� �+516:,

Lqdg

glwlrq

/iru

wkhvhfrq

gsduw

ri+5169,/

vrphri

wkhghs

hqghq

flhvfdq

ehuhp

ryhggxhwr

wkhPdunry

surs

huw|dq

gwkhlqghs

hqghqfh

riwkh

yduldeohv1

S

�O>{f> �{-� �

g�

�'2m{-� �

@S

�O

m{f> �{-� �

g�

�'� �

S

�{f> �{-� �

g�

�'2m{-� �

@S+O

m{f ,S�?|

:+{

f ,g

\�'2

S�?|

:

�{-� �

+516;,

Vlqfh

S+{

f ,lvwkhlqwulq

vlfsure

delolw|

z1u1w1

wkhjuds

kJ/uhfdoo

wkdw

lwzdv

vhwwr

dxqlirup

glvwule

xwlrq

S�?|

:+{

f@

�f ,

@�

��f� /zklfk

fdqwkhq

ehde

vruehg

lqwr

wkhqrup

dol}dwlrqfrq

vwdqw1

Sxwwlq

j+516:,

dqg+516;,

lqwr

+5169,jlyhv

Si |

:

�{-� �

@f�%-�

S�O>U

m{-� �

@f��%-�

[

%fc%-2c���c%

-g�

S�U

m{f >{

-�>===>{

-g� �S+O

m{f ,

g�

\�'2

S�?|

:

�{-� �

@f��%-�

[

+%fc%-�c%-2c���c%

-g� ,M7-

;i%-�j

S+O

m{f ,

g�

\�'2

S�?|

:

�{-� �

>+516<,

zkhuh

f��%-�

lvwkhds

surs

uldwhqrup

dol}dwlrqfrq

vwdqw1

Lqfrp

sdulvrq

/wkh

dqdorjrx

vh{suhvvlrq

iruh{wulq

vlfsure

delolw|

z1u1w1

wkhqrghU

lvjlyhq

77VRIWLWHUDWLYHGHFRGLQJ

e|

Si |

-

�{-� �

@f

�%-�

[

+%fc%-�c%-2c���c%

-g� ,M7-

;i%-�j

S�?|

-+{

f ,g

\�'2

S�?|

-

�{-� �

>+5173,

zklfk

lvdgluhfw

dssolfdwlrq

ri+515;,1

Frp

sdulq

jwkhvh

wzrhtxdwlrq

vvkrz

vwkdw

xvlq

jwkhqrghfrp

sxwd0

wlrq+5173,

zlwk

dqlqsxwri

S�?|

-+{

f ,vhw

wrS+O

m{f ,

jlyhvdq

lghqwlfdo

fdofxodwlrq

wrwkhh{wulq

vlfsure

delolw|

riwkhwz

r0qrghjuds

klq

+516<,1Khq

fh/wklv

vxjjhvwv

wkhiroorz

lqjvhw

ridvvljq

phqwv/

S�?|

-

�{-� �

#S

�?|

:

�{-� �

S�?|

u

�{u� �

#S

�?|

:

�{u� �

S�?|

-+{

f ,#

Si |

u+{

f ,+5174,

S�?|

u+{

f ,#

Si |

-+{

f ,

zkhuh

wkhh{wulq

vlfsure

delolw|

Si |

u+{

f ,lvvlp

so|

wkhqrup

dol}hgyhuvlrq

riS+O

m{f ,1

Lqrwk

huzrug

v/zhvwduw

zlwk

lqwulq

vlfsure

delolwlhv

z1u1w1

J/dq

gwkhq

dvvljqwkhp

wrwkhqrghv/

vrwkdw

S�?|

u

�{u� �

@S�?|

:

�{u� �1

Zlwk

wkhvh

lqwulq

vlfsure

delolwlhv

z1u1w1

O/wkhh{wulq

vlfsure

delolw|

iru{fz1u1w1

O

lvjlyhq

e|

Si |

u+{

f ,@f�%f

[

+%fc%u�c���c%

ug,M7u

;t%f�

g\�'�

S�?|

u

�{u� �

>

dvlq

+5159,1Wkhq

liwklv

h{wulq

vlfsure

delolw|

Si |

u+{

f ,lv

xvhg

dvwkh

lqwulq

vlfsure

delolw|

S�?|

-+{

f ,lq

+5173,/wkhuhvx

owlqjh{wulq

vlfsure

delolw|

z1u1w1

Uwxuqvrx

wwr

dovrehwkhh{wulq

vlfsure

delolw|

z1u1w1

wkhjuds

kJ/dffrug

lqjwr

+516<,/l1h1/

Si |

:

�{-� �

@Si |

-

�{-� �1

51517

H[WULQVLFIURPRQHQRGHLVLQWULQVLF

IRUWKHRWKHU

Wkhfkrlfh

riwkhhg

jh0yduldeoh{-�lq

+516<,dq

g+5173,

zdv

duelwudu|1

Vwduwlq

jzlwk

wkhdvvljq

phqwv

lq+5174,/

dqgds

so|lqjwkhxsgdwh

uxoh

+515;,wr

hdfkhgjh0ydulde

ohfrq

qhfwhg

wrqrghv

Odq

gU/wkhfrp

sx0

wdwlrqvdw

wkhwz

rqrghv

|lhog

h{wulq

vlfsure

delowlhv

wkdw

duhdovr

wkh

Page 17: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp 78

h{wulqvlf suredelolwlhv z1u1w1 wkh judsk J=

S �?|:

�{u�

�# S i |

u

�{u�

�iru l @ 4> 5> ===> N

S �?|:

�{-�

�# S i |

-

�{-�

�iru l� @ 4> 5> ===>N �

Wkhq lw lv qrw qhfhvvdu| wr xvh +5164, wr hydoxdwh wkh h{wulqvlf suredelolw|/exw lqvwhdg frpsxwdwlrqv fdq eh pdgh dw erwk qrghv1 Iru wkh fdvh rielqdu| yduldeohv/ wklv hqwdlov wzr vxppdwlrqv ri xs wr 5g dqg 5g

whupv/iru qrghv O dqg U/ uhvshfwlyho|/ udwkhu wkdq rqh vxppdwlrq ryhu 5gng�

whupv zkhq frqvlghulqj wkh hqwluh judsk1 Euhdnlqj xs wkh frpsxwdwlrqlq wklv zd| fdq wkxv uhgxfh wkh frpsoh{lw|1

Jlyhq wkh dvvljqphqwv lq +5174,/ zh fdq gh�qh phvvdjhv �u<- +{f,dqg �-<u +{f, ehwzhhq wkh qrghv/ dv vkrzq lq Iljxuh 5161 Wkh qrwdwlrq�u<- +{f, lqglfdwhv wkdw lw lv d phvvdjh iurp qrgh O wr qrgh U dorqjwkh hgjh dvvrfldwhg zlwk wkh yduldeoh {f1 +Lq dqdorj| wr wkh suredelolw|S +{f,/ wkh qrwdwlrq �u<- +{f, uhsuhvhqwv d ydoxh �u<- +{f @ �f, iruvrph lpsolflw ydoxh �f/ xvlqj wkh vlpsol�hg qrwdwlrq lq Vhfwlrq 514161,Wkh phvvdjhv fduu| wkh h{wulqvlf suredelolw| iurp rqh qrgh wr eh xvhgdv wkh lqwulqvlf suredelolw| dw wkh rwkhu qrgh1

�u<- +{f, @ S i |u +{f, @ S �?|

- +{f,

�-<u +{f, @ S i |- +{f, @ S �?|

u +{f,

Lq vkruw/ wkh h{wulqvlf suredelolw| iru rqh qrgh jlyhv wkh lqwulqvlf sure0

delolw| iru wkh rwkhu qrgh/ dqg wkh phvvdjhv �u<- +{f, dqg �-<u +{f,vhuyh wr sdvv wkhvh suredelolwlhv ehwzhhq wkh wzr qrghv1 Wkh edvlf lghdlv wr xvh wkh frpsxwdwlrq edvhg rq wkh orfdo frqvwudlqwv dw wkh qrghvdv vxeurxwlqhv lq wkh pdlq surjudp iru �qglqj wkh h{wulqvlf suredelo0lwlhv z1u1w1 J1 Wkh uhodwlrqvkls ri wkh ydulrxv lqwulqvlf dqg h{wulqvlfsuredelolwlhv lv dovr vkrzq lq Iljxuh 5171

Phvvdjh0sdvvlqj iru wzr qrghv

Vwhs 31 Lqlwldol}h

Lqsxw= S �?|

:

�{u�

�/ iru l @ 4> 5> ===>N/ dqg S �?|

:

�{-��

�/ iru l� @ 4> 5> ===>N �

Lqwhuqdo hgjh0yduldeoh vhw wr xqlirup= S �?|

: +{f @ �f, @�

��f�

Vwhs 4d1 Frpsxwh phvvdjh iurp O wr U

Lqsxw= S �?|

u

�{u�

�# S �?|

:

�{u�

�/ iru l @ 4> 5> ===> N

79 VRIW LWHUDWLYH GHFRGLQJ

L R( )0xRL→µ

0x

( )LxP 1int*

( )LL xP 1int

( )LKL xPint

( )LKxPint

*

( )0xPLext ( )0xPR

int

( )RR xP 1ext

( )RKR xP '

int

( )RKxP '

int*

*

( )RxP 1ext*

Iljxuh 5171 Phvvdjh0sdvvlqj iru wzr qrghv

Frpsxwdwlrq=

Si |

u +{f, @ f�

%f

[

+%fc%u� c���c%ug,M7u;t%f�

g\

�'�

S �?|

u

�{u��

Rxwsxw= �u<- +{f,# S i |

u +{f,

Vwhs 4e1 Frpsxwh phvvdjh iurp U wr O

Lqsxw= S �?|

-

�{-��

�# S �?|

:

�{-��

�/ iru l� @ 4> 5> ===>N �

Frpsxwdwlrq=

S i |

- +{f, @ f�%f

[

+%fc%-� c���c%-g�,M7-;t%f�

g�\

�'�

S �?|

-

�{-���

Rxwsxw �-<u +{f,# S i |

- +{f,

Vwhs 5d1 Frpsxwh h{wulqvlf rxwsxw iru O

Lqsxw= S �?|

u +{f,# �-<u +{f, dqg S �?|

u

�{u��# S �?|

:

�{u��

Page 18: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp 7:

Frpsxwdwlrq= iru l @ 4> 5> ===>N/

S i |

u

�{u��@ f�

%u�

[

+%f c%u� c���c%ug,M7u;i%u� j

S �?|

u +{f,g\

,'�

, �'�

S �?|

u

�{u,�

Rxwsxw= S i |

:

�{u�

�# S i |

u

�{u�

�iru l @ 4> 5> ===>N

Vwhs 5e1 Frpsxwh h{wulqvlf rxwsxw iru U

Lqsxw= S �?|

- +{f,# �u<- +{f,/ dqg S �?|

-

�{-��

�# S �?|

:

�{-��

Frpsxwdwlrq= iru l� @ 4> 5> ===>N �>

S i |

-

�{-���@ f�

%-��

[

+%fc%-� c���c%-g�,M7-;i%-

��j

S �?|

- +{f,g�\

,'�

, �'��

S �?|

-

�{-,�

Rxwsxw= S i |

:

�{-��

�# S i |

-

�{-��

�iru l� @ 4> 5> ===> N�

Lw vkrxog eh phqwlrqhg wkdw dqrwkhu frqvhtxhqfh ri wkh dvvljqphqwvlq +5174, lv wkdw wkh srvwhulru suredelolw| h{suhvvlrqv iru {f ehfrphfrqvlvwhqw1 Lq rwkhu zrugv/ wkh srvwhulru suredelolw| z1u1w1 wkh judsk Jlv htxdo wr wkh srvwhulru suredelolw| z1u1w1 wkh qrgh O dqg z1u1w1 wkhqrgh U= Xvlqj Ed|hv* wkhruhp dqg +5165,/ zh kdyh

STLt|

:+{f, @ S +{f m O>U,

@S +{f,

S +O>U,S +O>U m {f,

@S +{f,

S +O>U,S +O m {f,S +U m {f, =

Vlqfh S +{f, @ S �?|:

+{f, @�

��f �/ dqg S +O m {f, dqg S +U m {f, duh sur0

sruwlrqdo wr wkhlu uhvshfwlyh h{wulqvlf suredelolwlhv/ zh kdyh

STLt|

: +{f, @ f%fSi |u +{f,S

i |- +{f, = +5175,

Rq wkh rwkhu kdqg/ iurp wkh edvlf uhodwlrqvkls ri lqwulqvlf/ h{wulqvlf dqgsrvwhulru suredelolwlhv lq +5145,/ zh kdyh

STLt|

u+{f, @ f%fS

i |u +{f,S

�?|u +{f, +5176,

STLt|

-+{f, @ f%fS

�?|- +{f,S

i |- +{f, = +5177,

7; VRIW LWHUDWLYH GHFRGLQJ

Zlwk wkh dvvljqphqw ri S �?|

u +{f, @ S i |

- +{f, dqg S �?|

- +{f, @ S i |

u +{f, lq+5174,/ wkhvh wkuhh h{suhvvlrqv +5175,/ +5176, dqg +5177, iru wkh srvwhulrusuredelolw| ehfrph lghqwlfdo1 Wklv revhuydwlrq ohqgv dgglwlrqdo fuhghqfhwr wkh fkrlfh ri wkh dvvljqphqwv lq +5174,1

516 PDQ\ QRGHV

Wklv h{dpsoh ri wzr qrghv frqqhfwhg e| d vlqjoh hgjh fdq eh jhqhu0dol}hg wr d judsk ri pdq| qrghv/ nqrzq dv d prgxoh1

51614 RQH PRGXOH

D prgxoh frqvlvwv ri dq duelwudu| qrupdo judsk J/ dv gh�qhg lq Vhf0wlrq 4151 Wkh vhw ri qrghv Q: lv frpsrvhg ri qrghv Q� iru m @ 4> ===> mQ: m/zkloh wkh vhw ri hgjhv H: kdyh dvvrfldwhg yduldeohv {� iru l @ 4> ===> mH: m/zkhuh wkh yduldeoh {� ehorqjv wr vrph doskdehw D�1 Dq h{dpsoh zlwkmQ: m @ ; dqg mH: m @ 44 lv vkrzq lq Iljxuh 5181 Hdfk qrgh Q� lv frq0qhfwhg wr vrph vxevhw ri wkh hgjhv/ dqg hgjhv fdq eh frqqhfwhg wr rqhru wzr +glvwlqfw, qrghv1 Dq hgjh frqqhfwhg wr wzr qrghv lv dq lqwhuqdo

hgjh/ zkloh dq hgjh frqqhfwhg wr rqo| rqh qrgh lv dq h{whuqdo hgjh ri wkhprgxoh1 Wkh h{whuqdo hgjhv ri wkh prgxoh duh xvhg iru sdvvlqj lqwulqvlfsuredelolwlhv S �?|

:+{�, lqwr wkh prgxoh1 Rq wkh rwkhu kdqg/ wkh lqwulqvlf

suredelolwlhv z1u1w1 J iru wkh lqwhuqdo hgjhv duh vhw e| ghidxow wr xqlirupglvwulexwlrqv S �?|

:+{�� @ d, @ �

����

� iru doo d 5 D�� 1 Lq d vhqvh/ wkh prgxoh

fdq eh ylhz dv d elj qrgh zkrvh hgjhv duh jlyhq e| wkh h{whuqdo hgjhvri wkh prgxoh1

Uhfdoo wkdw hdfk qrgh kdv d orfdo frqvwudlqw/ zklfk vhohfwv d vxevhw riwkh srvvleoh frq�jxudwlrqv ri wkh qhljkerulqj hgjh0yduldeohv1 Dv iru wkhfrqvwudlqw vhw iru wkh prgxoh/ d frq�jxudwlrq ri ydoxhv iru hgjh0yduldeohvlq H: lv frqvlghuhg ydolg iru wkh hqwluh judsk J li wkh frq�jxudwlrq dovrvdwlv�hv doo ri wkh orfdo frqvwudlqwv dw wkh qrghv lq Q: 1 Ohw H��

uhsuhvhqwwkh vhw ri qhljkerulqj hgjhv iru wkh qrgh Q� lq wkh judsk1 Wkhq wkhfrqvwudlqw vhw V��

gh�qhv d vxevhw ri ydolg frq�jxudwlrqv rxw ri wkhgluhfw surgxfw ri wkh doskdehwv D� iru {� 5 H��

1 Lq rwkhu zrugv/ wkh vhwV: lv frpsrvhg ri frq�jxudwlrqv

���> ===> ��.: �

�5 D� � � � � �D�.: �

vxfk wkdw iru hdfk qrghv Q� +iru m @ 4> 5> ===> mQ: m,/ wkh vxevhw ri wkh frq0�jxudwlrq fruuhvsrqglqj wr H��

vdwlv�hv wkh frqvwudlqw vhw V��1 Khqfh/

wkh orfdo frqvwudlqwv ri wkh qrghv frpelqh wr gh�qh dq ryhudoo frqvwudlqwrq wkh hgjh0yduldeohv1

Xvlqj wklv frqvwudlqw vhw V: / lw ehfrphv srvvleoh wr fdofxodwh wkh h{0wulqvlf +dqg srvwhulru, suredelolwlhv iru doo wkh h{whuqdo hgjh0yduldeohv

Page 19: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp 7<

N1

N3

N4

Internal edgeExternal edge

N2 N5

*

N6

N8

N7

Node

Module

Iljxuh 5181 Dq h{dpsoh ri d prgxoh

edvhg rq wkh lqwulqvlf suredelolwlhv1 Zlwk wklv gh�qlwlrq ri wkh frq0

vwudlqw vhw V: / zh fdq frpsxwh wkh h{wulqvlf suredelolw| dv

Si |

: +{�, @ f�%�

[�%�c���c%�.: �

�M7:

;t%��

�.: �\

,'�

, �'�

S lqw

: +{,, > +5178,

zkhuh f�%� lv wkh dssursuldwh frqvwdqw1 +Iru wkh srvwhulru suredelolw|/ zh

qhhg wr pxowlso| e| S �?|

: +{�, dqg qrupdol}h djdlq1, Lq jhqhudo/ zkhqwkhuh duh pdq| qrghv dqg hgjhv lq wkh judsk J/ wklv h{suhvvlrq fdq ehtxlwh frpsolfdwhg wr hydoxdwh1

51615 WZR PRGXOHV

Lq wkh fdvh wkdw wkh judsk J fdq eh glylghg lqwr wzr vxejudskv vkdulqjd vlqjoh frpprq hgjh {f/ lw lv srvvleoh wr glylgh wkh sureohp lq +5178,lqwr wzr vxesureohpv/ dqg sdvv phvvdjhv ehwzhhq wkhp1 Odeho wkh wzrvxejudskv e| O dqgU/ iru ohiw dqg uljkw/ uhvshfwlyho|/ dv vkrzq lq Iljxuh5191 Wkh vhw ri qrghv Q: lv wkh glvmrlqw xqlrq ri wkh vhwv Qn dqg Q,1Phdqzkloh/ wkh vhw ri hgjhv H: lv wkh xqlrq ri Hn dqg H,/ zkhuh wkh

83 VRIW LWHUDWLYH GHFRGLQJ

/ 5

( )0x5/→µ

( )0x/5→µ

0x

/

Kx

/

1x

5

'Kx

5

1x

Iljxuh 5191 Wzr prgxohv

lqwhuvhfwlrq frqvlvwv ri d vlqjoh hgjh {f1

Qn ^Q, @ Q:

Qn _Q, @ >

Hn ^H, @ H:

Hn _H, @ i{fj

Lw lv frqyhqlhqw wr uhqdph wkh hgjh0yduldeohv lq H: edvhg rq wkh vxe0judsk wkdw wkh| ehorqj wr1 Lq dgglwlrq wr wkh frpprq hgjh {f/ ohwwkh prgxoh O frqwdlq wkh hgjh0yduldeohv {n

�> ===> {n

g/ dqg wkh prgxoh

U frqwdlq wkh hgjh0yduldeohv {,�> ===> {,

g� / vr wkdw mHnm @ N . 4 dqgmH,m @ N� . 41 Qrwh wkdw li zh fxw wkh judsk dw wkh hgjh {f/ wkhq wkhjudsk ehfrphv ghfrpsrvhg lqwr wzr prgxohv/ dqg wkh lqwhuqdo hgjh {fehfrphv wzr h{whuqdo hgjhv +dovr nqrzq dv kdoi0hgjhv,1 Lq Iljxuh 51:/wkh prgxoh J iurp Iljxuh 518 lv glylghg lqwr wzr prgxohv O dqgU/ zlwkmQnm @ 8/ mHnm @ ;/ mQ,m @ 6/ dqg mH,m @ 71

Zlwk wkhvh gh�qlwlrqv ri wkh prgxohv/ zh fdq ghulyh vlplodu h{suhv0vlrqv dv +516<, dqg +5173, li zh uhsodfh O dqg U e| wkhlu fdooljudsklfhtxlydohqwv O dqg U1 Iurp wklv zh frqfoxgh wkdw dv lq wkh wzr0qrgh vlw0xdwlrq/ vhwwlqj S �?|

, +{f, wr S +O m {f, lv wkh dssursuldwh fkrlfh wr pdnhwkh h{wulqvlf suredelolwlhv z1u1w1 U htxdo wr wkh h{wulqvlf suredelolwlhvz1u1w1 wkh zkroh judsk J1 Khqfh/ wklv vxjjhvwv wkh iroorzlqj vhw ridvvljqphqwv/

S �?|

,

�{,�

�# S �?|

:

�{,�

S �?|

n

�{n�

�# S �?|

:

�{n�

S �?|

, +{f,# S i |

n +{f, +5179,

S �?|

n +{f,# S i |

, +{f,

Page 20: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp 84

N1

N3

N4

N2 N5

/

N6

N8

N7

5

0x

Iljxuh 51:1 Wzr prgxohv vkdulqj dq hgjh

zkhuh wkh h{wulqvlf suredelolw| S i |

n+{f, lv vlpso| wkh qrupdol}hg yhuvlrq

ri S +O m {f,1Lq rwkhu zrugv/ wr ohyhudjh wkh frpsxwdwlrqv edvhg rq prgxoh O dqg

prgxoh U iru wkh frpelqhg judsk J/ zh vkrxog sdvv wkh iroorzlqj phv0vdjhv ehwzhhq wkh wzr prgxohv/ lq h{dfw dqdorj| wr wkh fdvh ri wzrqrghv=

�n<, +{f, @ S i |

n +{f, @ S �?|

, +{f,

�,<n +{f, @ S i |

, +{f, @ S �?|

n +{f, =

Wkh h{wulqvlf suredelolw| iru {f iurpprgxoh O jlyhv wkh sulru suredelolw|iru prgxoh U/ dqg wkh h{wulqvlf suredelolw| iru prgxoh U jlyhv wkhlqwulqvlf suredelolw| iru prgxoh O1 Lq vkruw/ iru d vlqjoh hgjh frqqhfwlqjwzr prgxohv/ wkh h{wulqvlf suredelolw| iurp rqh prgxoh jlyhv wkh lqwulqvlf

suredelolw| iru wkh rwkhu prgxoh1Jhqhudoo| vshdnlqj/ dq| phvvdjh dorqj dq hgjh lv h{wulqvlf lqirupd0

wlrq iurp wkh qrgh +ru prgxoh, zkhqfh lw fdph/ dqg lqwulqvlf lqirupdwlrqiru wkh qrgh +ru prgxoh, wr zklfk lw jrhv1 E| shuiruplqj frpsxwd0wlrqv vhsdudwho| iru wkh prgxohv O dqg U/ dqg wkhq sdvvlqj phvvdjhv�n<, +{f, dqg �,<n +{f,/ lw lv srvvleoh wr vroyh wkh odujhu sureohp rifrpsxwlqj wkh ryhudoo h{wulqvlf suredelolwlhv iru wkh frpelqhg judsk J1Wkh uhodwlrq ri wkh ydulrxv phvvdjhv duh vkrzq lq Iljxuh 51;/ +zklfklv wkh vdph dv Iljxuh 517/ h{fhsw wkdw wkh qrgh kdv ehhq uhsodfhg e|d prgxoh,1 Wkh surfhgxuh iru wzr prgxohv lv h{dfwo| dq uhsolfd ri wkh

85 VRIW LWHUDWLYH GHFRGLQJ

/( )0x

5/→µ

0x

( )/* 1xPint

( )// 1xPint

( )// KxPint

( )/* KxPint

( )0xPext/ ( )0xPint

5

( )5

5 1xPext

( )5

5 'KxPint

( )5

* 'KxPint

*

( )5

* 1xPext

5

Iljxuh 51;1 Phvvdjh0sdvvlqj ehwzhhq wzr prgxohv

surfhgxuh iru wzr qrghv/ vr wkdw wkh �surjudp� lq Vhfwlrq 51517 iru wzr

qrghv O dqg U fdq eh iroorzhg h{dfwo| iru wkh wzr prgxohv O dqg U1

517 PHVVDJH0SDVVLQJ DOJRULWKP

D f|foh lv d zdon dorqj wkh hgjhv ri d judsk wkdw ehjlqv dqg hqgv dwwkh vdph qrgh/ zlwkrxw wudyhuvlqj dq| hgjh pruh wkdq rqfh1 D judskzlwkrxw f|fohv lv d wuhh +ru lv frpsrvhg ri pxowlsoh wuhhv,1 Zh �uvwfrqvlghu wkh phvvdjh0sdvvlqj dojrulwkp iru judskv zlwkrxw f|fohv/ dqgwkhq wdon derxw lwv dssolfdwlrq wr judskv zlwk f|fohv1

51714 JUDSKV ZLWKRXW F\FOHV

Iru d f|foh0iuhh judsk/ d fxw dw dq| hgjh glylghv wkh judsk lqwr wzrglvfrqqhfwhg vxejudskv1 Iurp wklv revhuydwlrq/ zh fdq xvh wkh uhvxowv lqwkh suhylrxv vhfwlrqv wr ghulyh wkh phvvdjh0sdvvlqj dojrulwkp1 Vwduwlqjzlwk d judsk J/ zh slfn dq duelwudu| lqwhuqdo hgjh {�/ zklfk glylghv wkhjudsk lqwr wzr prgxohv O dqg U1 Wkhq wkh h{wulqvlf suredelolwlhv fdqeh fdofxodwhg iru wkh judsk J e| frpsxwlqj wkhp iru prgxohv O dqg Udqg sdvvlqj wkh dssursuldwh phvvdjhv dorqj wkh hgjh {�=

�n<, +{�, @ S i |

n +{�, @ S �?|

, +{�,

�,<n +{�, @ S i |

, +{�, @ S �?|

n +{�, =

Page 21: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvd

jh0sdvvlq

jdojrulwk

p86

Surfhhg

lqjuhfx

uvlyho|/zhfdq

wkhq

wdnhhdfk

riwkhvh

prgxohv

lqwxuq

+h1j1/O,/dqgfkrrvh

dqlqwhuq

dohgjh

+h1j1/{n�,zklfk

vsolwv

wkdw

prgxoh

+l1h1/O,lqwr

wzrvp

doohuprgxohv

+h1j1/ode

hohgOOdqgOU,1

E|sdvvlq

jwkhdssurs

uldwhphvvdjhv

lqerwk

gluhfwlrq

vdorq

jwkdw

lqwhuq

dohgjh

+l1h1/{n�,/wkhh{wulq

vlfsure

delolwlhv

iruwkhprgxoh+l1h1/

O,fdq

ehirx

qge|

frpsxwlq

jwkhh{wulq

vlfsure

delolwlhv

iruwkhwzrvp

doohuprgxohv

+l1h1/OO

dqgOU,1

Wdnlqjwklv

uhfxuvlyh

sulq

flsoh

wrlwv

orjlfdofrq

foxvlrq

jlyhvwkhuhvx

owwkdw

wkhh{dfw

h{wulq

vlfsure

delolwlhv

iruwkhjuds

kJfdq

ehrewdlq

hge|shuirup

lqjfrp

sxwdwlrq

viru

hdfklqglylgxdo

qrghdqg

wkhq

sdvvlq

jphvvdjhv

ehwz

hhqwkhqrghv1

Wkhsurri

rih{dfwq

hvviru

f|foh0iuhh

judskvlvdovr

irxqglq

^Zle<:`^N

IO34`^I

ru34`1Wruhfds

wklv

surfhg

xuh/

wkhlqsxwwr

wkhjuds

klv

dfroohfwlrq

rilqwulq

vlfsure

delolwlhv

iruwkhh{whuq

dohgjhv0ydulde

ohvS

�?|

:+{� ,1

Phdq

0zkloh/

lqwhuq

dohgjh0ydulde

ohvvwduw

r�e|ghidx

owzlwk

xqlirup

lqwulq

vlfsure

delolwlhv

S�?|

:+{��,@

���� � 1

+Qrwh

wkdw

zkhq

wkhjuds

kfrq

wdlqvf|0

fohv/wkhq

lwlv

dovrqhfhvvdu|

wrlqlwldol}h

phvvdjhv

lqerwk

gluhfwlrq

vdorq

jdoo

lqwhuq

dohgjhv/

dvglvfx

vvhglq

Vhfwlrq

517151,Iru

hdfkqrgh

Q� /

wkhh{wulq

vlfsure

delolwlhv

zlwk

uhvshfw

wrwkdw

qrghfdq

ehirx

qg

iurpwkhlqwulq

vlfsure

delolwlhv

zlwk

uhvshfw

wrwkdw

qrghxvlq

jhtxd0

wlrq+515;,/

zklfk

lvgxsolfdwhg

ehorz

lq+517:,1

Wklvh{suhvvlrq

lvdovr

nqrz

qdv

wkhvxp0surg

xfwxsgdwh

uxoh/vlq

fhwklvfrp

sxwdwlrq

frqvlvwv

riwkhvxp

risurgxfwv1

Dvduhvx

ow/wkhphvvdjh0s

dvvlqjdojrulwk

plv

dovrnqrz

qdv

wkhvxp0surgxfwdojrulwk

p1

Wkhvxp0surgxfw

xsgdwh

uxoh

lvuhs

hdwo|dssolhg

iruwkhqrghv/

dqg

phvvdjhv

duhsdvvhg

dorqjwkhhgjhv1

Wkhrug

hulq

zklfk

wkhphvvdjhv

duhsdvvhg

lvuhihuuhg

wrdv

wkhphvvd

jh0sdvvlq

jvfkhgxoh1

Wkhvwr

sslqj

fulwh

ulrqlvwkhwhvw

wkdw

ghwhup

lqhv

zkhwk

huwr

vwrssdvvlq

jphvvdjhv

dqgghfoduh

wkdw

wkhghfr

glqjlv

frpsohwh1

Wkhrxwsxwiurp

wklv

do0jrulwk

pfdq

wkhq

ehuhdg

dvwkhh{wulq

vlfsure

delolwlhv

iruwkhh{whuq

dohgjhv1

+Iru

vrphdssolfdwlrq

v/wkh�qdo

ghvluhg

dqvz

hupd|

ehwkhsrv0

whulrusure

delolw|/

zklfk

fdqehhdvlo|

fdofxodwhg

iurpsxwwlq

jwrjhwk

huwkhlqwulq

vlfdqgh{wulq

vlfsure

delolwlhv

iruwkhh{whuq

dohgjh0ydulde

ohvxvlq

j+5145,1,

Wkhphvvd

jh0sdvvlq

jdojrulwk

pfdq

ehvxppdul}hg

dviroorz

v=

Phvvdjh0sdvvlqjdojrulwkp

Vwhs

31Lqlwld

ol}h1H{whuq

dohg

jh0yduldeohv

vwduwr�

zlwk

lqsxwphv0

vdjhvfruuhvs

rqglqjwr

wkhlqwulq

vlfsure

delolw|

z1u1w1

wkhjuds

k1+Lq

judskvzlwk

f|fohv/

wkhlqwhuq

dohg

jh0yduldeohv

dovrvwduw

r�zlwk

gh0

idxow

phvvdjhv

lqerwk

gluhfwlrq

v1,

87VRIWLWHUDWLYHGHFRGLQJ

Vwhs

41Vxp0surgxfwxsgdwh

uxoh1Iru

dqrghQ

wkdw

lvfrq

qhfwhg

wrN

.4qrghv/

[f >[

� >===>[g

+yld

wkhhg

jh0yduldeohv

{f >{

� >===>{g/

uhvshfwlyho|

,/wk

hrx

wsxwphvvdjh

��<f

� +{� ,

irurq

hhg

jhlv

irxqg

iurpwk

hlqsxwphvvdjhv

�f, <

�+{

, ,dorq

jwk

hrwk

huN

hgjhv

+o@

3>4>5>===>N

dqgo9@l,1

��<f� +{

� ,@f%�

[

E%fc%

�c���c%

g�M7�

;t%� �

g\,'f

,�'�

�f

, <�+{

, ,+517:,

Wkhxsgdwh

irupxoddssolhv

irudoo

l@3>4>===>N

1

Vwhs

51Phvvdjh0sdvvlqj1Iru

hdfkhgjh/

wkhrxwsxwphvvdjh

iurprqhqrghlvsdvvhg

dorqjwkhhgjh

wrehfrp

hwkhlqsxwphvvdjh

iruwkhrwk

huqrgh1

Uhshdw

Vwhs

4iru

dooqrghvdffrug

lqjwr

wkhphvvd

jh0

sdvvlq

jvfkhgxoh

xqwlo

wkhvwr

sslqjfulwh

ulrqkdv

ehhq

uhdfkhg1

Pruh

jhqhudoo|/

wkhwhup

�phvvdjh0s

dvvlqjdojrulwk

p�fdq

uhihuwr

rwkhu

glvwule

xwhg

frpsxwdwlrq

vrq

judskvlqyroy

lqjwkhsdvvlq

jri

phv0

vdjhv/exwzhxvh

lwwr

uhihuvshfl�

fdoo|wr

wkhfdvh

zkhuh

wkhfrp

sx0

wdwlrqvduh

edvhg

rqwkhvxp0surgxfw

xsgdwh

uxoh

+517:,1Vrp

hrwk

huh{dp

sohv

duhvrog

lhu0frxqwlq

jdv

lqFkdswhu

4/dqgwkhplq0vx

pdojr0

ulwkp

^NIO34`1 :

Wkhuh

fdqehzlghydulde

lolw|lq

wkhphvvdjh0s

dvvlqjvfk

hgxoh/

dqg

wkhfkrlfh

rivfk

hgxohfdq

ghwhup

lqhwkhfrp

sxwdwlrq

doh!

flhqf|

riwkh

dojrulwkp1Lq

rqhvfk

hgxoh/

hdfkqrghxsgdwhv

lwvrxwjrlq

jphvvdjhv

zkhqhyhu

lwuhfhlyhv

dvxevhw

frqvlvwlq

jri

Nlqfrp

lqjphvvdjhv

+lilw

kdv

dwrwdo

riN

.4hgjhv,1

Wkhq

zkhqhyhu

dq|lqfrp

lqjphvvdjhv

jhwxsgdwhg

/wkhrxwjrlq

jphvvdjh

dovrjhw

xsgdwhg

1Wkxvwkhqrghv

dwwdfkhg

wrh{whuq

dohgjhv

riwkhprgxohduh

xsgdwhg

�uvw/

iroorzhg

e|

wkhqrghv

frqqhfwhg

wrwkhp

1Dqrwk

husrvvle

ohvfk

hgxohlvwkh� rrg0

lqjvfk

hgxoh/�

lqzklfk

phvvdjhv

duhsdvvhg

dorqjdoo

hgjhv

dwdoo

wlphv

vlpxowdq

hrxvo|1

Zkhq

wkhjuds

kfrq

vlvwvri

qrghv

duudqjhg

lqdolqhdu

idvklrq

/wkhuh

lvdqdwx

udovfk

hgxoh/

lqzklfk

dghfr

ghu

xvhv

dvlq

johiruz

dugvzhhs

dqgdvlq

johedfn

zdug

vzhhs

riphvvdjh0s

dvvlqj1

Wklv

�iruzdug

0edfn

zdug

�dojrulwk

plvglvfx

vvhglq

pruh

ghwdlo

lqFkdswhu

61Iru

judskvzlwk

rxwf|fohv/

lwwxuqvrxwwkdw

liwkhvwrs

slqjfulwhulrq

lvvxfk

wkdw

wkhghfr

ghu

sdvvhv

phvvdjhv

xqwlo

wkhuh

duhqrpruh

xsgdwhv

:Dvd

yduld

wlrq

rq

wkhphvvd

jh0s

dvvlq

jdojrulwk

p/rqh

pljkwfrqvlg

huwk

hhfr

qrp|

dvd

glvwule

xwhg

frpsxwdwlr

qlqyroylqj

wkh

h{fkdqjh

riqxphulf

dowrnhqvehwz

hhq

hqwlwlhv/

dq

dfwly

lw|zklfk

pljkwpruh

dffx

udwho|

ehwhup

hgwk

hprqh|0s

dvvlq

jdojrulwk

p1

Page 22: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvd

jh0sdvvlq

jdojrulwk

p88

wrehpdgh/

wkhq

uhjdugohvv

riwkhfkrlfh

riphvvdjh0s

dvvlqjvfk

hgxoh/

wkhuhvx

owlqjrxwsxwv

dorqjwkhh{whuq

dohgjhv

duhh{dfwo|

wkhh{wulq

vlfsure

delolwlhv

zlwk

uhvshfw

wrwkhhqwluh

judskJ/dv

vshfl�

hglq

+5178,1Wkhfrp

sxwdwlrq

doo|gl!

fxowsure

ohpri

hydoxdwlq

jwkhh{wulq

vlfsure

0delolwlhv

kdv

ehhq

eurnhq

grz

qlqwr

vlpsohvxe0frp

sxwdwlrq

vedvhg

rqorfdo

frqvwudlq

wvdw

hdfkri

wkhqrghv

lqwkhjuds

k1

51715

JUDSKVZLWKF\FOHV

Lidjuds

kfrq

wdlqvf|fohv

+dovrnqrz

qdv

�orrs|�juds

kv,/

zklfk

lvwkhfdvh

lqprvw

vlwxdwlrq

vri

lqwhuhvw/

wkhsuhvhq

fhri

f|fohv

lqydolg

dwhvwkhlqghshqghqfh

dvvxpswlrq

vlq

wkhphvvdjh0s

dvvlqjdojrulwk

p1Wkh

vlwxdwlrq

rijuds

kvzlwk

dvlq

johf|foh

kdv

ehhq

dqdo|

}hg+h1j1/

lqDml

hw

do1^DKP<;`

dqgZhlvv

^Zhl33`,/

exwlq

jhqhudo/

zkhq

wkhuh

duhf|fohv/

wkhphvvdjh0s

dvvlqjdojrulwk

pvhuyhv

rqo|

dvdq

dssur{

lpdwh

dojrulwkp

zlwk

rxwdjxdudq

whhrq

frqyhujhq

fh1Rqhgl�huhq

fhzlwk

wkhfdvh

rif|foh0iuhh

judskvlv

wkdw

irujuds

kv

zlwk

f|fohv/

lwlvqhfhvvdu|

wrkdyh

lqlwldo

phvvdjhv

+lqerwk

gluhfwlrq

v,iru

wkhlqwhuq

dohgjhv1

Dfrp

prq

vhwwlqjiru

wkhlqlwldo

ydoxhriwkhphv0

vdjhvlvwkhxqlirup

glvwule

xwlrq

��

� <�

�� +{�@d,@

��� � 1

Liwkhlqwhuq

do

hgjhv

grqrw

vwduwzlwk

phvvdjhv/

wkhq

wkhsuhvhq

fhrif|fohv

frxog

fdxvh

wkhphvvdjh0s

dvvlqjdojrulwk

pwr

vwdoo/dv

qrqrghlvuhfhly

lqjhqrxjk

phvvdjhv

wrshuirup

dfrp

sxwdwlrq

1Wklvlvhylghqwiurp

wkhh{dp

soh

riwkuhh

qrghv

lqIljx

uh51</

zkhuh

lqwkhjuds

krq

wkhohiw

vlgh/

wkh

vlqjoh

lqfrp

lqjphvvdjh

ohdgvwr

qrixuwk

hufrp

sxwdwlrq

v/vlq

fhwkdw

qrghrqo|

kdv

4lqfrp

lqjphvvdjh

exw6hgjhv1

Lqwkhjuds

krq

wkh

uljkwvlg

hri

Iljx

uh51</

wkhlqwhuq

dohgjhv

vwduwr�

zlwk

ghidx

owphv0

vdjh/vr

wkdw

wkhphvvdjh0s

dvvlqjdojrulwk

pfdq

surfhhg

dvxvxdo1

Qrwh

wkdw

irujuds

kvzlwk

f|fohv/

wkhfkrlfh

riphvvdjh0s

dvvlqjvfk

hgxohdqg

vwrsslqjfulwhulrq

fdqkdyh

dqlp

sdfw

rqwkhghfr

glqjfrp

soh{

lw|dqg

shuirup

dqfh1

Iruwx

qdwho|/

lwkdv

ehhq

irxqglq

sudfwlfh

wkdw

irupdq|vlwx

dwlrqv

rijuds

kvzlwk

f|fohv/

vxfk

lqwkhfdvhv

riwxuerdqgOGSF

frghv/

wkhphvvdjh0s

dvvlqjdojrulwk

psurgxfhv

dffxudwh

hvwlpdwhv

riwkhgh0

vluhgsure

delolwlhv

zlwk

pxfk

orzhu

frpsoh{

lw|wkdq

dqh{dfw

ghfr

glqj1

Zkloh

vkruw

f|fohv

+h1j1/ri

ohqjwk

7,fohduo|

fdxvh

wkhlqghshqghqfh

dvvxpswlrq

vwr

ehylrodwhg

/liwkhf|fohv

fdqehpdghodujh

hqrxjk/

wkhq

wkhlu

h�hfw

ehfrp

hvohvv

ghwulp

hqwdo1

Qrwh

wkdw

zkhq

f|fohv

duhsuhvhq

w/wkhPSDgrhv

qrw

dfwxdoo|

fdofxodwh

wkhwux

hh{wulq

vlfsure

delo0

lwlhvdv

lq+5178,/

exwhvvhq

wldoo|vroyhv

dgl�huhq

wsure

ohpzkrvh

dqvz

hulvdq

dssur{

lpdwlrq

riwkhghvluhg

sure

delolwlhv

lq+5178,1

E|h{fkdqjlq

jphvvdjhv

ehwz

hhqpxowls

ohvlp

sohghfr

ghuv/

wkhshuirup

dqfh

ridpruh

frpsolfdwhg

ghfr

ghu

fdqehdssurdfk

hg1

89

VRIWLWHUDWLYHGHFRGLQJ

Iljxuh

51<1

Lqlwld

ophvvd

jhyhfwruv

duh

qhfhvvd

u|iruwkhlqwhuqdohgjhvlq

judskvzlwk

f|fohv

51716

FRQWLQXRXVDOSKDEHWV

Qrwh

wkdwwkhderyhglvfx

vvlrqkdvehhq

lqwhup

vriglvfuhwh

doskdehwv

D1Lw

lvdovr

srvvle

ohwr

xvh

frqwlq

xrxv0y

doxhgdoskdehwv/

vxfk

dvwkhuhd

o

qxpehuvD@U1Vxssrvh

wkdwdyduld

eoh

i{

� jehor

qjvwr

wkhfrqwlq

rxv0

ydoxhg

doskdehwD@U1Wkhq

wkhphvvd

jh�f

�<�+{

�@

�,uhs

uhvhqwv

dsured

elolw|

ghqvlw|

ixqfwlr

qryhu

�5U

+udwkhu

wkdq

dsure

delolw|

pdvv

ixqfwlrq

,1Dvduhvx

ow/lw

vkrx

ogdfwx

doo|ehuhs

uhvhqwhg

dvdixqfwlrq

i+{

�@

�� ,udwk

huwkdq

e|S+{

�@

�� ,+z

klfk

lpsolhv

dsure

delolw|

,/vlq

fhvwulfwo|

vshdn

lqj/

wkhsure

delolw|

wkdw

{�htxdov

dq|

sduwlfx

oduydox

hlv

htxdo

wr31

Dvdq

dexvh

riqrwdwlrq

/zhzloo

pdlq

wdlqfrq

vlvwhqf|

dqgfrq

wlqxhwr

xvh

wkhqrwdwlrq

S+{

�@

�� ,wr

uhsuhvhq

wwkhsure

delolw|

ghq

vlw|ixqfwlrq

lqwkhfdvh

ridfrq

wlqxrx

vdos

kde

hwv1+Q

rwhwkdw

vlqfh

phvvdjh

yhfwruvduh

dvvxphg

wrehhtxlydohq

wxswr

pxowls

olfdwlrqe|d

frqvwdq

w/lwlv

wkhuhodwlyh

surs

ruwlrqwkdw

lvlp

sruwdq

wdq

|zd|

v1,Qrwh

wkdw

lqwkhvlwx

dwlrqvwkdw

zhfrq

vlghu/

hyhqliwkhydulde

ohvfrp

hiurp

dfrq

wlqxrx

vdos

kde

hw/wkhfrq

vwudlqwvhwv

V�

duhdoo

glvfuhwh

vhwv1Dvduhvx

ow/iru

wkhvxp0surgxfw

dojrulwkp/lwlvqrw

qhfhvvdu|

wruhs

odfhwkhvxppdwlrq

e|dq

lqwhjudo

+517:,1

51717

DSSOLFDWLRQ

Eurdg

o|vwdwhg

/wkhphvvdjh0s

dvvlqjdojrulwk

pfdq

ehds

solhg

wrdq

|sure

ohpri

wkhiroorz

lqjw|sh=

Frq

vlghu

dvhw

e

Hri

yduldeohv/

zkhuh

hdfk

yduldeoh

{�ehorq

jvwr

lwvuhvs

hfwlyhdos

kde

hwD� /

irul@

4>5>===> ��� eH ��� 1

Vxssrvh

wkdw

dfrq

vwudlqwvhw

eVgh�

qhv

wkhydolg

frq�jx

udwlrqvri

wkh

Page 23: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp 8:

ydoxhv iru wkhvh yduldeohv/ zkhuh

eV � D� �D2 � � � � �Dm e.m1

Ohw S �?| +{�, eh wkh lqwulqvlf suredelolw| iru {�/ dqg vxssrvh wkdw wkh

yduldeohv duh lqghshqghqw vr wkdw wkh mrlqw suredelolw| idfwruv lqwr wkh

surgxfw ri wkh lqglylgxdo suredelolwlhv1 Wkhq wkh srvwhulru suredelolw|

wdnlqj lqwr dffrxqw wklv frqvwudlqw eV wxuqv rxw eh htxdo wr wkh vxp ri

wkh surgxfw ri lqwulqvlf suredelolwlhv1

STLt| +{�, @ f�

[�%�c���c%� e.�

�Me7

;t%��

\

,

S lqw +{,, > +517;,

Wkh vxppdwlrq lv ryhu doo frq�jxudwlrqv ri ydoxhv lq wkh vhw eV1Wklv iudphzrun fdq eh xvhg wr ghvfuleh pdq| ghfrglqj sureohpv/

zkhuh wkh jrdo lv wr frpsxwh d srvwhulru suredelolw| edvhg rq lqwulqvlfsuredelolwlhv/ dv gh�qhg e| +517;,1 Lq wkh fdvh ri huuru0frqwuro frghv

+HFFv,/ wkh yduldeohv lq eH uhsuhvhqw wkh frghzrug v|perov/ wkh lqwulqvlf

suredelolwlhv duh ghulyhg iurp wkh uhfhlyhg vljqdo/ wkh vhw eV uhsuhvhqwvwkh vhw ri ydolg frghzrugv/ dqg wkh srvwhulru suredelolwlhv uhsuhvhqw wkhghfrghu rxwsxw/ zklfk fdq eh xvhg wr pdnh elw ghflvlrqv1

Dv wklv h{suhvvlrq +517;, fdq riwhq eh frpsolfdwhg wr hydoxdwh/ wkhlghd lv wr lqwurgxfh d judsk J wr h{suhvv wkh uhodwlrqvklsv h{suhvvhge| wkh frqvwudlqw vhw eV rq wkh yduldeohv lq eH lq rughu wr vlpsoli| wkhfrpsxwdwlrq1 Wkh judsk J lv fkrvhq vr wkdw lwv h{whuqdo hgjh0yduldeohvfruuhvsrqg h{dfwo| wr wkh yduldeohv lq eH/ nqrzq dv �v|pero yduldeohv1�Lq dgglwlrq/ wkh judsk lqfoxghv d vhw ri qrghv ghvfulelqj orfdo frqvwudlqwv/dv zhoo dv lqwhuqdo hgjh0yduldeohv/ nqrzq dv �vwdwh yduldeohv1�

Edvhg rq wklv vhw ri lqwhuqdo dqg h{whuqdo hgjh0yduldeohv H: dqg wkhvhw ri qrghv Q: / lw lv srvvleoh wr gh�qh dq ryhudoo frqvwudlqw vhw V: /

zklfk vkrxog pdwfk wkh ruljlqdo frqvwudlqw vhw eV zkhq uhvwulfwhg wr wkhh{whuqdo hgjh0yduldeohv1 +Dfwxdoo|/ wkh judsk J lv riwhq fkrvhq �uvw/ dqg

wkhq wkh vhw eV ri ydolg frghzrugv lv gh�qhg iurp wkh frqvwudlqw vhw V: e|orrnlqj dw wkh h{whuqdo hgjh0yduldeohv1, Zlwk wkh judsk uhsuhvhqwdwlrqJ/ wkh vroxwlrq ri +517;, fdq wkhq eh irxqg e| dsso|lqj wkh phvvdjh0sdvvlqj dojrulwkp1

61 QRGHV DQG PRGXOHV

Kdylqj vhw xs wkh iudphzrun iru vriw lwhudwlyh ghfrglqj/ zh glvfxvv wzredvlf w|shv ri qrghv/ wkh htxdolw| qrgh dqg wkh sdulw|0fkhfn qrgh/ zklfkirup wkh edvlf exloglqj eorfnv iru orz0ghqvlw| sdulw| fkhfn +OGSF,

8; VRIW LWHUDWLYH GHFRGLQJ

frghv1 Qrghv duh wkhq rujdql}hg lqwr prgxohv/ dqg vrph nh| h{dpsohvduh lqwurgxfhg1

614 HTXDOLW\ QRGH

Lq wkh htxdolw| qrgh / slfwxuhg lq Iljxuh 5143/ wkh frqvwudlqw vshfl�hvwkh yduldeohv wr eh htxdo1 +Lq wkh fdvh ri elqdu| OGSF frghv/ wkh htxdolw|qrgh lv dovr nqrzq dv d �elw qrgh/� vlqfh lw fruuhvsrqgv wr d vlqjoh elw1,Vxssrvh wkdw wkh qrgh Q lv frqqhfwhg wr N . 4 hgjhv/ zlwk dvvrfldwhgyduldeohv {f> {�> ===> {g wkdw ehorqj wr wkh vdph doskdehw D1 Wkhq wkhfrqvwudlqw vhw iru wklv qrgh lv

V� @ i+{f> {�> ===> {g, m {f @ {� @ � � � @ {gj >

dqg frqwdlqv mDm hohphqwv1 Vxssrvh wkdw wkh lqsxw phvvdjh yhfwruv duh�f�<�

+{�, iru l @ 3> 4> ===> N1 Zlwkrxw orvv ri jhqhudolw|/ ohw xv frqvlghuwkh rxwsxw dorqj wkh hgjh0yduldeoh {f1 Wkh xsgdwh htxdwlrq iroorzv iurp+517:,/

��<ff+{f @ �, @ f%f

[

E%fc%�c���c%g�M7�;t%f�

g\,'�

�f,<� +{,,

@ f%f

g\,'�

�f,<� +{, @ �, +517<,

zkhuh f%f lv wkh qrupdol}dwlrq idfwru/ zklfk lv htxdo wr

f%f @

3C[

lM�

g\,'�

�f,<� +{, @ �,

4D3�

=

Wkh vxppdwlrq lv ryhu doo hohphqwv ri wkh doskdehw D iru {f1 Wkh vxp risurgxfwv uhgxfhv wr d surgxfw ri wkh lqwulqvlf suredelolwlhv fruuhvsrqglqjwr wkh vdph ydoxh1 Wklv htxdwlrq +517<, fdq eh dssolhg vlploduo| wrfrpsxwh wkh rxwsxw dorqj dq| hgjh0yduldeoh {�/ edvhg rq wkh lqsxwvdorqj wkh uhpdlqlqj hgjh0yduldeohv {, iru 3 � o � N/ dqg o 9@ l1

Page 24: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp 8<

=xK

x1

x2

...

N

x0X0

X1

XK

X2

Iljxuh 51431 Phvvdjh0sdvvlqj dw dq htxdolw| qrgh

Lq wkh fdvh ri d elqdu| doskdehw D/ zh fdq uhzulwh +517<, xvlqj wkhorj0olnholkrrg udwlr +OOU,1

OOU�<ff+{f, @ orj

��<ff+{f @ 4,

��<ff+{f @ 3,

@ orj

Tg

�'��f�<�

+{� @ 4,Tg

�'��f�<�

+{� @ 3,

@g[

�'�

OOUf�<�+{�, +5183,

Iru dq htxdolw| qrgh/ wkh rxwjrlqj phvvdjh dorqj rqh hgjh lv wkh vxpri wkh lqfrplqj phvvdjhv dorqj doo ri wkh rwkhu hgjhv lq whupv ri orj0olnholkrrg udwlrv1 +Revhuyh wkh vlplodulw| ri htxdwlrq +5183, wr wkh h{0dpsoh ri vroglhu0frxqwlqj jlyhq lq Fkdswhu 4/ zkhuh wkh rxwsxw phvvdjhlv d vxp ri wkh lqfrplqj phvvdjhv1,

Wkh lqsxw dqg rxwsxw iru dq htxdolw| qrgh duh vxppdul}hg e| htxd0wlrq +517<, iru suredelolwlhv dqg +5183, iru orj0olnholkrrg udwlrv1 Ohw xvfrqvlghu vrph h{dpsohv xvlqj elqdu| yduldeohv/ zklfk duh ghslfwhg lqIljxuh 5144=

Htxdolw| qrgh zlwk wzr hgjhv1 Frqvlghu dq htxdolw| qrgh Q rqwzr elqdu| yduldeohv/ {�/ {21 Wklv fruuhvsrqgv wr d uhshwlwlrq frgh/zkhuh wkh frqvwudlqw vhw V @ i+3> 3, > +4> 4,j1 Vxssrvh wkdw wkh lqsxwv

93 VRIW LWHUDWLYH GHFRGLQJ

=

=

1

11

p

p

1

11

p

p

2

21

p

p

1

11

p

p

( )( )

−−

21

21 11

pp

ppc

Iljxuh 51441 Htxdolw| qrghv zlwk wzr dqg wkuhh hgjhv

wr wkh qrghv duh ghqrwhg e|

�$�f�<� +{�, @

�4� s�s�

�$�f2<�+{2, @

�4� s2s2

�>

uhsuhvhqwlqj lqwulqvlf suredelolwlhv zlwk uhvshfw wr wkh qrgh Q 1 Wkhqiroorzlqj +517<,/ wkh rxwsxwv iurp wkh qrgh duh jlyhq e|

��<f�+{�, @ �f2<�

+{2,

��<f2+{2, @ �f�<� +{�, =

Wkh srvwhulru suredelolw| iru {� lv wkhq jlyhq dv iroorzv/

�$S TLt| +{�, @ f � +�$� �<f�

+{�,��$� f�<�

+{�,,

@ f

�+4� s�, +4� s2,

s�s2

�/ +5184,

zkhuh wkh qrupdol}dwlrq frqvwdqw lv jlyhq e|

+f,3� @ s�s2 . +4� s�, +4� s2, =

Dq lghqwlfdo h{suhvvlrq wr +5184, krogv iru wkh srvwhulru suredelolw|iru {21

Page 25: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp 94

Htxdolw| qrgh zlwk wkuhh hgjhv1 Qh{w/ frqvlghu dq htxdolw| qrghQ zlwk wkuhh elqdu| yduldeohv/ {f/ {�/ dqg {21 Vxssrvh wkh lqwulqvlfsuredelolwlhv rq {� dqg {2 duh jlyhq e| s� dqg s2/ uhvshfwlyho|1 Wkhqwkh rxwsxw h{wulqvlf suredelolw| yhfwru zlwk uhvshfw wr Q lv jlyhq e|

�$

S i | +{f, @ f

�+4� s�, +4� s2,

s�s2

�>

zklfk lv wkh vdph dv wkh srvwhulru suredelolw| iru wkh fdvh ri dqhtxdolw| qrgh zlwk wzr hgjhv lq +5184,1 Lq whupv ri orj0olnholkrrgudwlrv/ zh kdyh

OOU�<ff+{f, @ OOUf�<�

+{�, . OOUf2<�+{2,

orjs�s2

+4� s�, +4� s2,@ orj

�s�

4� s�

�. orj

�s2

4� s2

�=

615 SDULW\0FKHFN QRGH

Lq d sdulw|0fkhfn qrgh +ru �fkhfn qrgh�,/ wkh orfdo frqvwudlqw vshfl�hvwkdw wkh yduldeohv vxp wr 31 Frqvlghu d sdulw|0fkhfn qrgh Q zkhuh wkhqhljkerulqj hgjhv kdyh dvvrfldwhg yduldeohv {f> {�> ===> {g wkdw ehorqj wrwkh vdph doskdehw D/ dv vkrzq lq Iljxuh 51451 Vxssrvh wkdw wkh doskdehwD lv dq dgglwlyh jurxs xqghu wkh rshudwlrq �1 Wkhq wkh frqvwudlqw vhwV� lv pdgh xs ri doo frq�jxudwlrqv wkdw vdwlvi| dq dgglwlyh frqvwudlqw1

V� @ i+{f> {�> ===> {g, m {f � {� � � � � � {g @ 3j

Wkh xsgdwh htxdwlrq iru {f lv jlyhq e| +517:,/

��<ff+{f, @ f%�

[

E%f c%�c���c%g�M7�;t%f�

g\

,'�

�f,<� +{,, > +5185,

zklfk lqyroyhv d vxppdwlrq ri doo frq�jxudwlrqv lq V� wkdw duh frqvlv0whqw zlwk dq lpsolhg ydoxh ri {f @ �f> iroorzlqj wkh vlpsol�hg qrwdwlrq1

Lq wkh fdvh ri elqdu| yduldeohv/ wkh doskdehw lv D @ ]2/ wkh dgglwlyhjurxs ri wzr hohphqwv1 Iru d sdulw|0fkhfn qrgh/ wkh frqvwudlqw vhw V�frqvlvwv ri 5g frq�jxudwlrqv iru wkh N.4 yduldeohv/ vlqfh wkh frqvwudlqwholplqdwhv h{dfwo| kdoi ri wkh srvvleoh frq�jxudwlrqv1 Wkhq lq +5185,/ wkhvxppdwlrq lv ryhu doo hohphqwv ri V� wkdw duh frqvlvwhqw zlwk d �{hgydoxh iru {�1 Wklv vxppdwlrq ryhu 5g3� hohphqwv dsshduv frpsolfdwhg/exw iruwxqdwho|/ lw wxuqv rxw wkdw wkhuh duh zd|v wr vlpsoli| wklv frpsx0wdwlrq1

95 VRIW LWHUDWLYH GHFRGLQJ

+xK

x1

x2

...

N

x0X0

X1

XK

X2

Iljxuh 51451 Phvvdjh0sdvvlqj dw d sdulw|0fkhfn qrgh

Zlwk uhvshfw wr wkh sdulw| fkhfn qrgh Q / wkh lqsxw phvvdjhv fruuh0vsrqg wr wkh lqwulqvlf suredelolwlhv s� @ S �?| +{� @ 4,1

�f�<�+{� @ 3, @ 4� s�

�f�<�+{� @ 4, @ s�

Wkh rxwsxw phvvdjh dorqj hgjh0yduldeoh {f vkrxog eh wkh h{wulqvlf sure0delolw| zlwk uhvshfw wr Q 1 Wklv lv jlyhq e| wkh iroorzlqj suredelolwlhv=

��<ff+{f @ 3, @ S +{� � � � � � {g @ 3,

��<ff+{f @ 4, @ S +{� � � � � � {g @ 4,

Wr hydoxdwh wkhvh h{suhvvlrqv/ zh �uvw frqvlghu wkh fdvh ri wkuhh hgjhv+vr N @ 5,1 Lq wklv fdvh/ {� � {2 htxdov 3 h{dfwo| zkhq erwk yduldeohvkdyh wkh vdph ydoxh/ vr wkdw

��<ff+{f @ 3, @ S +{� � {2 @ 3,

@ s�s2 . +4� s�, +4� s2, +5186,

dqg

��<ff+{f @ 4, @ s� +4� s2, . +4� s�, s2>

dv vkrzq lq Iljxuh 51461Wklv h{suhvvlrq +5186, fdq eh uhzulwwhq dv

5S +{� � {2 @ 3,� 4 @ +4� 5s�, +4� 5s2, > +5187,

Page 26: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp 96

+

2

21

p

p

1

11

p

p

( )( )( ) ( )

−+−

−−+

2121

2121

11

11

pppp

pppp

Iljxuh 51461 H{dpsoh ri d sdulw|0fkhfn qrgh

zklfk fdq eh jhqhudol}hg iru N A 5 dv iroorzv1 Jlyhq wkh iroorzlqjgh�qlwlrqv/

Pg @ S +{� � {2 � � � � � {g @ 3,

Pg3� @ S +{� � {2 � � � � � {g3� @ 3,

sg @ S �?| +{g @ 4,

vxssrvh e| lqgxfwlrq wkdw wkh h{suhvvlrq +5187, lv wuxh iru wkh fdvh riN � 4 yduldeohv/ vr wkdw

5Pg3� � 4 @g3�\

�'�

+4� 5s�, =

Wkhq wkh h{suhvvlrq {� � {2 � � � � � {g fdq eh rewdlqhg dv wkh vxp ri{g dqg {� � {2� � � � � {g3�1 Iroorzlqj d fdofxodwlrq vlplodu wr +5186,/zh rewdlq

Pg @ +4� sg,Pg3� . sg +4�Pg3�,

@ +4� 5sg,Pg3� . sg >

zklfk fdq eh uhzulwwhq dv

5Pg � 4 @ 5 +4� 5sg,Pg3� . 5sg � 4

@ +4� 5sg, +5Pg3� � 4, >

wkxv |lhoglqj wkh ghvluhg uhvxow/

5Pg � 4 @g\

�'�

+4� 5s�, = +5188,

97 VRIW LWHUDWLYH GHFRGLQJ

E| wkh lqgxfwlrq k|srwkhvlv/ wklv h{suhvvlrq +5188, lv ydolg iru doo N � 41Vroylqj iru Pg / zh rewdlq

Pg @4 .Tg

�'�+4� 5s�,

5>

vr wkdw wkh rxwsxw phvvdjhv fdq eh zulwwhq lq whupv ri wkh lqsxw phv0vdjhv s� @ �f�<�

+{� @ 4, dv iroorzv/ jlylqj d vlpsol�fdwlrq ri wkh xs0gdwh htxdwlrq +5185,1

��<ff+{f @ 3, @

4 .Tg

�'�+4� 5s�,

5+5189,

��<ff+{f @ 4, @

4�Tg

�'�+4� 5s�,

5+518:,

Wkhvh htxdwlrqv +5189, dqg +518:, fdq dovr eh irxqg xvlqj d Irxulhuwudqvirup dssurdfk wkdw fdq eh jhqhudol}hg wr qrq0elqdu| doskdehwv/ dvvkrzq lq Dsshqgl{ E1

Vlqfh wkh yduldeohv duh elqdu|/ wkh h{suhvvlrqv +5189, dqg +518:, fdqeh frpelqhg xvlqj wkh orj0olnholkrrg udwlr +OOU,1

OOU�<ff+{f, @ orj

��<ff+{f @ 4,

��<ff+{f @ 3,

@ orj4�Tg

�'�+4� 5s�,

4 .Tg

�'�+4� 5s�,

+518;,

Wkh k|shuerolf wdqjhqw ixqfwlrq dqg lwv lqyhuvh/

wdqk�{5

�@h% � 4

h% . 4

5 wdqk3� +|, @ orj4� |

4 . |>

fdq eh dssolhg wr vlpsoli| wklv h{suhvvlrq1 Iluvw/ wkh uhodwlrq ri wkh vriwelw dqg OOU +fi1 +5146,, jlyhv wkh iroorzlqj=

4� 5s� @ � wdqk

�4

5OOU+s�,

Page 27: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp 98

Wkhq wkh rxwjrlqj phvvdjh iurp wklv fkhfn qrgh +518;, fdq eh fdvwsxuho| lq whupv ri orj0olnholkrrg udwlrv/

OOU�<ff+{f, @ orj

4� +�4,gTg

�'� wdqk��2OOU+s�,

�4 . +�4,g

Tg

�'� wdqk��2OOU+s�,

�@ +�4,g3� 5 wdqk3�

#g\�'�

wdqk

�4

5OOUf�<�

+{�,

�$>

+518<,

zkhuh wkh idfw wkdw wkh lqyhuvh k|shuerolf wdqjhqw wdqk3� +�, lv dq rgg

ixqfwlrq lv xvhg wr pryh wkh vljq +�4,g3� rxwvlgh wkh ixqfwlrq1Zh uhpdun wkdw li zh kdg gh�qhg wkh orj0olnholkrrg udwlr zlwk wkh

rssrvlwh frqyhqwlrq +l1h1 orj � E%'f�� E%'��, dv lq ^Jdo96`/ wkhq wkh idfwru ri

+�4,g3� glvdsshduv lq +518<,1 Zh kdyh fkrvhq rxu OOU gh�qlwlrq lqrughu wr kdyh d srvlwlyh OOU fruuhvsrqg wr { @ 4 dqg qhjdwlyh OOU wr{ @ 31

616 PRGXOHV

Kdylqj gh�qhg vhyhudo qrghv/ zh fdq rujdql}h wkhp lqwr prgxohv1 Dvzlwk qrghv/ hdfk prgxoh lv d �vriw0lq/ vriw0rxw� ghfrghu/ zkhuh iru hyhu|h{whuqdo hgjh ri wkh prgxoh/ wkh lqsxw lv wkh lqwulqvlf suredelolw| dqg wkhrxwsxw lv wkh h{wulqvlf suredelolw| zlwk uhvshfw wr wkh prgxoh1 Vlqfh wkhprgxoh fdq eh xvhg d sduw ri d odujhu v|vwhp/ wkh prgxoh lv dovr nqrzqdv d �judsk iudjphqw� lq ^Iru34`1 Lq rwkhu zrugv/ vlqfh wkh hqwluh v|vwhpfdq eh uhsuhvhqwhg e| dq ryhudoo qrupdo judsk/ d prgxoh lv vlpso| d zd|wr vshfli| d iudjphqw ri wkdw odujhu judsk wkdw uhsuhvhqwv d sduwlfxoduixqfwlrq1 Dv ghvfulehg lq Vhfwlrq 61614/ lw lv srvvleoh wr frqqhfw prgxohvwrjhwkhu xvlqj pxowlsoh hgjhv1

Wklv xvh ri prgxohv frlqflghv yhu| qdwxudoo| zlwk wkh xvxdo eorfn gl0djudp qrwdwlrq iru frppxqlfdwlrq v|vwhpv/ zkhuh wkh hgjhv ehwzhhqeorfnv uhsuhvhqw d vwuhdp ri vljqdov ru elwv/ dv lq Fkdswhu 41 Wkhvheorfn gldjudpv fdq wkhq eh lqwhusuhwhg xvlqj wkh vriw lwhudwlyh ghfrg0lqj iudphzrun/ zkhuh hdfk eorfn lv d prgxoh dqg vljqdov duh uhsodfhge| phvvdjhv fruuhvsrqglqj wr suredelolw| glvwulexwlrqv1 Zlwk wklv uhs0uhvhqwdwlrq/ wkh eorfn gldjudpv iru wkh hqfrghu dqg ghfrghu duh riwhqyhu| vlplodu/ dqg lq pdq| fdvhv duh lghqwlfdo�wkh rqo| gl�huhqfh lv wkdwwkh hqfrghu vwduwv zlwk phvvdjhv fruuhvsrqglqj wr wkh nqrzq xvhu gdwddw rqh hqg ri wkh eorfn gldjudp/ zkloh ghfrghu vwduwv zlwk phvvdjhvfruuhvsrqglqj wr d qrlv| uhfhlyhg vljqdo dw wkh rwkhu hqg ri wkh eorfngldjudp1

99 VRIW LWHUDWLYH GHFRGLQJ

/ 5

x1

x2

x3

xK

/ 5

x1

x2

x3

xK

x1

x2

x3

xK

Iljxuh 51471 Frqqhfwlqj wzr prgxohv/ lq zklfk wzr exqgohv ri h{whuqdo hgjhv frp0

elqh wr irup rqh exqgoh ri lqwhuqdo hgjhv1

61614 PXOWLSOH HGJHV EHWZHHQ PRGXOHV

Lw lv srvvleoh wr kdyh pxowlsoh hgjhv frqqhfwlqj wzr prgxohv/ dv vkrzqlq Iljxuh 51471 D vhw ri sdudooho hgjhv lv uhihuuhg wr dv d exqgoh ri hgjhv1Wkh wrs sduw ri wkh �jxuh vkrzv wzr prgxohv/ hdfk zlwk d exqgoh rih{whuqdo hgjhv1 Wkhvh fdq eh frpelqhg wr irup d vlqjoh exqgoh ri hgjhv/dv vkrzq rq wkh erwwrp1 Lq wkh judsk frpsrvhg ri wkh frpelqdwlrq riwkhvh wzr prgxohv/ wkh wzr exqgohv ri h{whuqdo hgjhv kdyh qrz ehfrphd vlqjoh exqgoh ri lqwhuqdo hgjhv1 Dv d qrwdwlrqdo frqyhqlhqfh/ zh fdqdovr uhsuhvhqw wkh exqgoh e| d wklfn olqh/ dv vkrzq lq Iljxuh 51481 +Do0whuqdwlyho|/ wkh exqgoh fdq eh uhsuhvhqwhg e| d olqh zlwk d vodvk wkurxjklw/ lq dffrugdqfh zlwk wkh qrwdwlrq iru zluhv lq d flufxlw gldjudp1,

Lw vkrxog eh qrwhg wkdw iru d exqgoh ri hgjhv/ wkh phvvdjh lv kdqgohgvhsdudwho| iru hdfk hgjh lq wkh exqgoh1 Dv vkrzq lq Iljxuh 5149/ mxvw dviru d qrgh/ wkh rxwsxw dorqj hdfk hgjh ri d prgxoh lv d ixqfwlrq ri wkhlqsxwv dorqj doo wkh rwkhu hgjhv/ lqfoxglqj wkh rwkhu hgjhv lq wkh vdphexqgoh1 Vxssrvh wkdw wkh exqgoh frqvlvwv ri N hgjhv wkdw duh dvvrfldwhgzlwk elqdu| yduldeohv {�/ {2/ 111/ {g +zkhuh wkh doskdehw lv D @ i3> 4j,1Wkhq wkh exqgoh fduulhv d vhsdudwh phvvdjh yhfwru iru hgjh {�/ vr wkdwwkh phvvdjh iru wkh exqgoh lv dfwxdoo| d froohfwlrq ri N vhsdudwh yhfwruv=

��4� s�s�

�>

�4� s2s2

�> ===>

�4� sgsg

��

Page 28: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp 9:

/ 5

/ 5

Iljxuh 51481 Frqqhfwlqj wzr prgxohv/ dv vkrzq xvlqj d wklfn olqh wr uhsuhvhqw wkh

d exqgoh ri hgjhv

Wkhuh lv 4 ghjuhh ri iuhhgrp lq hdfk hgjh dqg wkxv d wrwdo ri N ghjuhhvri iuhhgrp1 Qrwh wkdw xvlqj exqgohv ri hgjhv pdnhv lw olnho| wkdw wkhjudsk frqwdlqv f|fohv/ vr wkdw phwkrgv vkrxog eh xvhg +h1j1/ wkh shupxwhuprgxoh lq Vhfwlrq 61616, wr lqfuhdvh wkh ohqjwk ri wkh f|fohv vr wkdw wkhluh�hfw lv plwljdwhg1

Rq wkh rwkhu kdqg/ dq dowhuqdwlyh lv wr wuhdw doo wkh elqdu| yduldeohvlq wkh exqgoh dv d vlqjoh yduldeoh/ lq zklfk fdvh wkh mrlqw doskdehw zrxogeh D�D�� � ��D @ i3> 4jg dqg wkh phvvdjh yhfwru zrxog eh ri ohqjwk5g / kdylqj 5g�4 ghjuhhv ri iuhhgrp/ udwkhu wkdq N ghjuhhv ri iuhhgrp1Wklv dssurdfk dyrlgv wkh lqwurgxfwlrq ri f|fohv lqwr wkh judsk/ exw iruprvw ghfrglqj dssolfdwlrqv/ wklv ohyho ri ghwdlo lv h{fhvvlyh dqg xqzlhog|1

61615 HTXDOLW\ PRGXOH

Uhfdoo wkdw dq htxdolw| qrgh lv xvhg iru h{suhvvlqj dq htxdolw| uhodwlrqehwzhhq hgjh0yduldeohv1 Htxdolw| qrghv fdq eh frpelqhg wrjhwkhu lqwrdq htxdolw| prgxoh wkdw vhuyhv wkh sxusrvh ri h{suhvvlqj dq htxdolw|uhodwlrq ehwzhhq wkh fruuhvsrqglqj hgjhv ri pxowlsoh exqgohv1

Iru dq htxdolw| prgxoh zlwk wkuhh exqgohv ri hgjhv/ dv vkrzq lq Iljxuh514:/ vxssrvh wkh lqsxw phvvdjhv iurp wkh ohiw dqg wrs duh

+�$� +{�, >�$� +{2, > ===>

�$� +{g,,

9; VRIW LWHUDWLYH GHFRGLQJ

* *

Iljxuh 51491 Wkh rxwsxw rq dq hgjh ghshqgv rq wkh lqsxwv rq doo rwkhu hgjhv

dqg ��$��{

�>�$��{

2

�> ===>�$��{

g

��>

uhvshfwlyho|/ zkhuh hdfk �$� +{�, lv d phvvdjh yhfwru ri wkh irup

�$� +{�, @

59997

� +{� @ d�,� +{� @ d2,

111

��{� @ d���

6:::8

Doo hgjh0yduldeohv i{�j dqg i{�

�j duh dvvxphg wr wkh vdph doskdehw D1

Wkhq wkh rxwsxw phvvdjh rq wkh uljkw lv jlyhq e| wkh iroorzlqj froohfwlrqri phvvdjh yhfwruv�f��$� +{�,�

�$��{�

�> f2�$� +{2,�

�$��{�

2

�> ===> fg

�$� +{g,��$��{�

g

��>

zkhuh � uhsuhvhqwv wkh srlqwzlvh surgxfw ri wkh wzr phvvdjh yhfwruv/dqg f� lv wkh qrupdol}dwlrq frqvwdqw vr wkdw hdfk ri wkh uhvxowlqj N

phvvdjh yhfwruv vxp wr 41

61616 SHUPXWHU PRGXOH

Wkh shupxwhu prgxoh lv riwhq fdoohg d udqgrp �lqwhuohdyhu� lq wxuerfrgh ghvljq1 Zkhq d exqgoh ri hgjhv h{lvwv ehwzhhq wzr prgxohv/ wkhv|vwhp ghvljqhu qhhgv wr eh fduhixo qrw wr lqwurgxfh vkruw f|fohv lqwrwkh ryhudoo judsk ri wkh wzr prgxohv1 Iru h{dpsoh/ gluhfwo| frqqhfwlqjwzr prgxohv zkhuh wkhuh lv d uhodwlrq ehwzhhq dgmdfhqw hohphqwv +h1j1/{� dqg {�n�, frxog ohdg wr vkruw f|fohv1 Dv d uhvxow/ vrph udqgrpl}dwlrqri wkh rughu pd| eh qhfhvvdu|/ zklfk fdq eh dffrpsolvkhg wkurxjk dshupxwhu prgxoh/ dovr nqrzq dv dq �lqwhuohdyhu� lq wxuer frglqj1

Dv vkrzq lq Iljxuh 514:/ wkhuh duh wzr exqgohv ri hgjhv wr wklv prgxoh1Wklv prgxoh wkhq shupxwhv wkh vhw ri phvvdjhv ehwzhhq wkh wzr exqgohv1

Page 29: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvd

jh0sdvvlq

jdojrulwk

p9<

Equa

lity=

==

==

=

Pe

rmute

Multiple

x

Iljxuh

514:1

Prgxohv

Ohw�$�

+{� ,ghqrwh

dphvvdjh

yhfwruiru

wkhydulde

oh{

�rq

wkhohiw

exqgoh1

Liwkhlqsxwphvvdjhv

iurpwkhohiw

duh

+ �$�+{

� ,> �$�

+{2 ,>===> �$�

+{g,,>

wkhq

wkhrx

wsxwphvvdjhv

rqwkhuljk

wduh

��$��{

ZE�� �> �$�

�{ZE2� �>===> �$�

�{ZEg

� ��>

zkhuh

�uhs

uhvhqwv

dshup

xwdwlrq

riwkhvhw

i4>5>===>N

j1Qrwh

wkdw

dvhw

riphvvdjh

yhfwruvsdvvhg

lqwkhuhyhuvh

gluhfwlrq

zloo

h{shulhq

fhdq

lqyhuvh

shup

xwdwlrq

+ghq

rwhg�3�,

iurpwkhphvvdjh

yhfwruvlq

wkh

iruzdug

gluhfwlrq

1Wkhuh

duhqrqrghv

lqyroyhg

lqwklvshup

xwhu

prgxoh/

dqgqrfrp

sxwdwlrq

lvuht

xluhg

1Wklv

shup

xwhu

fdqxvh

dshup

xwdwlrq

�wkdw

lvwdloruhg

irudsduwlfx

oduvlwx

dwlrq>lq

vrphds

solfdwlrq

v/zlwk

kljk

sure

delolw|/

dudq

grp

o|vhohfwhg

shup

xwdwlrq

vx!fhv1

61617

PXOWLSOH[PRGXOH

Wkhpxowls

oh{prg

xoh

frpelqhv

wzr+ru

pruh,

exqgohv

rihgjhv

lqwr

dvlq

johexqgoh/

dqglq

uhyhuvh/lw

wdnhvdexqgoh

dqgvsolwv

lwlqwr

wzr

+rupruh,

exqgohv1

Dvvkrz

qlq

Iljx

uh514:/

liwkhuh

duhwz

rexqgohv

rq

:3VRIWLWHUDWLYHGHFRGLQJ

wkhohiw

riwkhpxowls

oh{prgxohfrq

vlvwlqjri

+�$�

+{� ,

>�$�

+{2 ,

>===>�$�

+{g,,

>

dqg

��$�

�{

�� �>�$�

�{

�2 �>===>�$�

�{

�g� ��

>

wkhq

wkhphvvdjh

yhfwruviru

wkhfrp

elqhg

exqgohrq

wkhuljk

wduh

��$�

+{� ,

>�$�

+{2 ,

>===>�$�

+{g,>�$�

�{

�� �>�$�

�{

�2 �>===>�$�

�{

�g� ��

=

Lqwkhuhyhuvh

gluhfwlrq

/wklvpxowls

oh{prgxohvsolwv

wkhfrp

elqhg

exq0

gohri

hgjhv

lqwr

wzrvp

doohuexqgohv1

61618

SUREDELOLVWLFFLUFXLWV

Dqdqdorj|

fdqehpdghehwz

hhqwkhqrup

dojuds

kdqgdgljlwdo

flufxlw1

Wkhqrghv

fruuhvsrqgwr

wkhedvlf

orjlfjdwhv/

dqgwkhhgjhv

fruuhvsrqgwr

wkhzluhv

riwklvv|vwhp

1Wkhh{fkdqjh

risure

delolvwlf

phvvdjhv

dorqjwkhhgjhv

riwkhjuds

kfruuhvs

rqgvwr

wkh rz

rihohfwulf

fxuuhq

wwkurx

jkwkhzluhv

lqdflufx

lw1Dprgxohlvdfroohfwlrq

riqrghv

dqghgjhv

wkdw

pd|

ehxvhg

dvdfrp

srqhqwlq

dv|vwhp

/lq

dqdorj|

wrwkhzd|

dqlqwhjudwhg

flufxlw/

frpsrvhg

riwudq

vlvwruvdqgzluhv/

fdqeh

xvhg

dvdfrp

srqhqwrq

derdug

1;

Wdeoh

515vxjjhvwv

vrphdqdorjrx

vihdwx

uhvehwz

hhqgljlwdo

flufxlwv

dqgwkhvh

qrup

dojuds

kv/zklfk

zhpljk

wwklqnridv

�sure

delolvwlf

flu0fxlwv1�

Lqgljlwdo

orjlf/wkhphvvdjhv

vhqwrq

wkhzluhv

duhdvvx

phg

wrehelqdu|

+zlwk

kljk

yrowdjhiru

4dqgorz

yrowdjhiru

3,/zkloh

lqd

judsk/wkhphvvdjhv

duhyhfwruv

risure

delolwlhv1

Lqvwhdg

riorjlf

jdwhv/wkhphvvdjh0s

dvvlqjdojrulwk

pkdv

qrghv

wkdw

rshudwh

rqsure

delolwlhv1

Wkhh{whuq

dohgjhv

dfwdv

wkhlqsxw2rx

wsxw+L2R

,lqwhuidfh

iruwkhprg0

xoh/

zkloh

dexqgohri

hgjhv

dfwvolnh

dgdwd

exvfduu|

lqjdsdudooho

vhwri

vljqdov1

Ilqdoo|/

wkhsdulw|

0fkhfn

qrghfruuhvs

rqgvwr

dq[RUjdwh/

vlqfh

lwh{suhvvhv

duhodwlrq

zkhuh

wkhlqyroyhg

yduldeohv

vxp

wr3/dqg

wkhhtxdolw|

qrghfruuhvs

rqgvwr

duhjlvwhu

+h1j1/iru

vwrulqjdvlq

johelw,/

vlqfh

wkhfrq

vwudlqwvhw

vshfl�

hvwkdw

wkhhgjh0ydulde

ohvvkrxogehhtxdo/

zkhuhdv

doozluhv

frqqhfwhg

wrduhjlvwhu

kdyh

wkhvdp

hydox

h1Rqhgl�huhq

fh/krz

hyhu/lvwkdw

lqdgljlwdo

flufxlw/

wkhfxuuhq

w rz

vlq

rqhgluhfwlrq

dorqjwkhzluh/

zkhuhdv

wkhsure

delolvwlf

phvvdjhv

rz

lqerwk

gluhfwlrq

vdorq

jwkhhgjhv

riwkhjuds

k1+Qrwh

wkdw

wkhlpsoh0

phqwdwlrq

riphvvdjh0s

dvvlqjghfr

ghuv

xvlq

jdqdorj

flufxlwv

wdnhvwklv

dqdorj|

ehwz

hhqflufx

lwvdqgjuds

kvrqhvwhs

ixuwk

hu1^OOKW34`1,

;Wkhvh

gd|

v/dpruh

dssursuld

whdqdorj|pljkwehdqLS

+lqwhoohfwx

dosurshuw|

,fr

uhwk

dwlv

lqwhj

udwhg

lqwr

dqDVLF

ghvlj

q1

Page 30: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp :4

gljlwdo flufxlw qrupdo judsk

jdwh qrgh +ixqfwlrq,zluh hgjh +yduldeoh,vljqdo phvvdjh yhfwru +suredelolwlhv,lqwhjudwhg flufxlw +LF, prgxoh +judsk iudjphqw,

lqwhuqdo zluh lqwhuqdo hgjhL2R lqwhuidfh h{whuqdo hgjhgdwd exv exqgoh ri hgjhv[RU jdwh sdulw|0fkhfn qrghuhjlvwhu htxdolw| qrgh

Wdeoh 5151 Dqdorj| ehwzhhq flufxlwv dqg qrupdo judskv

71 ORZ0GHQVLW\ SDULW\0FKHFN FRGHV

Kdylqj suhvhqwhg wkh iudphzrun ri qrupdo judskv dqg wkh phvvdjh0sdvvlqj dojrulwkp/ lw lv vwudljkwiruzdug wr glvfxvv orz0ghqvlw| sdulw|0fkhfn +OGSF, frghv dqg wkhlu ghfrglqj dojrulwkp1 OGSF frghv +dovrnqrzq dv Jdoodjhu frghv, duh huuru0fruuhfwlqj frghv zlwk vsduvh sdulw|0fkhfn pdwulfhv/ zklfk duh dphqdeoh wr ghfrglqj xvlqj wkh phvvdjh0sdvvlqj dojrulwkp1 Wkh| zhuh lqwurgxfhg e| Jdoodjhu ^Jdo95`^Jdo96`lq wkh hduo| 4<93*v/ uhylvlwhg e| Wdqqhu ^Wdq;4`/ dqg rqo| diwhu wkh dg0yhqw ri wxuer frghv/ zhuh wkh| uhglvfryhuhg e| PdfNd| dqg Qhdo ^PQ<:`^Pdf<<`/ dqg Vlsvhu dqg Vslhopdq ^VV<9`1

714 GHILQLWLRQ

D elqdu| orz0ghqvlw| sdulw| fkhfn +OGSF, frgh lv d elqdu| olqhduhuuru0fruuhfwlqj frgh gh�qhg e| d vsduvh sdulw|0fkhfn pdwul{ K 1 Li wkhsdulw|0fkhfn pdwul{ K kdv Q froxpqv dqg P urzv/ wkh frghzrugv frq0vlvw ri vhtxhqfhv y ri Q elwv wkdw vdwlvi| d vhw ri P elqdu| sdulw| fkhfnvgh�qhg e| wkh sdulw|0fkhfn htxdwlrq Ky @ 3/ dv vkrzq lq Iljxuh 514;1Wkh qxpehu ri phvvdjh elwv lv N @ Q �P / dqg wkh udwh ri wkh frghlv g

�/ dvvxplqj wkdw wkh pdwul{ lv ri ixoo udqn1 +Vhh Dsshqgl{ E iru wkh

h{whqvlrq ri wklv gh�qlwlrq wr qrq0elqdu| doskdehwv1,D vwudljkwiruzdug phwkrg iru jhqhudwlqj wkh vhw ri ydolg frghzrugv lv

wr uhgxfh K wr v|vwhpdwlf irup Kr+r @ ^L� m S ` e| shuiruplqj elqdu|Jdxvvldq holplqdwlrq dqg uhrughulqj wkh froxpqv dssursuldwho|1 +Lw zlooeh dvvxphg wkdw K kdv ixoo udqn1 Li vrph urzv ri K duh olqhduo| gh0shqghqw/ uhpryh wkh h{wud urzv/ dqg ohw P htxdo wkh uhvxowlqj qxpehuri lqghshqghqw urzv1, Iru K lq v|vwhpdwlf irup/ wkh v|vwhpdwlf elwv xirup d vxevhw ri wkh frghzrug elwv y1 +Lq prvw fdvhv/ wkh v|vwhpdwlfelwv duh xvhg dv wkh xvhu elwv1, D fruuhvsrqglqj jhqhudwru pdwul{ lv

:5 VRIW LWHUDWLYH GHFRGLQJ

3DULW\ FKHFNV

=

0

0

0

0

0

0

010001100

001100010

100010001

100100100

010010010

001001001

9

3

2

1

v

v

v

v

%LWV

+ + + + + +

= = = = = = = = =

jC

iB

Iljxuh 514;1 Sdulw|0fkhfn pdwul{ dqg lwv dvvrfldwhg judsk

Jr+r @��S

A m L�3��1 Vlqfh wkh pdwul{ S lv olnho| wr eh xqlirupo|

ghqvh +zlwk kdoi 4*v dqg kdoi 3*v,/ wkh ghqvhqhvv ri S pdnhv hqfrglqjxvlqj Jr+r yld pdwul{ pxowlsolfdwlrq kljk lq frpsoh{lw|1 Phwkrgv kdyhehhq irxqg iru juhdwo| uhgxflqj wkh frpsoh{lw| ri wkh hqfrglqj surfhvv+uhi1 Ulfkdugvrq dqg Xuedqnh ^UX34e`,1 Xvlqj frghv wkdw kdyh vwuxf0wxuh +dv lq Fkdswhu ;, lv dqrwkhu zd| wr uhgxfh wkh frpsoh{lw| ri wkhhqfrglqj surfhvv1 Dqrwkhu dssurdfk xvhg e| Fkhqj ^Fkh<:` lv orz0ghqvlw| jhqhudwru pdwul{ +OGJP, frghv/ lq zklfk K lv fkrvhq wr eherwk v|vwhpdwlf dqg vsduvh1 Wkhvh duh hdvlo| hqfrghg/ exw lq jhqhudo grqrw shuirup dv zhoo dv pdwulfhv wkdw duh qrw uhvwulfwhg wr eh v|vwhpdwlf1

Wkh judsk frqvlvwv ri �elw qrghv/� zklfk duh htxdolw| qrghv +iurpVhfwlrq 614, wkdw uhsuhvhqw wkh elwv ri wkh frghzrug/ dqg �fkhfn qrghv�+iurp Vhfwlrq 615, wkdw h{suhvv wkh sdulw|0fkhfn uhodwlrqvklsv1 Wklv sdu0lw| fkhfn pdwul{ K surylghv wkh vwuxfwxuh iru wkh ghfrglqj dojrulwkp1Dv vkrzq lq Iljxuh 514;/ wkhuh lv dq hgjh lq wkh judsk frqqhfwlqj wkh elwdqg fkhfn qrghv h{dfwo| zkhq wkhuh lv d �4� lq wkh sdulw| fkhfn pdwul{/vr wkdw dq dvvrfldwlrq fdq eh pdgh ehwzhhq hgjh0yduldeohv dqg wkh qrq0}hur hqwulhv ri wkh sdulw| fkhfn pdwul{1 Wkh hgjh0yduldeohv lq wkh vdphurz ri wkh sdulw|0fkhfn pdwul{ K duh frqqhfwhg wr wkh vdph fkhfn qrgh/

Page 31: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp :6

me

ssage

parity

=

=

=

=

=

=

=

=

Π

+

+

+

+

+

+

Iljxuh 514<1 Prgxoh0edvhg uhsuhvhqwdwlrq iru OGSF frgh

dqg vr vdwlvi| d sdulw|0fkhfn frqvwudlqw1 Phdqzkloh/ wkh hgjh0yduldeohvlq wkh vdph froxpq ri wkh sdulw|0fkhfn pdwul{ K duh frqqhfwhg wr wkhvdph elw qrgh/ dqg vr vdwlvi| dq htxdolw| frqvwudlqw1 Wkh judsk iru dqOGSF frgh fdq dovr eh uhsuhvhqwhg xvlqj d shupxwhu prgxoh � wr gh0vfuleh wkh frqqhfwlrqv ehwzhhq wkh elwv dqg fkhfnv/ dv vkrzq lq Iljxuh514< +uhi1 ^Iru34`,1 Lq rwkhu zrugv/ wkh ghwdlov ri krz wkh hgjhv frqqhfwwrjhwkhu duh devruehg e| wkh shupxwhu prgxoh1

Wklv judsk uhsuhvhqwdwlrq frpsohwho| ghvfulehv doo wkh uhodwlrqv riwkh frgh/ dqg fdq eh xvhg iru ghfrglqj xvlqj wkh phvvdjh0sdvvlqj dojr0ulwkp1 Vwduwlqj zlwk wkh lqsxw frqvlvwlqj ri lqwulqvlf suredelolwlhv iru wkhelwv/ wkh phvvdjh0sdvvlqj dojrulwkp xvhv wkh sdulw|0fkhfn uhodwlrqvklsvdprqjvw wkh elwv wr lwhudwlyho| sdvv phvvdjhv ehwzhhq wkh elw qrghv dqgfkhfn qrghv wr rewdlq h{wulqvlf suredelolwlhv iru wkh elwv1 Wkh lqwulqvlfdqg h{wulqvlf suredelolwlhv fdq wkhq eh frpelqhg wr jlyh srvwhulru sure0delolwlhv iru wkh frghzrug elwv wkdw lqfrusrudwh wkh nqrzohgjh jdlqhgiurp wkh OGSF frgh1 Ghvslwh wkh idfw wkdw wkh judskv dvvrfldwhg wrOGSF frghv pd| srvvhvv pdq| f|fohv/ iru udqgrpo| fkrvhq vsduvh pd0wulfhv zlwk odujh eorfn ohqjwkv/ lw wxuqv wkh dojrulwkp frpsxwhv dffxudwhhvwlpdwhv ri wkh h{wulqvlf dqg srvwhulru suredelolwlhv1 Vhh Ulfkdugvrqdqg Xuedqnh ^UX34d` dqg Fkxqj hw do1 ^FUX34` iru dqdo|vhv ri wkhfdsdflw| ri OGSF frghv1

Lq whupv ri wkh ghvljq ri orz0ghqvlw| sdulw|0fkhfn frghv/ lw zdv vkrzqe| Jdoodjhu ^Jdo96` wkdw iru odujh eorfn ohqjwkv/ wkh plqlpxp glvwdqfhri wkh frgh jurzv olqhduo| zlwk eorfnohqjwk/ vr wkdw d udqgrpo| fkrvhqOGSF frgh zloo kdyh odujh glvwdqfh zlwk kljk suredelolw|/ pdnlqj lw d

:7 VRIW LWHUDWLYH GHFRGLQJ

srzhuixo huuru0fruuhfwlqj frgh1 Khqfh/ wkh �uvw ghvljq sulqflsoh lv wrxvh d yhu| orqj eorfn ohqjwk li srvvleoh/ vr wkdw zlwk kljk suredelolw|/fkrrvlqj d udqgrp frgh zloo jlyh jrrg shuirupdqfh1 Qrwh wkdw wkhuhfdq eh pdq| htxlydohqw sdulw|0fkhfn pdwulfhv iru wkh vdph frgh/ dqg lwlv suhihudeoh wr xvh d sdulw| fkhfn pdwul{ zlwk orz ghqvlw| vr dv wr dyrlgvkruw f|fohv lq wkh judsk +zklfk ylrodwh wkh lqghshqghqfh dvvxpswlrqvri wkh phvvdjh0sdvvlqj dojrulwkp,1 Lq sduwlfxodu/ f|fohv ri ohqjwk 7/ lqzklfk wzr elwv duh erwk frqqhfwhg e| hgjhv wr wkh vdph wzr fkhfnv/vkrxog hvshfldoo| eh dyrlghg1Wkh fkrlfh ri wkh ghjuhhv ri wkhvh qrghv +fruuhvsrqglqj wr wkh qxpehu

ri rqhv lq d froxpq ru urz ri wkh sdulw| fkhfn pdwul{, ghwhuplqhv wkhshuirupdqfh ri wkh OGSF ghfrghu1 Frghv zlwk wkh vdph ghjuhh dw hdfkqrgh duh nqrzq dv �uhjxodu� OGSF frghv/ zlwk wkh rwkhuv nqrzq dv�luuhjxodu1� Lw kdv ehhq irxqg wkdw zlwk fduhixo fkrlfh ri wkh glvwulexwlrqri ghjuhhv +uhi1 Ulfkdugvrq hw do1 ^UVX34` dqg Oxe| hw do1 ^OPVV34e`,dqg vx!flhqwo| orqj eorfn ohqjwkv/ lw lv srvvleoh wr ghvljq luuhjxodu OGSFfrghv wkdw doorz iru huuru0iuhh wudqvplvvlrq dw udwhv wkdw duh h{wuhpho|forvh wr wkh Vkdqqrq fdsdflw| ri wkh DZJQ fkdqqho/ shuiruplqj hyhqehwwhu wkdq wxuer frghv1Zkloh lw kdv ehhq irxqg wkdw fhuwdlq luuhjxodu frqvwuxfwlrqv shuirup

ehwwhu wkdq rwkhuv/ iru rxu vlpxodwlrqv/ zh �qg lw frqyhqlhqw wr xvhpdwulfhv wkdw duh froxpq0uhjxodu1 Wkhvh sdulw|0fkhfn pdwulfhv duh frq0vwuxfwhg e| udqgrpo| vhohfwlqj wS orfdwlrqv lq hdfk froxpq dqg sxwwlqjd 4 lq wkdw orfdwlrq/ dqg vshfldo fduh kdv ehhq wdnhq wr dyrlg f|fohv riohqjwk 71

715 OGSF GHFRGHU XVLQJ

SUREDELOLWLHV

Xvlqj wkh xsgdwh htxdwlrqv iru wkh elw qrghv +d1n1d1 htxdolw| qrghv,dqg fkhfn qrghv +d1n1d1 sdulw|0fkhfn qrghv, jlyhq deryh/ zh fdq shuirupghfrglqj rq wkh judsk iru dq OGSF frgh/ dv vkrzq lq Iljxuh 51531 Irud sdulw|0fkhfn pdwul{ K ri vl}h P �Q / vxssrvh wkdw wkh elw qrghv duhE� iru l @ 4> 5> ===> Q / dqg wkh fkhfn qrghv duh F� iru m @ 4> 5> ===>P 1 Lidq hgjh h{lvwv ehwzhhq wkh qrghv E� dqg F� / wkhq wkdw hgjh lv odehohge| wkh yduldeoh h��1 Wkhvh duh wkh lqwhuqdo hgjhv ri wkh judsk iru wkhOGSF frgh1 Lq dgglwlrq/ wkhuh lv dovr dq hgjh iru hdfk elw qrgh E�

wkdw lv dvvrfldwhg zlwk wkh yduldeoh y�1 Wkhvh hgjhv fruuhvsrqg wr wkhh{whuqdo hgjhv ri wkh judsk iru wkh OGSF frgh1 Lq dgglwlrq/ lw zloo ehfrqyhqlhqw wr lqwurgxfh d qrgh Q� +zklfk ehorqjv wr vrph rwkhu ghfrghuru prgxoh, wkdw lv frqqhfwhg wr E� yld wkh hgjh0yduldeoh y�1 Zlwk wklvqrgh Q�/ wkhq wkh lqwulqvlf suredelolw| z1u1w1 wkh OGSF ghfrghu fdq ehghqrwhg dv wkh phvvdjh ���<��

+y�, @ S �?|

w#��+y�,1

Page 32: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp :8

( )iBNi vpii →= µ

( )iNB vii →µ

iN

=

=

=

=

=

=

=

=

=

=

+

+

+

+

+

iBjC

LDPC decoder

ji CBjiq →= µ

ij BCjir →= µ

ivjie

Iljxuh 51531 Judsk iru dq OGSF ghfrghu

Wkh phvvdjh0sdvvlqj dojrulwkp vwduwv zlwk d vhw ri lqfrplqj suredelo0lwlhv ���<��

+y�, fruuhvsrqglqj wr S �?|

w#��+y�,1 Dffruglqj wr +517<,/ wkh

phvvdjh iurp elw qrghv E� wr fkhfn qrghv F� lv jlyhq e| wkh h{suhvvlrqv/

���<��+h�� @ 3, @ f�����<��

+y� @ 3,\

� �M�E��.t��

����<��+h�� @ 3,

+5193,

���<��+h�� @ 4, @ f�����<��

+y� @ 4,\

� �M�E��.t��

����<��+h�� @ 4,

+5194,

zkhuh P+l, lv wkh vhw ri sdulw| fkhfnv lq zklfk elw y� lv lqyroyhg1 Wklvlv dovr wkh vhw ri urz orfdwlrqv lq wkh l0wk froxpq ri wkh sdulw| fkhfnpdwul{ wkdw frqwdlq d 41 Wkh vhw P+l,q imj phdqv wkh vhw P+l, zlwk wkhhohphqw m rplwwhg1 +Qrwh wkdw wkh qrupdol}dwlrq frqvwdqw f�� lv rswlrqdokhuh vlqfh lw fdq eh uhvwruhg dw wkh hqg ri wkh ghfrglqj surfhgxuh1,

:9 VRIW LWHUDWLYH GHFRGLQJ

FKHFN

ELW

ULM

+

=

=

=

=

=

ELW

FKHFN

SULRU

TLM

=

++

++

+

SL

jC

iB jC

iB

Iljxuh 51541 Phvvdjh0sdvvlqj iurp elw qrgh wr fkhfn qrgh/ dqg ylfh yhuvd

Rq wkh rwkhu kdqg/ dffruglqj wr +5189, dqg +518:,/ wkh phvvdjhv iurpfkhfn qrgh F� wr elw qrgh E� duh jlyhq e| wkh h{suhvvlrqv/

���<��+{f @ 3, @

4

5

3C4 .

\��MwE��.t��

�4� 5��

��<��

+{�� @ 4,�4D +5195,

���<��+{f @ 4, @

4

5

3C4�

\��MwE��.t��

�4� 5����<��

+{�� @ 4,�4D +5196,

zkhuh O+m, lv wkh vhw ri elw qrghv frqqhfwhg wr wkh m0wk sdulw| fkhfn/ ruwkh vhw ri froxpqv lq wkh m0wk urz wkdw frqwdlq d 41

Wkhvh wzr sdluv ri htxdwlrqv +5193/ 5194, dqg +5195/ 5196, ghvfulehwkh phvvdjh0sdvvlqj dojrulwkp iru wkh orz0ghqvlw| sdulw|0fkhfn frgh1Wklv uhsuhvhqwdwlrq fdq eh uhodwhg wr wkh qrwdwlrq xvhg hovhzkhuh +h1j1/^Pdf<<`, e| lqwurgxflqj wkh iroorzlqj yduldeohv=

sK� @ ���<��+y� @ e,

tK�� @ ���<��+h�� @ e,

uK�� @ ���<��+h�� @ e,

iru e @ 3> 4/ dv vkrzq lq Iljxuh 51541 Wkhq wkh xsgdwh htxdwlrqv +5193,dqg +5194, iru wkh phvvdjhv iurp elw qrgh E� wr wkh fkhfn qrgh F� pd|

Page 33: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp ::

eh uhzulwwhq dv

tf�� @ f�

�� � sf

\

��M�E��.t��

uf��� +5197,

t��� @ f��� � s��

\

��M�E��.t��

u����> +5198,

zkhuh f��� lv d qrupdol}dwlrq frqvwdqw1 Wkh xsgdwh htxdwlrqv +5195, dqg+5196, iru wkh phvvdjh vhqw iurp fkhfn qrgh F� wr elw qrgh E� pd| ehuhzulwwhq dv

uf�� @4

5

3C4 .

\��MwE��.t��

�t��

4D +5199,

u��� @4

5

3C4�

\��MwE��.t��

�t��

4D > +519:,

zkhuh �t�� @ tf�� � t��� @ 4 � 5t���1 Wkh ghfrglqj dojrulwkp vwduwv zlwk

wkh lqwulqvlf suredelolwlhv�sf� > s

��

�/ dqg xqlirup glvwulexwlrqv iru u�� +l1h1/

uf�� @ u��� @�2,1 Wkhq wkh uhodwlrqv +5197, dqg +5198, |lhog wkh phvvdjhv t��

iurp elwv wr fkhfnv1 +Iru wkh �uvw lwhudwlrq/ wkh phvvdjhv duh tK�� @ sK� 1,Wkh phvvdjhv t�� duh wkhq xvhg lq +5199, dqg +519:, wr fdofxodwh wkhphvvdjhv u�� iurp fkhfnv wr elwv1

Wkhvh wzr vwhsv +xsgdwlqj t�� dqg xsgdwlqj u��, frpsulvh d vlqjohlwhudwlrq ri wkh phvvdjh sdvvlqj dojrulwkp1 Wkh h{wulqvlf suredelolw|z1u1w1 wkh OGSF ghfrghu S i |

w#�� +y�, pd| eh irxqg e| frpsxwlqj wkhrxwjrlqj phvvdjh iru hdfk elw qrgh y�/

���<��+y� @ 3, @ f�� �

\

�M�E��

uf��

���<��+y� @ 4, @ f�� �

\

�M�E��

u���>

zkhuh f�� lv d qrupdol}lqj frqvwdqw1 Ilqdoo|/ wkh hvwlpdwh ri wkh srvwhulrusuredelolw| fdq eh irxqg e| pxowlso|lqj wklv h{wulqvlf suredelolw| zlwkwkh lqwulqvlf suredelolw| dv lq +5145,/

tK� @ f��� ���<��+y� @ e,���<��

+y� @ e,

:; VRIW LWHUDWLYH GHFRGLQJ

iru e @ 3> 41 Lq rwkhu zrugv/ wkh srvwhulru suredelolwlhv duh

tf�@ f� � sf

\

�M�E��

uf��

t�� @ f� � s�\

�M�E��

u���>

zkhuh wkh frqvwdqw f� lv d qrupdol}dwlrq frqvwdqw1 Lw lv krshg wkdw diwhu dqxpehu ri lwhudwlrqv ri wkh phvvdjh0sdvvlqj dojrulwkp/ wkhvh hvwlpdwhvri wkh srvwhulru suredelolwlhv

�tf� > t

��

�frqyhujh wr wkh dfwxdo srvwhulru

suredelolwlhv z1u1w1 wkh OGSF frgh1

716 OGSF GHFRGHU XVLQJ OOUV

Vlqfh wkh yduldeohv duh elqdu|/ wkhvh frpsxwdwlrqv fdq dovr eh gh0vfulehg lq whupv ri wkh orj0olnholkrrg udwlrv1 Wkh phvvdjh OOU+t��, iurpd elw qrgh wr d fkhfn qrgh pd| eh irxqg xvlqj +5183,1

OOU+t��, @[

� �M�E��.t��

OOU�u���

�.OOU+s�, +519;,

Iroorzlqj +518<,/ wkh phvvdjh OOU+u��, iurp d fkhfn qrgh wr d elw qrghfdq eh zulwwhq dv

OOU+u��, @ +�4,�wE��� 5 wdqk3�

3C \

��MwE��.t��

wdqk

�4

5OOU+t���,

�4D =

+519<,+Djdlq/ li wkh rssrvlwh frqyhqwlrq zhuh fkrvhq iru wkh orj0olnholkrrg

udwlr/ wkhq wkh idfwru ri +�4,�wE��� frxog eh rplwwhg1, Wkh srvwhulru OOUlv jlyhq e| vxpplqj ryhu doo wkh fkhfnv wkdw frqwdlq wkh l0wk elw1

TLt|ih�Lh} �� ~OOU+t�, @

i |h�?t�U} �� ~[��M�E��

OOU�u���

�.

�?|h�?t�U} �� ~OOU+s�, +51:3,

Zlwk uhvshfw wr wkh OGSF ghfrghu/ wkh lqwulqvlf/ h{wulqvlf/ dqg srvwhulruOOUv vdwlvi| wkh uhodwlrq +5147,1 Edvhg rq wklv srvwhulru suredelolw|OOU+t�,/ elw ghflvlrqv fdq eh pdgh dqg d ghfrghg zrug ey fdq eh irxqg1Li wklv zrug vdwlv�hv wkh sdulw|0fkhfn pdwul{ +KeyA @ 3,/ wkhq lw lv dfrghzrug/ jlylqj d qdwxudo vwrsslqj fulwhulrq/ zklfk uhgxfhv wkh dyhudjhqxpehu ri lwhudwlrqv1

Wr uhgxfh wkh frpsoh{lw| iru kdugzduh lpsohphqwdwlrq/ lw lv ghvludeohwr wudqvirup wkh pxowlsolfdwlrqv lq htxdwlrq +519<, lqwr dgglwlrqv1 Iluvw/

Page 34: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvd

jh0sdvvlq

jdojrulwk

p:<

revhuyh

wkdw

dq|surgxfw

riuhdo

qxpehuv

fdqehzulwwhq

dviroorz

v/

\

d�@ #\

vjq+d

� , $h{s #[

orj+md� m, $

=+51:4,

E|lqwur

gxflq

jwkhixqfwlrq

�+{,@

�orj �

wdqk+{5, �

@orj

4.h{s+�

{,

4�h{s+�

{,>

zklfk

lvgh�

qhg

iru{A

3/wkhhtxdwlrq

+519<,fdq

ehghfrp

srvhg

lqwr

vljqdq

gdp

solwx

ghxvlq

j+51:4,/

OOU+u�� ,@+�

4,�w

E���

v���� 3C

[��MwE��.

t�� � � ��O

OU �

t��� � �� � 4D

+51:5,

zkhuh

wkhvljq

lvjlyhq

e|

v��@

\��MwE��.

t�� vjq �O

OU �

t��� ��

=+51:6,

Dvre

vhuyhge|Jdoodjhu

^Jdo96`/

wkhixqfwlrq

�+{,lv

htxdo

wrlwv

rzq

lqyhuvh�

3�+{

,ryhu

wkhudq

jh{

A31

Wkhixqfwlrq

�+{,lv

ghslfwhg

lqIljx

uh51551

Htxdwlrq

+51:5,wkxvsury

lghv

dqxsgdwh

htxdwlrq

xv0

lqjdvlq

johwudq

virupdwlrq

ixqfwlrq

�dq

gvrp

hdg

glwlrq

vdq

gelqdu|

rshudwlrq

v+wr

frpsxwh

wkhvljq

,1Qrwh

lq+51:5,/

lwlvwkhodujh

ydoxhv

wkdw

grp

lqdwh

lqwkhvxppdwlrq

1Lq

idfw/liwkhuh

lvdvlq

johodujh

ydoxh�� ��O

OU �

t��� � �� �/

wkhvxppdwlrq

zloo

ehdodujh

srvlwlyh

qxpehu/

dqgwkhrwk

huwhup

vgrqrw

pdwwhu1

Vlqfh

wkhixqfwlrq

�+{,lv

prq

rwrqlfdoo|

ghfuhdvlq

jiru

{A3/

dodujh

ydoxhri

�� ��O

OU �

t��� � �� �

fruuhvsrq

gvwr

dvp

dooydox

hri ��O

OU �

t��� � ��1

Khqfh/

dvdq

dssur{

lpdwlrq

/lw

lvsrvvle

ohwr

uhsodfh

wkhvxppdwlrq

e|wkhplqlp

xp

ydoxhri ��O

OU �

t��� � ��

ryhudoo

l �lq

wkhuhohydq

wudq

jh+l �5O+m,q

ilj,/

xvlq

jwkhidfw

wkdw

�+�

+{,,@

{1Lq

rwkhu

zrug

v/wkh

h{suhvvlrq

+51:5,fdq

ehds

sur{

lpdwhg

e|

OOU+u

�� ,�+�

4,�wE���v���

plq

��MwE��.

t�� � ��O

OU �

t��� � �� �

>+51:7,

zkhuh

wkhvljq

v��lvfdofx

odwhgdv

ehiruh

lq+51:6,1

Wklv�plq�

dssur{l0

pdwlr

qfdq

|lhog

dvljq

l�fdq

wghfuhdvh

lqfrp

soh{

lw|zlwk

rqo|

dvp

dooghjudg

dwlrqlq

shuirup

dqfh/

dvvkrz

qlq

Irvvrulhu

hwdo1^IPL<<`1

Sxwwlq

jwklv

doowrjhwk

hu/zhvxppdul}h

wkhghfr

glqjri

orz0ghq

vlw|sdulw|

0fkhfn

frghv1

Wkhdojrulwk

plv

jlyhqlq

dirup

vxlwde

ohiru

lp0

sohp

hqwdwlrq

/l1h1/

xvlq

jorj0olnholk

rrgudwlrv

dqgwkhixqfwlrq

�+{,wr

frqyhuw

pxowls

olfdwlrqvlqwr

dgglwlrq

v1

;3

VRIWLWHUDWLYHGHFRGLQJ

00.5

11.5

22.5

33.5

40

0.5 1

1.5 2

2.5 3

3.5 4Ψ

(x) = -log(tanh(x/2))

x

Ψ(x)

Iljxuh

51551

Wkhixqfwlr

q[E%�irufrqyhuwlq

jsurgxfwv

lqwr

vxpvlq

wkhOOU

irup

riwkhOGSF

ghfrglqj1

Ghfrglqjdojrulwk

piruOGSF

frghvxvlq

jOOUv1

Vwhs

31Lqlwld

ol}h1Wkhlqsxwwr

wkhghfr

glqjdojrulwk

plvdphvvd

jh

yhfwr

u�sf� >s�� �

iruhdfk

elw

y� /

jlylqjwkhlqwulq

vlfsure

delolw|

zlwk

uhvshfw

wrwkhghfr

ghu

iruhdfk

elw1

Dvdorj0olnholk

rrg

udwlr/wklv

lqwulq

vlflqirup

dwlrqfdq

ehuhs

uhvhqwhg

dv

OOU+s

� ,@

OOU�?|

w#��+y

� ,

@orj

S�?|

w#��+y

�@

4,

S�?|

w#��+y

�@

3, 1+51:8,

Lqdg

glwlrq

/wkhfkhfn

0wr0elwphvvdjh

yhfwruvvwduw

r�vhw

wrdxqlirup

glvwule

xwlrq

/vr

wkdw �

uf�� >u��� �

@��2>�2 �/

dqg

OOUEf�+u

�� ,@

orju���

uf��

@3=

Wkhlwhudwlrq

qxpehu

nvwduwv

dw41

Page 35: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp ;4

Vwhs 41 Elw0wr0fkhfn phvvdjhv1 Wkh phvvdjhv iurp elw qrghv wrfkhfn qrghv duh jlyhq e| +519;,/

OOUE&� +t��, @[

��M�E��.t��

OOUE&3���u���

�. OOU+s�, =

Vwhs 51 Fkhfn0wr0elw phvvdjhv1 Wkh phvvdjhv iurp fkhfn qrghvwr elw qrghv duh jlyhq e| +51:5,/

OOUE&� +u��, @ +�4,�wE���

3C \

��MwE��.t��

vjq�OOUE&�

�t�����4D�

3C [

��MwE��.t��

�����OOUE&�

�t��������4D >

zkhuh �+{, @ � orj�wdqk+%2,

�1

Vwhs 5�1 Fkhfn0wr0elw phvvdjhv +�plq� dssur{lpdwlrq,1 Dvdq dowhuqdwlyh wr Vwhs 5/ wkh phvvdjhv iurp fkhfn qrghv wr elw qrghvfdq eh dssur{lpdwhg xvlqj wkh plqlpxp ixqfwlrq +51:7,=

OOUE&� +u��, @ +�4,�wE���

3C \

��MwE��.t��

vjq�OOUE&�

�t�����4D�

plq��MwE��.t��

����OOUE&��t��������

Vwhs 61 Frpsxwh rxwsxw1 Wkh rxwsxw phvvdjh iurp wkh ghfrghulv wkh h{wulqvlf lqirupdwlrq/

OOUi |w#�� +y�, @

[

� �M�E��

OOUE&��u���

�>

dqg wkh hvwlpdwh ri wkh srvwhulru lqirupdwlrq OOUTLt| +y�, lv jlyhq e|

OOUE&� +t�, @[

��M�E��

OOUE&��u���

�.OOU+s�, >

iroorzlqj +51:3,1 Wklv lv xvhg wr frpsxwh wkh elw0e|0elw hvwlpdwh riwkh frghzrug=

eyE&��

@

�4 li OOUE&� +t�, A 3

3 li OOUE&� +t�, ? 3

;5 VRIW LWHUDWLYH GHFRGLQJ

Vwhs 71 Uhshdw xqwlo grqh1 Fkhfn li wkh vwrsslqj frqglwlrq kdvehhq uhdfkhg +h1j1/ d �{hg qxpehu ri lwhudwlrqv n4@ kdv ehhq uhdfkhg/ru wkh ghflvlrq zrug eyE&� vdwlv�hv wkh sdulw|0fkhfn pdwul{ K,1 Li qrw/wkhq lqfuhphqw n# n . 4/ dqg uhshdw Vwhsv 4/ 5 +ru 5�,/ dqg 61

Lq wkh qh{w vhfwlrq/ zh glvfxvv krz wr frpsxwh wkh lqwulqvlf OOUv lq+51:8,1 Iru uhihuhqfh/ iru frpprq fkdqqho prghov/ wkh lqwulqvlf OOUvfdq eh irxqg dv iroorzv=

iru d elqdu| v|pphwulf fkdqqho zlwk furvvryhu suredelolw| s dqg uh0fhlyhg elw |�/

OOU�?|w#�� +y�, @

+orj �3R

Rli |� @ 4

orj R

�3Rli |� @ 3

iru dq DZJQ fkdqqho zlwk qrlvh yduldqfh �2 dqg uhfhlyhg vljqdo |�/

OOU�?|w#�� +y�, @

5

�2|�1

717 PHPRU\OHVV FKDQQHOV

Qrz zh frqvlghu xvlqj wkh OGSF ghfrghu lq frqmxqfwlrq zlwk d fkdq0qho ghfrghu1 Frqvlghu wkh vlwxdwlrq vkrzq lq Iljxuh 5156 +vhh dovr Iljxuh4144,/ zkhuh wkh HFF hqfrghu +h1j1/ OGSF hqfrghu, �uvw wdnhv d xvhu elwvhtxhqfh x wr dq hqfrghg elw vhtxhqfh y/ zklfk lv wkhq sdvvhg wkurxjkd pdsshu wkdw pdsv wkh elwv wr d vhtxhqfh ri v|perov { wr eh wudqvplw0whg1 Diwhu wkhvh vljqdov sdvv wkurxjk wkh fkdqqho/ wkh uhfhlyhg vhtxhqfh| pxvw wkhq eh ghfrghg e| d fkdqqho ghfrghu1

Lq wklv vhfwlrq/ zh frqvlghu wkh wkuhh phpru|ohvv fkdqqhov suhvhqwhglq wkh Lqwurgxfwlrq +¢40514,/ dqg frqvlghu prgxohv wkdw xvh wkh uhfhlyhgydoxh |� wr �qg dq h{wulqvlf suredelolw| S i |

U�@?+{�, rq wkh wudqvplwwhg

v|pero {�1 +Lq odwhu fkdswhuv +uhi1 ¢60416,/ fkdqqhov zlwk phpru| duhfrqvlghuhg1, Wklv prgxoh iru kdqgolqj wkh uhfhlyhg ydoxhv lv d �fkdq0qho ghfrghu/� dovr nqrzq dv d �fkdqqho ghwhfwru1� Iru vrph fkdqqhov/wkh rshudwlrq ri wklv prgxoh pd| wxuq rxw wr eh udwkhu wulyldo/ exw zhqhyhuwkhohvv frqwlqxh wr fdoo lw d �ghfrghu� iru frqvlvwhqf|1

Iroorzlqj rxu glvfxvvlrq derxw prgxohv lq Vhfwlrq 516/ iru dq| hgjhehwzhhq prgxohv/ wkh h{wulqvlf suredelolw| z1u1w1 rqh prgxoh jlyhv wkhlqwulqvlf suredelolw| z1u1w1 wkh rwkhu prgxoh1 Lq sduwlfxodu/ wklv dssolhviru sdvvlqj phvvdjhv ehwzhhq wkh wkuhh prgxohv/ wkh fkdqqho ghfrghu/ dghpdsshu dqg dq HFF ghfrghu/ vkrzq lq 51561 Qrwh wkdw lq wkh elqdu|0lqsxw fkdqqhov frqvlghuhg lq wklv vhfwlrq/ wkh pdsshu lv dq dvvljqphqw

Page 36: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp ;6

channel

u

u

ECCencoder

v

ECCdecoder

vchanneldecoder

y

mapper

x

demapper

x

Iljxuh 51561 Fkdqqho ghfrghu/ ghpdsshu dqg HFF ghfrghu

ri d elw wr d elqdu| vljqdo/ vr wkdw wkh ghpdsshu mxvw qhhgv wr sdvvwkh phvvdjhv vwudljkw wkurxjk/ dfwlqj olnh dq htxdolw| prgxoh zlwk wzrexqgohv1

Vwduwlqj iurp wkh fkdqqho ghfrghu/ wkh h{wulqvlf suredelolw| S i |U�@?

+{�,lv wdnhq dv dq lqwulqvlf suredelolw| e| wkh ghpdsshu S �?|

_i4@T+{�,/ zklfk

wkhq +wulyldoo|, pdsv lw wr d suredelolw| S i |_i4@T

+y�, iru y�/ zklfk lv lq

wxuq wdnhq dv dq lqwulqvlf suredelolw| e| wkh HFF ghfrghu S �?|,��

+y�,/ dvlq +51:8,1 +Wr frpsohwh wkh surfhvv/ wkh HFF ghfrghu frpsxwhv h{wulqvlfsuredelolwlhv S i |

,��+y�,/ zklfk fdq wkhq eh xvhg wr jhqhudwh ghflvlrqv iru

wkh xvhu elwv x�1, Lq wklv vhfwlrq/ zh xvh �HFF� udwkhu wkdq �OGSF� lqrughu wr lqfoxgh rwkhu huuru0frqwuro frghv lq wklv glvfxvvlrq1

71714 ELQDU\ V\PPHWULF FKDQQHO

Iru wkh elqdu| v|pphwulf fkdqqho +EVF,/ zlwk furvv0ryhu suredelolw|s/ li wkh lqsxw wr wkh fkdqqho lv d elqdu| yduldeoh {� dqg wkh rxwsxw lv delqdu| yduldeoh |�/ wkhq wkh frqglwlrqdo suredelolwlhv lqwurgxfhg e| wkhfkdqqho duh jlyhq e|

S +|� @ d m {� @ d, @ 4� s

S +|� 9@ d m {� @ d, @ s

;7 VRIW LWHUDWLYH GHFRGLQJ

iru d @ 3> 41 Wkhq jlyhq wkh uhfhlyhg lqirupdwlrq |�/ wkh srvwhulru sure0delolw| fdq eh uhzulwwhq xvlqj Ed|hv* wkhruhp/

S +{� @ d m |� @ e, @S +|� @ e m {� @ d,S +{� @ d,

S +|� @ e,=

Vlqfh S +{� @ d, lv wkh lqwulqvlf suredelolw| iru {�/ wkh h{wulqvlf sured0elolw| iurp wklv prgxoh lv sursruwlrqdo wr S +|� @ e m {� @ d,/ zklfk lqwklv fdvh jlyhv

S i |

�5�+{�, @

�4� s li |� @ {�

s li |� 9@ {�

1

Lq whupv ri orj0olnholkrrg udwlrv/ wklv jlyhv

OOUi |

�5�+{�, @

+orj �3R

Rli |� @ 4

orj R

�3Rli |� @ 3

/

zklfk fdq eh xvhg dv OOU�?|

,��+y�, iru lqsxw wr wkh vriw ghfrghu iru wkh

HFF1Dfwxdoo|/ lqvwhdg ri wkh phvvdjh0sdvvlqj dojrulwkp iru OGSF frghv/

wkhuh h{lvw kdug0ghflvlrq yhuvlrqv ri wkh ghfrglqj dojrulwkp wkdw fdqwdnh dv lqsxw wkh uhfhlyhg elwv |� gluhfwo| iurp wkh elqdu| v|pphwulffkdqqho1 Wkhvh kdug0ghflvlrq dojrulwkpv/ lqfoxglqj rqh lqwurgxfhg e|Jdoodjhu ^Jdo96` dqg dqrwkhu e| Vlsvhu dqg Vslhopdq ^VV<9`/ duh pruhdssursuldwh iru wkh elqdu| v|pphwulf fkdqqho +dqg duh frpsxwdwlrqdoo|vlpsohu, wkdq wkh vxp0surgxfw xsgdwh uxoh1

71715 ELQDU\ HUDVXUH FKDQQHO

Rq wkh rwkhu kdqg/ iru wkh elqdu| hudvxuh fkdqqho +EHF,/ zlwk hudvxuhsuredelolw| s/ wkh frqglwlrqdo suredelolwlhv lqwurgxfhg e| wkh fkdqqho duh

S +|� @ 4 m {� @ 4, @ 4� s

S +|� @ H m {� @ 4, @ s

S +|� @ H m {� @ 3, @ s

S +|� @ 3 m {� @ 3, @ 4� s

zkhuh H lv wkh hudvxuh v|pero1 Djdlq/ wkh h{wulqvlf suredelolw| lv sur0sruwlrqdo wr wkhvh wudqvlwlrq suredelolwlhv/ vr wkdw

S i |

�,�+{� @ 4, @

;?=

4� s li |� @ 4s li |� @ H

3 li |� @ 3

S i |

�,�+{� @ 3, @

;?=

3 li |� @ 4s li |� @ H

4� s li |� @ 3=

Page 37: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp ;8

Lq whupv ri orj0olnholkrrg udwlrv/ wklv jlyhv

OOUi |

�,�+{�, @

;?=

.4 li |� @ 43 li |� @ H

�4 li |� @ 31

Wklv uh hfwv wkh idfw wkdw uhfhlylqj |� @ 3 ru 4 phdqv wkdw wkh lqsxwkdg wr eh {� @ 3 ru 4/ uhvshfwlyho|1Wkhq wkh lqsxwv wr wkh HFF ghfrghu zrxog eh

OOU�?|

,��+y�, @ OOUi |

�,�+{�, >

zklfk olhv lq wkh udqjh i�4> 3>.4j1 Wkhvh lq�qlwh ydoxhv pd| qrweh vxlwdeoh iru wkh OGSF ghfrglqj dojrulwkp vlqfh wkh| fdxvh htxd0wlrqv +519;, dqg +519<, wr ehfrph qrw zhoo gh�qhg1 Ghfrglqj dojrulwkpvghvljqhg iru wkh elqdu| hudvxuh fkdqqho duh suhvhqwhg lq Oxe| hw do1

^OPVV34d`1

71716 DZJQ FKDQQHO

Ilqdoo|/ zh frqvlghu wkh DZJQ fkdqqho/ lq zklfk dgglwlyh zklwhJdxvvldq qrlvh lv dgghg wr wkh wudqvplwwhg vljqdo1 Wkh vljqdov duh uhdo0ydoxhg +zklfk lv lqglfdwhg e| wkh xvh ri yduldeohv zlwk erog irqw,/ dqgwkh uhfhlyhg vljqdo |� lv jlyhq e|

|�@ {�.q�>

zklfk frqvlvwv ri wkh wudqvplwwhg vljqdo {� zlwk dgglwlyh qrlvh q� fkrvhqiurp d }hur0phdq Jdxvvldq glvwulexwlrq

S? +d, @4s5��

h{s

�� d2

5�2

�>

zkhuh � lv wkh vwdqgdug ghyldwlrq ri wkh qrlvh1 Wkh wudqvplwwhg vljqdo{� lv riwhq uhvwulfwhg wr d glvfuhwh vljqdoolqj frqvwhoodwlrq/ vxfk dv sxovhdpsolwxgh prgxodwlrq +SDP,,1 D frpsoh{0ydoxhg DZJQ fkdqqho fdqdovr eh gh�qhg lq dq dqdorjrxv zd|1Zh frqvlghu wkh xvh ri d elqdu| SDP frqvwhoodwlrq/ zklfk lv fdoohg wkh

elqdu|0lqsxw DZJQ fkdqqho +vrphwlphv deeuhyldwhg �ELDZJQF�,1Wkh hqfrghg elwv y� 5 i3> 4j duh pdsshg wr wkh frqvwhoodwlrq srlqwv{� 5 i�4> 4j iru wudqvplvvlrq/ xvlqj wkh pdsslqj {� @ 5y� � 41< Vlqfh

wkh fdsdflw| ri wkh DZJQ fkdqqho lv dfklhyhg e| d wudqvplw vljqdo wkdw

< Lw vkrxog eh qrwhg wkdw wkh rssrvlwh frqyhqwlrq/ � ' E3���l ' �3 2��/ lv frpprqo| xvhg

lq frppxqlfdwlrqv1

;9 VRIW LWHUDWLYH GHFRGLQJ

PAMsignal

mapper

AWGNchannel

xu

u~

ECCencoder

v

demapper

x̂ECC

decoder

v̂ AWGNchanneldecoder

y

Iljxuh 51571 Eorfn gldjudp ri v|vwhp zlwk dq DZJQ fkdqqho

dovr kdv d Jdxvvldq glvwulexwlrq/ wkh olplwdwlrq ri wkh wudqvplwwhg vljqdowr elqdu| SDP uhgxfhv wkh fdsdflw|1 Wkh fdofxodwlrq ri wkh fdsdflw| riwkh elqdu|0lqsxw DZJQ fkdqqho lv jlyhq lq Dsshqgl{ 51D/ dqg wkh EL0DZJQF fdsdflw| dqg DZJQ fkdqqho fdsdflw| duh frpsduhg lq Iljxuh51581

Lq wkh fdvh ri dq DZJQ fkdqqho/ zkhuh wkhuh lv qr lqwhuv|pero lqwhu0ihuhqfh +LVL,/ wkh fkdqqho uhfhlyhu fdq hvwlpdwh wkh fkdqqho orj0olnholkrrgudwlr dv iroorzv/

OOUTLt|�`B� +{�, @ orjS +{� @ 4 m |�,

S +{� @ �4 m |�,

@ orjS +|� m {� @ 4,

S +|� m {� @ �4,. orj

S +{� @ 4,

S +{� @ �4,

@ orjh{s

�� �2j2

+|� � 4,2�

h{s��

2j2+|� . 4,2

� . OOUlqw

�`B� +{�,

@5

�2|� .OOU�?|

�`B� +{�, >

vr wkdw wkh h{wulqvlf orj0olnholkrrg udwlr zlwk uhvshfw wr wkh fkdqqhoghfrghu lv

OOUi |

�̀ B� +{�, @5

�2|�1 +51:9,

Page 38: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvd

jh0sdvvlq

jdojrulwk

p;:

-10

12

34

56

0.2

0.4

0.6

0.8 1

1.2

1.4

1.6

1.8 2

Eb /N

0

bits per signal

capacity of AW

GN

channel

binary-input arbitrary input

Iljxuh

51581

Fdsdflwlhv

ri+due

lwudu|0lq

sxw,

DZJQ

fkdqqho

dqg

elqdu|

0lqsxw

DZJQ

fkdq

qho

Wkhghp

dsshu

vlpso|dvvlj

qvwkhsuredelolwlhv

iru{

�wr

y� 1Irud50SDP

frqvwhood

wlrq/wklvzrxog

vlpso|

frqvlvw

ridvvlj

qphqwv

riwkhirup

�f

�<T� +y�@4,

@��

�<f

� +{�@4,

�f

�<T� +y�@3,

@��

�<f

� +{�@�4,

=

Wkhq

wkhlqwulq

vlfOOUviruwkhHFFghfr

ghuduh

jlyhqe|

OOU�?|

,��+y� ,@OOUi |

�̀B�+{

� ,@

5�2|

� =

Lqwhup

vri

sure

delolwlhv/

wklvfdq

ehzulwwhq

dv

S�?|

,��+y� ,@

h{s�

2j2|� �

4.h{s�2j2|� �1

+51::,

Wkhvh

sure

delolwlhv

rqy�fdq

wkhq

ehwdnhq

dvwkhlqsxwwr

wkhOGSFvriw

ghfr

glqjdojrulwk

p1+Q

rwhwkdw

wkhDZJQfkdqqhoghfr

ghu

ghshqgvrq

dqhvwlp

dwhiru

wkhfkdqqhoqrlvh

sdudp

hwhu�21

Lqsudfwlfdo

lpsohp

hq0

wdwlrqv/wklvsdudp

hwhuqhhg

vwr

ehhvwlp

dwhg/dqgghyldwlrq

viurp

wkh

dfwxdo

ydoxhri

�2pd|

d�hfw

wkhshuirup

dqfh1

Lqwkhvh

vlpxodwlrq

v/lw

lvdvvx

phg

wkdw

wklvqrlvh

sdudp

hwhulvnqrz

qshuihfwo|1,

;;

VRIWLWHUDWLYHGHFRGLQJ

Wkhtxdolw|

riwkhfkdqqholvphdvx

uhglq

whupvri

wkhvljq

dowr

qrlvh

udwlr+VQU,/zklfk

lvgh�qhg

dv

HK

Qf

@S

5U�2>

+51:;,

zklfk

uhsuhvhq

wvwkhhqhuj|

qrup

dol}hgshu

xvhu

elw1

Wkhsrz

hulvdv0

vxphg

wrehS

@4/

wkhfrghudwh

lvU

/dqgwkhqrlvh

yduldqfh

lv�21

Dgmxvwlq

je|wkhfrghudwh

Ulq

wkhghqrp

lqdwru

wxuqvrxwwr

jlyhd

idlufrp

sdulvrq

ehwz

hhqfrghv

rigl�huhq

wudwhv1

Wkhvljq

do0wr0qrlvh

ud0wlr

lvxvxdoo|

phdvx

uhglq

ghfle

hov+gE,/

vrwkdw

wkhVQU

lvhtxdo

wr

43orj�f �

.K

�f �

gE1

Zhvkrz

vhyhudo�jxuhv

jlylqjwkhshuirup

dqfh

ridOGSF

frghrq

dLVL0iuhh

fkdqqho

zlwk

dgglwlyh

zklwh

Jdxvvldq

qrlvh

+DZJQ,1

Wkh

ghidx

owsdudp

hwhuvduh

dviroorz

v=wkhudwh

riwkhOGSFfrghlvU

@Hb /

wkheorfn

ohqjwk

lvQ

@577;/

wkhsdulw|

0fkhfn

pdwulfhv

duhfrq

vwuxfwhg

wrkdyh

wS@

6elwv

shu

froxpq+dq

gqr70f|

fohv,/dqgwkhghfr

glqjlv

shuirup

hgiru

dpd{lp

xp

ri53

lwhudwlrqv1

Iljx

uh5159

frqvlg

huvwkhshuirup

dqfh

irugl�huhq

weorfn

ohqjwk

v1Lq

dgglwlrq

/wkhelqdu|

0lqsxwfds

dflw|irup

xod

lvxvhg

wrsorw

dfds

dflw|fxuyh

vkrz

lqjwkhplqlp

xp

VQU

wkdw

lvuht

xluhg

wrdfk

lhyhwkdw

huuruudwh

zlwk

dudwh

Hbfrgh1

Dfrghri

ohqjwk

Q@

7;<9uht

xluhv

716gEwr

dfklhyh

delwhuuru

udwhri

433D1

Zkloh

wklvshuirup

dqfh

lvfrq

vlghude

o|ehwwhu

wkdq

wkhxqfrghg

v|vwhp

/wkhuh

lvvwloo

djds

wrwkhplqlp

xp

VQU

+HK @Q

f�

6gE,uht

xluhg

iruuholde

ohwudq

vplvvlrq

ryhuwkhelqdu|

0lqsxw

DZJQfkdqqhozlwk

dudwh

Hbfrgh1

Wkhshuirup

dqfh

fdqehlp

suryhg

e|

lqfuhdvlq

jwkheorfn

ohqjwk

dqgrswlp

l}lqjwkhghvljq

riwkhsdulw|

0fkhfn

pdwul{

1Iljx

uh515:

frqvlg

huvOGSFfrghv

rigl�huhq

wudwhv

U1Qrwh

wkdw

wkh

gh�qlwlrq

riHK @Q

fkdv

douhdg|wdnhq

lqwr

dffrxqwwkhgl�huhq

fhvlq

udwh/vr

wkdw

wkh�jxuh

jlyhvdidlu

frpsdulvrq

riwkhgl�huhq

wfrglqjudwhv1

Iljx

uh515;

frqvlg

huvwkhh�hfw

rifkdqjlq

jwkhghvljq

riwkhsdulw|

0fkhfn

pdwul{

wrkdyh

wS@

5>6>7elwv

shu

froxpq1Lw

fdqehvhhq

wkdw

kdylqjrqo|wS@

5elwv

shu

froxpqgrhv

qrw

zrun

yhu|zhoo/

zkloh

wS@

7

shuirup

vvoljk

wo|ehwwhu

wkdq

wS@

61Iljx

uh515<

sorwv

wkhelw

huuruudwh

fxuyh

diwhugl�huhq

wqxpehuv

rilwhudwlrq

v1Lw

fdqehvhhq

wkdw

lqwklv

fdvh/hyhq

rqhlwhudwlrq

riwkh

OGSFghfr

glqjdojrulwk

pjlyhv

vljql�fdq

wfrglqjjdlq

ryhudq

xqfrghg

v|vwhp

/dqgwkdw

doorzlqjdpd{lp

xp

ri7lwhudwlrq

vjhwv

zlwk

lq318

gE

riwkhshuirup

dqfh

ri53

lwhudwlrqv1

Iljx

uh5163

vkrz

vwkheorfn

huuruudwh

dvdixqfwlrq

riwkhVQU/iru

gl�huhq

wqxpehuv

rilwhudwlrq

v/dv

vhhqlq

1Wkheorfn

huuruudwh

fxuyhv

duhvwhhs

huwkdq

wkhelwhuuru

udwhfxuyhv1

Page 39: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvd

jh0sdvvlq

jdojrulwk

p;<

23

45

67

8

10-5

10-4

10-3

10-2

10-1

Eb /N

0

bit error rate

AW

GN

, vary N, t=

3, R=

8/9

uncoded N

=1224

N=

2448 N

=4896

capacity

Iljxuh

51591

Ydu|lqjwkheorfn

ohqjwk�

23

45

67

89

10-6

10-5

10-4

10-3

10-2

Eb /N

0

bit error rate

AW

GN

, vary rate, t=3, N

=2448

1/2 2/3 7/9 8/9 16/17 uncoded

Iljxuh

515:1

Ydu|lqjwkhudwh-

<3

VRIWLWHUDWLYHGHFRGLQJ

23

45

67

8

10-5

10-4

10-3

10-2

Eb /N

0

bit error rate

AW

GN

, vary t, N=

2448, R=

8/9

uncoded t=2 t=3 t=4 capacity

Iljxuh

515;1

Ydu|lqjwkhqxpehurielwv

shufroxpq

|S

3.54

4.55

5.56

6.57

7.58

8.5

10-5

10-4

10-3

10-2

Eb /N

0

bit error rate

bit error rate for different numbers of iterations

uncoded 1 iteration 2 iterations 3 iterations 4 iterations 10 iterations20 iterations

Iljxuh

515<1

Elwhuur

uudwh

yv1

VQU/sorwwh

girugl�huhqwqxpehuv

rilwh

udwlr

qv

Page 40: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp <4

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

10-2

10-1

100

Eb/N

0

bloc

k er

ror

rate

block error rate for different numbers of iterations

uncoded 1 iteration 2 iterations 3 iterations 4 iterations 10 iterations20 iterations

Iljxuh 51631 Eorfn huuru udwh yv1 VQU/ sorwwhg iru gl�huhqw qxpehuv ri lwhudwlrqv

718 QXPHULFDO H[DPSOH

Wr looxvwudwh wkh ghfrglqj ri OGSF frghv/ ohw xv frqvlghu wkh iroorzlqj

sdulw| fkhfn pdwul{=

K @

574 4 3 4 3 3

3 4 4 3 4 3

4 3 4 3 3 4

68 =

Vxssrvh wkdw wkh lqsxw phvvdjh lv x @�4 4 3

�1 Wkhq wdnlqj wkh

�uvw wkuhh elwv dv wkh v|vwhpdwlf phvvdjh elwv/ wkh fruuhvsrqglqj hqfrghg

frghzrug lv

y @�4 4 3 3 4 4

�1

Wkh wudqvplwwhg zrug lv wkhq

{ @�4 4 �4 �4 4 4

�=

Vxssrvh wkdw gxh wr qrlvh rq dq DZJQ fkdqqho/ wkh uhfhlyhg yhfwru lv

| @��4 5 �5 �5 5 3

�1

<5 VRIW LWHUDWLYH GHFRGLQJ

Wkh h{wulqvlf suredelolw| z1u1w1 wkh DZJQ ghfrghu lv jlyhq e| +51:9,/zklfk jrhv wkurxjk wkh ghpdsshu wr jlyh wkh lqwulqvlf suredelolwlhv z1u1w1wkh OGSF ghfrghu=

OOU�?|w#�� +y�, @ OOUi |

U�@? +{�, @5

�2|�

Wkhq dvvxplqj wkdw wkh DZJQ fkdqqho lv nqrzq wr kdyh d qrlvh sd0udphwhu ri �2 @ 5/ iru wklv uhfhlyhg zrug |/ wkh lqsxw wr wkh OGSFghfrghu kdv d orj0olnholkrrg udwlr ri

OOU+s�, @ OOU�?|w#�� +y�,@

��4 5 �5 �5 5 3

�>

fruuhvsrqglqj wr wkh suredelolwlhv

s� @�3=59; 3=;;4 3=44< 3=44< 3=;;4 3=8

�1

Wkh �uvw elw kdv ehhq fruuxswhg/ dqg wkh odvw elw kdv ehhq hudvhg1

71814 PHVVDJH0SDVVLQJ DOJRULWKP

Xvlqj wkh pdwul{ K dqg wkh lqwulqvlf lqirupdwlrq OOU+s�,/ zh fdqdsso| wkh orj0grpdlq yhuvlrq ri wkh phvvdjh0sdvvlqj dojrulwkp1 VlqfhOOUEf� +u��, lv lqlwldoo| vhw wr }hur/ dsso|lqj htxdwlrq +519;, jlyhv wkh

elw0wr0fkhfn phvvdjhv OOUE�� +t��, dv OOU+s�,1

OOU+t��, @

57�4 5 �5

5 �5 5�4 �5 3

68

Qrwlfh wkdw wkh duud| iru OOU+t��, lv ohiw eodqn zkhq wkh fruuhvsrqglqjhqwu| ri pdwul{ K lv }hur1 Wkh ghshqghqflhv ri wkh hqwulhv duh vkrzq lqIljxuh 51641

Wkhq wkh �uvw fkhfn0wr0elw phvvdjhv OOUE�� +u��, duh frpsxwhg xvlqjhtxdwlrq +519<,/ zkhuh hdfk phvvdjh OOU+u��, lv d ixqfwlrq ri doo wkhrwkhu phvvdjhv OOU+t���, lq wkh vdph urz1 Iru h{dpsoh/ wkh �uvw hqwu|lq wkh �uvw urz lv jlyhq e|

OOU+u��, @ +�4, 5 wdqk3��wdqk

�4

5� 5

�wdqk

�4

5� �5

��

@ 5wdqk3�+3=:952, @ 4=658=

Vlploduo|/ lw lv srvvleoh wr frpsxwh wkh uhvw ri wkh phvvdjhv iurp fkhfnv

wr elwv1

OOUE�� +u��, @

57 4=658 �3=:68 3=:68

4=658 �4=658 4=6583 3 �3=:68

68

Page 41: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvd

jh0sdvvlq

jdojrulwk

p<6

3633

31

2523

22

1412

11

qq

q

qq

q

qq

q

3633

31

2523

22

1412

11

rr

r

rr

r

rr

r

3633

31

2523

22

1412

11

qq

q

qq

q

qq

q

3633

31

2523

22

1412

11

rr

r

rr

r

rr

r

=

+

= =

14

q

12

q 11

r

=1

1q

3p

31

r

++

[]6

54

32

1p

pp

pp

p

Iljxuh

51641

Lwhudwlq

jsure

delolwlhv

lqwkhOGSF

ghfr

glqjdojrulwk

p+vwduw

dwwkh

xsshu

ohiwpdwul{

dqgsurfhhg

frxqwhu0for

fnzlvh,

Wkhq

vxpplqjxswkhfrox

pqvjlyhv

wkhh{wulq

vlfrx

wsxwiru

wkh�uvw

lwhudwlrqri

wkhghfr

ghu/

dqgdg

glqjwkhlqwulq

vlflqsxwv

OOU+s� ,

dvlq

htxdwlrq

+51:3,jlyhv

wkhsrvwhulru

lqirup

dwlrqdv

OOU

E��+t� ,@ �

3=658

5=8<3

�6=658

�4=598

6=658

�3=:68 �

=

Pdnlqjdkdug

0ghflvlrq

edvhg

rqwkhvljq

vri

wkhvh

OOUvdv

iroorzv/

ey�@ �

4liOOU+t� ,A3

3liOOU+t� ,?3

>

jlyhvdq

lqlwldo

hvwlpdwh

riwkhwudq

vplwwhg

frghz

rug/diwhu

rqhlwhudwlrq

riphvvdjh0s

dvvlqj=ey

E��@ �

44

33

43 �

Frp

sdulq

jzlwk

wkhruljlq

dofrghz

rugy@ �

44

33

44 �/

zh

vhhwkdw

wkh�uvw

elwkdv

ehhq

fruuhfwhg/exwwkhodvw

elwlvvwloo

lqhuuru1

Surfhhg

lqjzlwk

dqrwk

hulwhudwlrq

riwkhphvvdjh0s

dvvlqjdojrulwk

p/

htxdwlrq

+519;,frp

sxwhv

wkhphvvdjh

OOU

E2�+t�� ,

e|vxpplqjdoo

wkh

phvvdjhv

OOU

E�� �u��� �

lqwkhvdp

hfrox

pq

+h{fhs

wiru

wkhrq

hlq

wkh

m0wkurz

,dq

gdovr

dgglqjwkhlqwulq

vlflqirup

dwlrqOOU+s

� ,1Dvdq

h{0

dpsoh/

wkh�uvw

hqwu|

OOU

E2�+t

�� ,

fdqehirx

qg

e|vxpplqjryhu

doo

<7VRIWLWHUDWLYHGHFRGLQJ

OOUE�� �

u��� �

lqwkh�uvw

froxpq/h{fhs

wiru

OOUE��+

u�� ,

+vlqfh

wkhlq0

frplqjlqwulq

vlfphvvdjh

pxvwehh{fox

ghg

zkhq

frpsxwlq

jwkhh{wulq

vlfphvvdjh,1

OOUE2�+t�� ,@ [�

��'�

OOUE�� �

u��� �

.OOU+s� ,

@OOUE��+

u�� ,.OOU+s� ,

@3.+�4,

@�4

Frqwlq

xlqjwklv

frpsxwdwlrq

/doo

wkhphvvdjhv

iurpelwv

wrfkhfn

vfdq

ehxsgdwhg

dviroorz

v=

OOUE2�+

t�� ,@ 57

�4

6=658

�5

4=597

�5

53=658

�6=658

3 68

Qrwh

wkdw

liwkhsrvwhulru

lqirup

dwlrqOOUE��+

t� ,

kdv

douhdg|ehhq

fdo0fxodwhg

/wkhq

wkhelw0wr0fk

hfnphvvdjhv

fdqdovr

ehirx

qgdv

OOUE2�+

t�� ,@OOUE��+t� ,�OOUE��+

u�� ,

=

Qh{w/

dsso|lqjhtxdwlrq

+519<,jlyhv

dqxsgdwhg

vhwri

fkhfn

0wr0elw

phvvdjhv=

OOUE2�+

u�� ,@ 57

4=:9<

�3=:68

3=<53

4=658

�3=<44

3=<44

33

3=635 68

Dsso|lqjhtxdwlrq

+51:3,iru

wkhvhfrq

gwlp

h|lhog

v

OOUE2�+

t� ,@ �

3=:9<

5=8<3

�5=<44

�4=3;3

5=<44

3=635 �

=

Wdn

lqjkdug

ghflvlrq

vwkhq

jlyhvdfruuhfw

hvwlpdwh

riwkhwudq

vplwwhg

frghz

rug/

eyE2�@ �

44

33

44 �

=

Wklvvkruw

sdulw|

0fkhfn

frghkdv

fruuhfwhgdq

huurudq

gdq

hudvxuh

xvlq

jwz

rlwhudwlrq

vri

wkhphvvdjh0s

dvvlqjdojrulwk

p1

71815

_PLQ%DSSUR[LPDWLRQ

Rqwkhrwk

hukdq

g/wkh�p

lq�ds

sur{

lpdwlrq

lq+51:7,

r�huv

vlplodu

ghfr

glqjshuirup

dqfh

zlwk

vlpsol�

hglp

sohp

hqwdwlrq

1Zhvwduw

djdlqzlwk

wkhlqwulq

vlfsure

delolwlhv

iurpwkhfkdq

qho/

zklfk

|lhog

vwkhlqlwldo

Page 42: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Phvvdjh0sdvvlqj dojrulwkp <8

phvvdjhv t�� iurp elwv wr fkhfnv1

OOU +s�, @��4 5 �5 �5 5 3

OOUE�� +t��, @

57 �4 5 �5

5 �5 5�4 �5 3

68

Dsso|lqj +51:7, |lhogv wkh phvvdjhv iurp fkhfnv wr elwv xvlqj wkh plql0pxp ixqfwlrq=

OOUE�� +u��, @

57 5 �4 4

5 �5 53 3 �4

68

Vxpplqj xs wkh froxpqv ri OOUE�� +u��, dv zhoo dv wkh lqwulqvlf lqiru0pdwlrq OOU+s�, xvlqj +51:3, jlyhv wkh srvwhulru OOU dv

OOUE�� +t�,@�4 6 �7 �4 7 �4

�=

Wdnlqj kdug0ghflvlrqv jlyhv wkh yhfwru eyE�� @�4 4 3 3 4 3

�/

zklfk lv lqfruuhfw lq wkh odvw elw1Wkh qh{w lwhudwlrq ehjlqv e| dsso|lqj +519;, wr rewdlq wkh xsgdwhg

phvvdjhv iurp elwv wr fkhfnv1

OOUE2� +t��, @

57 �4 7 �5

4 �5 54 �7 3

68

Wkhq dsso|lqj +51:7, rqfh pruh jlyhv

OOUE2� +u��, @

57 5 �4 4

5 �4 43 3 4

68 >

vr wkdw vxpplqj wkh froxpqv ri OOUE2� +u��, dqg OOU+s�, jlyhv

OOUE2� +t�,@�4 6 �6 �4 6 4

�>

zklfk fruuhvsrqgv wr wkh fruuhfw frghzrug diwhu pdnlqj d kdug0ghflvlrq=

eyE2� @ �4 4 3 3 4 4

�=

DSSHQGL[ 51D= Fdsdflw| ri elqdu|0lqsxw fkdqqho

Wkh hydoxdwlrq ri wkh fdsdflw| ri d elqdu|0lqsxw fkdqqho zlwk dggl0wlyh zklwh Jdxvvldq qrlvh lv dv iroorzv1 +Wklv ghvfulswlrq lv wdnhq iurp

<9 VRIW LWHUDWLYH GHFRGLQJ

^Iuh<;`,1 Iluvw/ iru d jlyhq qrlvh ohyho/ uhsuhvhqwhg e| wkh qrlvh yduldqfh�2/ wkh pxwxdo lqirupdwlrq lv jlyhq e|

P��2�@ L +{�> |�, @

[%�M�

]+�

S +{�> |�, orj2S +{�> |�,

S +{�,S +|�,g|�

@[%�M�

]+�

S +{�> |�, orj2 S +|� m {�, g|� �]+�

S +|�, orj2 S +|�,g|�

zkhuh D @ i�4>.4j lv wkh SDP frqvwhoodwlrq iru elqdu| vljqdoolqj1Wkh �uvw whup lv wkh qhjdwlyh ri wkh hqwurs| ri d Jdxvvldq glvwulexwlrq/��

2orj2

�5��2h

�/ zkloh wkh vhfrqg whup lv jlyhq e|

]+�

3C[

%�M�

h{s�� �

2j2+|� � {�,

2

5s5��2

4D orj

2

3C[

%�M�

h{s�� �

2j2+|� � {�,

2

5s5��2

4Dg|�

zklfk fdq eh hydoxdwhg xvlqj qxphulfdo lqwhjudwlrq1Wkhq wkh pxwxdo lqirupdwlrq P

��2�jlyhv wkh fdsdflw|/ zklfk uhsuh0

vhqwv wkh pd{lpxp udwh U wkdw doorzv uholdeoh wudqvplvvlrq1 Xvlqj wkhgh�qlwlrq +51:;,/ wkh plqlpxp vljqdo0wr0qrlvh udwlr +HK@Qf, wkdw doorzvuholdeoh wudqvplvvlrq lv wkhq jlyhq e|

VQU4�? @4

5P +�2,�2>

zkhuh S @ 4/ dvvxplqj elqdu| SDP1

Page 43: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Fkdswhu ;

RWKHU WRSLFV

Zh forvh zlwk vrph wrslfv uhodwhg wr wkh dssolfdwlrq ri vriw lwhudwlyhghfrglqj whfkqltxhv wr frppxqlfdwlrqv1

41 PXOWLOHYHO PRGXODWLRQ

Wr h{whqg wkhvh lghdv ri vriw lwhudwlyh ghfrglqj wr rwkhu frppxql0fdwlrqv fkdqqhov/ lw lv qhfhvvdu| wr frqvlghu pxowlohyho prgxodwlrq1 Lqwkh frqwh{w ri frppxqlfdwlrqv/ prgxodwlrq uhihuv wr krz wkh vljqdo lvvkdshg iru wudqvplvvlrq ryhu d fkdqqho1 Wkh prgxodwlrq frqvlvwv ridvvljqlqj wkh elwv wr vrph frqvwhoodwlrq srlqwv/ zklfk uhsuhvhqw wkh vlj0qdoolqj zdyhirupv wkdw duh xvhg wr wudqvplw wkh vljqdo1 Zh uhvwulfw rxudwwhqwlrq wr Sxovh Dpsolwxgh Prgxodwlrq +SDP,/ lq zklfk wkh vljqdofrqvwhoodwlrqv duh srlqwv rq wkh uhdo olqh/ vlqfh Txdgudwxuh DpsolwxghPrgxodwlrq +TDP, lq wkh frpsoh{ sodqh fdq riwhq eh lpsohphqwhg vrwkdw wkh uhdo dqg lpdjlqdu| frpsrqhqwv pd| eh ghfrghg vhsdudwho|1

Wr hqfrgh wkhvh frqvwhoodwlrqv/ wkhuh duh ydulrxv srvvlelolwlhv iru dv0vljqlqj elwv wr wkh frqvwhoodwlrq srlqwv1 Li zh wkhq dsso| dq huuru0frqwurofrgh rq wkhvh elwv/ wklv frq�jxudwlrq lv nqrzq dv frghg prgxodwlrq1 Lizh xvh Jud| frglqj iru wkh odehov/ vr wkdw odehov iru dgmdfhqw frqvwho0odwlrq srlqwv rqo| gl�hu lq d vlqjoh elw/ wkhq wklv w|sh ri frghg prg0xodwlrq lv fdoohg Elw0Lqwhuohdyhg Frghg Prgxodwlrq +ELFP,1 +Li zh xvhdqrwkhu odehoolqj wkdw glylghv wkh frqvwhoodwlrq lqwr frvhwv/ dqg wkhq ds0so| d frqyrqxowlrqdo frgh rq wkh elwv wkdw fkrrvh wkh frvhw/ wklv lv fdoohgWuhoolv0Frghg Prgxodwlrq +WFP,1, Wkh ghwdlov ri wkh vlwxdwlrq zloo gh0whuplqh wkh w|sh ri frghg prgxodwlrq wkdw lv dssursuldwh1 +Uhihu wr^Fkh<:`^QOl33` iru pruh ghwdlov rq ELFP dqg WFP zlwk vriw lwhudwlyhghfrglqj1,

538

539 VRIW LWHUDWLYH GHFRGLQJ

mapper/demapper

x

1w

2w

Lw

Iljxuh ;141 Prgxoh iru pdsslqj dqg ghpdsslqj

Wkh ghpdsshu wkhq kdv wkh wdvn ri �qglqj wkh h{wulqvlf suredelolwlhvedvhg rq wkh uhfhlyhg vljqdov1 Xqwlo qrz/ zh kdyh rqo| frqvlghuhg elqdu|SDP +zkhuh z @ i3> 4j pdsv wr { @ i�4> 4j,/ vr wkdw wkh pdsslqj dqgghpdsslqj kdyh ehhq wulyldo1

Iru wkh jhqhudo fdvh/ dv vkrzq lq Iljxuh ;14/ zh fdq wuhdw wklv vljqdopdsshu dv d orfdo frqvwudlqw V/ zklfk sxwv d uhvwulfwlrq rq wkh lqsxwelwv +z�> z2> ===> zu, dqg wkh rxwsxw vljqdo {/ zklfk lv d uhdo0ydoxhg frq0vwhoodwlrq srlqw +lq wkh fdvh ri SDP, ru d frpsoh{ frqvwhoodwlrq srlqw+lq wkh fdvh ri TDP,1 Lq rwkhu zrugv/ iru hdfk vhw ri lqsxw elwv/ wkhuh lvd xqltxh rxwsxw { fruuhvsrqglqj wr wkh srlqw wkdw lv odehoohg e| wkhvhlqsxw elwv1

Wklv jlyhv xv wkh ghvfulswlrq ri wkh ghpdsshu/ zkrvh xsgdwh htxdwlrqlv dv iroorzv/

S i |_i4@T +z�, @ f�

��

[

E�� c���c�uc �M7;t���

S�?|_i4@T +{,

\

�� �'�

S�?|_i4@T +z��, > +;14,

zkhuh f���lv d qrupdol}lqj frqvwdqw1 Li wkhuh duh qr lqwulqvlf suredelo0

lwlhv rq wkh elwv +l1h1 wkh lqwulqvlf suredelolwlhv duh doo htxdo,/ wkhq wkhh{suhvvlrq ehfrphv vlpsohu |hw=

S i |_i4@T +z�, @ f����

[

E��c���c�uc �M7;t���

S �?|_i4@T +{, +;15,

Lq wkh fdvh ri 70SDP/ wkh frqvwudlqw vhw V pljkw frqvlvw ri hohphqwv+z�> z2>{, dv iroorzv=

V @ i+3> 3>.6, > +4> 3>.4, > +4> 4>�4, > +3> 4>�6,j

Page 44: Fkdswhu 4 LQWURGXFWLRQ - Stanford University · 2001. 5. 15. · zrw vroglhvu rq wkh hqg/ zklfk rqo| hkdy zrw hgjhv/ d|q phvdjhv kh| uhfhlyh droqj rqh hgjh vl gluhfwo| sdvvhg dorqj

Rwkhu wrslfv 53:

mapper1+=x

11 =w

02 =w

demapperx

−=

1

11

1

p

pw

−=

2

22

1

p

pw

Iljxuh ;151 Pdsslqj dqg ghpdsslqj iru 70SDP

Lq wklv fdvh/ wkh odeho xvhv Jud| frglqj/ vr wkdw wklv ghpdsshu fruuh0vsrqgv wr elw0lqwhuohdyhg frghg prgxodwlrq +ELFP,1 Pdsslqj dqg vriwghpdsslqj lq wklv h{dpsoh lv ghslfwhg lq Iljxuh ;151

Lq wkh fdvh ri dq DZJQ fkdqqho/ wkhq wkh suredelolw| ghqvlw| ixqfwlrqrq { zloo eh d Jdxvvldq glvwulexwlrq1 Lq rwkhu zrugv/ li wkh ydoxh | zdvuhfhlyhg/ dqg lw lv nqrzq wkdw wkh qrlvh kdv yduldqfh �

2+ / wkhq { kdv d

Jdxvvldq glvwulexwlrq zlwk phdq ydoxh htxdo wr | dqg yduldqfh �2+ 1

S�?|_i4@T +{, @

4s5��+

h{s

��

4

5�2++{� |,2

�=

Lq wkh h{suhvvlrq iru wkh h{wulqvlf suredelolw| +;15,/ wkh odujhvw whup zloowhqg wr grplqdwh wkh vxppdwlrq1 Lq rwkhu zrugv/ wkh SDP frqvwhoodwlrqsrlqw wkdw lv forvhvw wr wkh uhfhlyhg vljqdo | zloo eh wkh grplqdwlqj whup/

Si | +z� @ $�, @ f��

[

E��c���c�uc �M7;t���

S�?| +{,

� f��

4s5��+

h{s

�� 4

5�2+

+[+c��'/� � |,2�

53; VRIW LWHUDWLYH GHFRGLQJ

zkhuh wkh srlqw

[+c��'/�@ duj plq

vxfk wkdw

+��c���c�uc ,M7dqg ��'/�

+{� |,2

uhsuhvhqwv wkh forvhvw srlqw { wr wkh uhfhlyhg vljqdo | rxw ri doo wkh srlqwv

lq wkh SDP frqvwhoodwlrq wkdw kdyh odehov zlwk elw z� htxdo wr $�1 Wkhqwkh orj0olnholkrrg udwlr lv jlyhq dv iroorzv/

OOUi | +z�, @h{s

��

2j2++[+c��'� � |,2

h{s��

2j2++[+c��'f � |,2

� >

zklfk fdq eh vlpsol�hg dv

OOUi | +z�, @ �4

5�2+

�+[+c��'�,

2� +[+c��'f,

2

. 5 +�[+c��'f .[+c��'�,|, =

+Qrwh wkdw iru 50SDP/ wklv h{suhvvlrq vlpsol�hv wr

OOUi | +z�, @5

�2+|

vlqfh [+c��@ 4/ zklfk frlqflghv zlwk wkh h{suhvvlrq ghulyhg lq +51:9,1,

Dv d uhvxow ri wklv dssur{lpdwlrq/ wkh orj0olnholkrrg udwlr lv vlpso| dolqhdu ixqfwlrq ri wkh uhfhlyhg vljqdo |/ pdnlqj wkh frpsoh{lw| ri wklvdssur{lpdwh vriw ghpdsslqj rshudwlrq ihdvleoh iru sudfwlfdo lpsohphqwd0wlrqv1 Lq dgglwlrq/ zh phqwlrq wkdw lw lv srvvleoh wr lwhudwh vriw lqirupd0wlrq zlwk wklv ghpdsshu1 Lq rwkhu zrugv/ xsgdwhg h{wulqvlf suredelolwlhviru z iurp wkh HFF ghfrglqj frxog eh xvhg dv lqwulqvlf suredelolwlhv iruwkh ghpdsshu wr lpsuryh wkh ghpdsslqj/ xvlqj htxdwlrq +;14,1 Lq wklvzd|/ wkh ghpdsshu fdq dovr eh lqwhjudwhg lqwr wkh vriw lwhudwlyh ghfrglqjiudphzrun iru dgglwlrqdo frglqj jdlq/ dv ghprqvwudwhg lq Pxtxhw hw do1^PPGF33`1

51 SRWHQWLDO DSSOLFDWLRQV RI

FRQVWUDLQHG FRGLQJ

Zh eulh | olvw vrph vlwxdwlrqv rwkhu wkdq wkh vwrudjh fkdqqho zkhuhwkh lghdv uhodwhg wr frqvwudlqhg frglqj pd| srwhqwldoo| eh dssolfdeoh=

Frqvwhoodwlrq0vkdslqj frghv wkdw fkrrvh wkh glvwulexwlrq ri frqvwhood0

wlrq srlqwv vxfk wkdw wkh uhvxowlqj glvwulexwlrq ri wkh wudqvplw vljqdo


Recommended