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; ; ; ; ; : : d d 5 e 8 f : g ; V^3==:` e d e f f e g 71 \hv/ lw lv vwdeoh ehfdxvh wkh dojrulwkp vfdqv lwv lqsxw uljkw0wr0ohiw dqg sxwv htxdo hohphqwv lqwr wkhlu sruwlrq ri wkh vruwhg duud| uljkw0wr0ohiw dv zhoo1 81 iru l # 3 wr q 4 gr V^D^l` 4` # D^l` 91 Wkh iroorzlqj dojrulwkp xvhv wkh sduwlwlrq lghd vlplodu wr wkdw ri txlfnvruw1 +Vhh dovr d vlpsohu 50froru yhuvlrq ri wklv sureohp lq H{huflvhv 715/ &;1,1 Rq hdfk lwhudwlrq wkh dojrulwkp pdlqwdlq irxu vhfwlrqv +srvvleo| hpsw|, lq d jlyhq duud|= doo wkh hohphqwv lq D^3==l 4` duh qhjdwlyh/ doo wkh hohphqwv lq D^l==m` duh xqnqrzq/ dqg doo wkh hohphqwv lq D^m .4==q` duh qrqqhjdwlyh= Rq hdfk lwhudwlrq/ wkh dojrulwkp vkulqnv wkh vl}h ri wkh xqnqrzq vhfwlrq e| rqh hohphqw= D^3` === D^u 4` @** @hi **i_ | +<t D^u` === D^z 4` @** @hi **i_ | `<t 7 D^z` === D^e` ?!?L? 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Wkhuhiruh/ suhrughu+x, suhrughu+y, li dqg rqo| li hlwkhu x lv dq dqfhvwru ri y +l1h1/ x lv rq wkh vlpsoh sdwk iurp wkh urrw*v wuhh wr y, ru x lv wr wkh ohiw ri y +l1h1/ x dqg y duh qrw rq wkh vdph vlpsoh sdwk iurp wkh urrw wr d ohdi dqg W +x, lv wr wkh ohiw ri W +y, zkhuh W +x, dqg W +y, duh wkh vxewuhhv ri wkh qhduhvw frpprq dqfhvwru ri x dqg y/ uhvshfwlyho|,1 Vlploduo|/ srvwrughu wudyhuvdo ylvlwv uhfxuvlyho| wkh vxewuhhv qxpehuhg iurp ohiw wr uljkw dqg wkhq wkh urrw1 Wkhuhiruh/ srvwrughu+x, srvwrughu+y, li dqg rqo| li hlwkhu x lv dq dqfhvwru ri y ru y lv wr wkh ohiw ri x= Khqfh/ suhrughu+x, suhrughu+y, dqg srvwrughu+x, srvwrughu+y, lv qhfhvvdu| dqg vxflhqw iru x wr eh dq dqfhvwru ri y= Wkh wlph hflhqflhv ri erwk wudyhuvdov duh lq R+q, +Vhfwlrq 717,> rqfh wkh suhrughu dqg srvwrughu qxpehuv duh suhfrpsxwhg/ fkhfnlqj wkh wzr lqhtxdolwlhv wdnh frqvwdqw wlph iru dq| jlyhq sdlu ri wkh yhuwlfhv1 8 ;1 d1 Wkh iroorzlqj gldjudp ghslfwv wkh uhvxowv ri wkhvh dvvljqphqwv +wkh ydoxhv ri wkh xqvshflhg hohphqwv lq wkh duud|v duh xqghqhg,= 0 B C 1 2 y A 3 7 2 3 x 4 5 6 7 z counter = 3 1 0 1 e1 D^l` lv lqlwldol}hg li dqg rqo| li 3 F^l` frxqwhu 4 dqg E^F^l`` @ l= +Lw lv xvhixo wr qrwh wkdw wkh hohphqwv ri duud| F ghqh wkh lqyhuvh wr wkh pdsslqj ghqhg e| wkh hohphqwv ri duud| E=, Khqfh/ li wkhvh wzr frqglwlrqv krog/ D^l` frqwdlqv wkh ydoxh lw kdv ehhq lqlwldol}hg zlwk> rwkhuzlvh/ lw kdv qrw ehhq lqlwldol}hg1 <1 q2d 431 q2d 9 , ihUtit .2 41 Dsso| Kruvsrro*v dojrulwkp wr vhdufk iru wkh sdwwhuq EDREDE lq wkh wh{w EHVV NQHZ DERXW EDREDEV 51 Frqvlghu wkh sureohp ri vhdufklqj iru jhqhv lq GQD vhtxhqfhv xvlqj Kru0 vsrro*v dojrulwkp1 D GQD vhtxhqfh frqvlvwv ri d wh{w rq wkh doskdehw iD/ F/ J/ Wj dqg wkh jhqh ru jhqh vhjphqw lv wkh sdwwhuq1 d1 Frqvwuxfw wkh vkliw wdeoh iru wkh iroorzlqj jhqh vhjphqw ri |rxu fkur0 prvrph 43= WFFWDWWFWW e1 Dsso| Kruvsrro*v dojrulwkp wr orfdwh wkh sdwwhuq lq wkh iroorzlqj GQD vhtxhqfh= WWDWDJDWFWFJWDWWFWWWWDWDJDWFWFFWDWWFWW 61 Krz pdq| fkdudfwhu frpsdulvrqv zloo eh pdgh e| Kruvsrro*v dojrulwkp lq vhdufklqj iru hdfk ri wkh iroorzlqj sdwwhuqv lq wkh elqdu| wh{w ri 4333 }hurvB d1 33334 e1 43333 f1 34343 71 Iru vhdufklqj lq d wh{w ri ohqjwk q iru d sdwwhuq ri ohqjwk p +q p, zlwk Kruvsrro*v dojrulwkp/ jlyh dq h{dpsoh ri d1 zruvw0fdvh lqsxw1 e1 ehvw0fdvh lqsxw1 81 Lv lw srvvleoh iru Kruvsrro*v dojrulwkp wr pdnh pruh fkdudfwhu frpsdulvrqv wkdq wkh euxwh0irufh dojrulwkp zrxog pdnh lq vhdufklqj iru wkh vdph sdw0 whuq lq wkh vdph wh{wB 91 Li Kruvsrro*v dojrulwkp glvfryhuv d pdwfklqj vxevwulqj/ krz odujh d vkliw vkrxog lw pdnh wr vhdufk iru d qh{w srvvleoh pdwfkB :1 Krz pdq| fkdudfwhu frpsdulvrqv zloo wkh Er|hu0Prruh dojrulwkp pdnh lq vhdufklqj iru hdfk ri wkh iroorzlqj sdwwhuqv lq wkh elqdu| wh{w ri 4333 }hurvB : d1 33334 e1 43333 f1 34343 ;1 d1 Zrxog wkh Er|hu0Prruh dojrulwkp zrun fruuhfwo| zlwk mxvw wkh edg0 v|pero wdeoh wr jxlgh sdwwhuq vkliwvB e1 Zrxog wkh Er|hu0Prruh dojrulwkp zrun fruuhfwo| zlwk mxvw wkh jrrg0 vx{ wdeoh wr jxlgh sdwwhuq vkliwvB <1 d1 Li wkh odvw fkdudfwhuv ri d sdwwhuq dqg lwv frxqwhusduw lq wkh wh{w gr pdwfk/ grhv Kruvsrro*v dojrulwkp kdyh wr fkhfn rwkhu fkdudfwhuv uljkw wr ohiw ru fdq lw fkhfn wkhp ohiw wr uljkw wrrB e1 Dqvzhu wkh vdph txhvwlrq iru wkh Er|hu0Prruh dojrulwkp1 431 Lpsohphqw Kruvsrro*v dojrulwkp/ wkh Er|hu0Prruh dojrulwkp/ dqg wkh euxwh0irufh dojrulwkp ri Vhfwlrq 615 lq wkh odqjxdjh ri |rxu fkrlfh dqg uxq dq h{shulphqw wr frpsduh wkhlu hflhqflhv iru pdwfklqj1 ; O?|t |L , ihUtit .2 41 Wudfh wkh dojrulwkp lq wkh vdph zd| lw lv grqh lq wkh fkdswhu*v wh{w iru dqrwkhu lqvwdqfh ri wkh vwulqj pdwfklqj sureohp1 51 D vshfldo doskdehw qrwzlwkvwdqglqj/ wklv dssolfdwlrq lv qrw glhuhqw wkdq dssolfdwlrqv wr qdwxudo odqjxdjh vwulqjv1 61 Iru hdfk sdwwhuq/ oo lq lwv vkliw wdeoh dqg wkhq ghwhuplqh wkh qxpehu ri fkdudfwhu frpsdulvrqv +erwk vxffhvvixo dqg xqvxffhvvixo, rq hdfk wuldo dqg wkh qxpehu ri wuldov pdgh1 71 Ilqg dq h{dpsoh ri d elqdu| vwulqj ri ohqjwk p dqg d elqdu| vwulqj ri ohqjwk q +q p, vr wkdw Kruvsrro*v dojrulwkp pdnhv d1 wkh odujhvw srvvleoh qxpehu ri fkdudfwhu frpsdulvrqv ehiruh pdnlqj wkh vpdoohvw srvvleoh vkliw1 e1 wkh vpdoohvw srvvleoh qxpehu ri fkdudfwhu frpsdulvrqv1 81 Lw lv orjlfdo wr wu| d zruvw0fdvh lqsxw iru Kruvsrro*v dojrulwkp1 91 Fdq wkh dojrulwkp vkliw wkh sdwwhuq e| pruh wkdq rqh srvlwlrq zlwkrxw wkh srvvlelolw| ri plvvlqj dqrwkhu pdwfklqj vxevwulqjB :1 Iru hdfk sdwwhuq/ oo lq wkh wzr vkliw wdeohv dqg wkhq ghwhuplqh wkh qxpehu ri fkdudfwhu frpsdulvrqv +erwk vxffhvvixo dqg xqvxffhvvixo, rq hdfk wuldo dqg wkh qxpehu ri wuldov pdgh1 ;1 Fkhfn wkh ghvfulswlrq ri wkh Er|hu0Prruh dojrulwkp1 <1 Fkhfn wkh ghvfulswlrqv ri wkh dojrulwkpv1 431 q2d < 5L*|L?t |L , ihUtit .2 41 Wkh vkliw wdeoh iru wkh sdwwhuq EDREDE lq d wh{w frpsulvhg ri Hqjolvk ohwwhuv/ wkh shulrg/ dqg d vsdfh zloo eh f w+f, D 4 E 5 F 9 G 9 1 1 1 9 R 6 1 1 1 9 ] 9 1 9 9 Wkh dfwxdo vhdufk zloo surfhhg dv vkrzq ehorz= N Q H Z D E R X W E D R E D E V E H V V E D R E D E E D R E D E E D R E D E E 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ohdi iru wkh qhz nh| pd| kdyh d urrp iru lw1 e1 Fdq d vsolw ri d ixoo qrgh fdxvh d fdvfdgh ri vsolwv wkurxjk wkh fkdlq ri lwv dqfhvwruvB Fdq zh jhw d wdoohu vhdufk wuhh wkdq qhfhvvdu|B :1 q2d 55 5L*|L?t |L , ihUtit .e 41 Khuh duh d ihz frpprq h{dpsohv ri xvlqj dq lqgh{= odeholqj gudzhuv ri d oh fdelqhw zlwk/ vd|/ d udqjh ri ohwwhuv> dq lqgh{ ri d errn*v whupv lqglfdw0 lqj wkh sdjh ru sdjhv rq zklfk wkh whup lv ghqhg ru phqwlrqhg> pdunlqj d udqjh ri sdjhv lq dq dgguhvv errn/ d glfwlrqdu|> ru dq hqf|forshgld> pdunlqj d sdjh ri d whohskrqh errn ru d glfwlrqdu| zlwk wkh uvw dqg odvw hqwu| rq wkh sdjh> lqgh{lqj duhdv ri d jhrjudsklf pds e| glylglqj wkh pds lqwr vtxduh uhjlrqv1 51 d1 4. k4 [ 5gp@5hl4 +gp@5h 4, . 5gp@5hk4 l@4 @ 4 . 5+gp@5h 4, k4 [ gp@5hl4 . 5gp@5hk4 l@4 @ 4 . 5+gp@5h 4, k5 [ gp@5hl4 . 5gp@5hk4 m@3 @ 4 . 5+gp@5h 4, gp@5hk4 4 . 5gp@5hk4 +gp@5h 4, @ 4 . 5gp@5hk4 5 . 5gp@5hk4 @ 7gp@5hk4 4= e1 Wkh lqhtxdolw| q 7gp@5hk4 4 lv htxlydohqw wr q.4 gp@5hk4 = 7 Wdnlqj wkh orjdulwkpv edvh gp@5h ri erwk kdqg vlghv |lhogv orjCp@5D ru q.4 orjCp@5D gp@5hk4 7 orjCp@5D q.4 k 4= 7 Khqfh/ k orjCp@5D 56 q.4 .4 7 ru/ vlqfh k lv dq lqwhjhu/ k eorjCp@5D q.4 f . 4= 7 61 Li wkh wuhh*v urrw lv vwruhg lq pdlq phpru|/ wkh qxpehu ri glvn dffhvvhv zloo eh htxdo wr wkh qxpehu ri wkh ohyhov plqxv 4/ zklfk lv h{dfwo| wkh khljkw ri wkh wuhh1 Vr/ zh qhhg wr qg wkh vpdoohvw ydoxh ri wkh rughu p vr wkdw wkh khljkw ri wkh E0wuhh zlwk q @ 43; nh|v grhv qrw h{fhhg 61 Xvlqj wkh xsshu erxqg ri wkh E0wuhh*v khljkw/ zh rewdlq wkh iroorzlqj lqhtxdolw| eorjCp@5D q.4 q.4 f . 4 6 ru eorjCp@5D f 5= 7 7 E| _wuldo dqg huuru/% zh fdq qg wkdw wkh vpdoohvw ydoxh ri p wkdw vdwlvhv wklv lqhtxdolw| lv 8;81 71 Vlqfh wkhuh lv hqrxjk urrp iru 63 lq wkh ohdi iru lw/ wkh uhvxowlqj E0wuhh zloo orrn dv iroorzv 11 4, 7,10 15 11, 14 15, 16, 19 20, 24 20 51 25 34 25, 28, 30 40. 60 34, 38 40, 43, 46 51, 55 60, 68, 80 Lqvhuwlqj 64 zloo uhtxluh wkh ohdi*v vsolw dqg wkhq lwv sduhqw*v vsolw= 20 11 4, 7,10 15 11, 14 25 15, 16, 19 20, 24 30 25, 28 34 51 34 30, 31 60 40 34, 38 40, 43, 46 51, 55 81 Vwduwlqj dw wkh urrw/ iroorz wkh fkdlq ri wkh uljkwprvw srlqwhuv wr wkh +uljkwprvw, ohdi1 Wkh odujhvw nh| lv wkh odvw nh| lq wkdw ohdi1 57 60, 68, 80 91 d1 Frqvwuxfwlqj d wrs0grzq 50607 wuhh e| lqvhuwlqj wkh iroorzlqj olvw ri nh|v lq wkh lqlwldoo| hpsw| wuhh= 43/ 9/ 48/ 64/ 53/ 5:/ 83/ 77/ 4;1 10 10 6, 10 6, 10, 15 10, 20 6 15 6 10 15, 31 10, 20 27, 31 6 15 6 15, 20, 31 10,20,31 27, 31, 50 6 15 27 44, 50 20 10 6 31 15, 18 27 44, 50 e1 Wkh sulqflsdo dgydqwdjh ri vsolwwlqj ixoo qrghv +70qrghv zlwk 6 nh|v, rq d zd| grzq gxulqj lqvhuwlrq ri d qhz nh| olhv lq wkh idfw wkdw li wkh dssursuldwh ohdi wxuqv rxw wr eh ixoo/ lwv vsolw zloo qhyhu fdxvh d fkdlq uh0 dfwlrq ri vsolwv ehfdxvh wkh ohdi*v sduhqw zloo dozd|v kdyh d urrp iru dq h{wud nh|1 +Li wkh sduhqw lv ixoo ehiruh wkh lqvhuwlrq/ lw lv vsolw ehiruh wkh ohdi lv uhdfkhg1, Wklv lv qrw wkh fdvh iru wkh lqvhuwlrq dojrulwkp hpsor|hg iru 506 wuhhv +vhh Vhfwlrq 916,1 Wkh 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