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Optimal memory minimization algorithms for the multifrontal method

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(1)Optimal memory minimization algorithms for the multifrontal method Abdou Guermouche, Jean-Yves l’Excellent. To cite this version: Abdou Guermouche, Jean-Yves l’Excellent. Optimal memory minimization algorithms for the multifrontal method. [Research Report] RR-5179, LIP RR-2204-26, INRIA, LIP. 2004. �inria-00071409�. HAL Id: inria-00071409 https://hal.inria.fr/inria-00071409 Submitted on 23 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Optimal memory minimization algorithms for the multifrontal method Abdou Guermouche (ENS Lyon) — Jean-Yves L’Excellent (INRIA). N° 5179 May 2004. ISSN 0249-6399. ISRN INRIA/RR--5179--FR+ENG. THÈME Num. apport de recherche.

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(73) lnm,|+ik„M®wz{¼€~†{m'9„†ˆ ~Uˆ m,xywyl:wyˆ€~U|+wz„†ˆ |+um'm ?y”B  Aâ½Ä¾ibwz}i wz{:~ |+u~Uˆk{+wy|+wy²£mcum'7pk},|+wz„†ˆ „Uƒ-|+ikm±l‡~U|+u+w´Ç°¢žu~U‚bi ~Uˆk9wz{#|+ikm—{l‡~Uxyxzm'{+|:k~U|~ {|+u+pk},|+pbumnum,‚bum'{m,ˆ¯|+wyˆb¢ bm,‚ m,ˆkbm,ˆk},wzm'{¼ m,|â¾^m'm,ˆ®|+ikm‡„†‚m,u~U|+wz„†ˆk{R· 8Eˆ¿‚bu~†},|+wz}'mž½¾³mnpk{m‡~ {|+u+pk},|+pbum }~Uxyxzm' #-#'     ½„†¼b|~Uwyˆkm' ¼¯›°lnm,u+¢žwyˆb¢Iˆk„¯bm'{n„Uƒ|+ikm m,xywyl:wyˆ€~U|+wz„†ˆ |+um'm ¾ik„ž{m:}'„†u+um'{+‚„†ˆk7wyˆb¢ }'„†xypbl:ˆk{0¼ m,xz„†ˆb¢ |„c|+ikmn{~Ulnm:{+pb‚ m,u+ˆk„Mbgm ?ÓB“ Aâ·:Ž {+pb‚ m,u+ˆk„Mbm#wz{~ }'„†ˆ¯|+wy¢žpk„†pk{u~Uˆb¢£m#„Uƒ³}'„†xypbl:ˆk{ < wyˆ.|+ikm(ƒe~†},|„†u-l‡~U|+u+w´/Ç =i€~'²Mwyˆb¢¦|+ikm#{~Ulnm(xz„¾^m,u7w ~U¢£„†ˆ€~Ux ˆk„†ˆbÅm,u„‡{+|+u+pk},|+pbumž· 8. T. 5 6 1. 2. 3. 4. X X. 5. 6. 5,6. X X X X. X. X X X X. X. X X X X. X X. 1 2 5. X X. X X X X X F X F X. 3 4 6 1,2. 3,4. wy¢žpbumn”ŠŽ l‡~U|+u+w´Ç@~Uˆkc|+ikm~†{{„M},w ~U|m'@~†{{m,l¼bxy›±|+um'mž·. X X X X X F X F X. :. h ikm:ƒ ~†},|„†u+wyÅQ~U|+wz„†ˆ9„Uƒ|+ikm:l‡~U|+u+w´Ç¿wz{g|+ikm,ˆIb„†ˆkm‡¼¶›º‚ m,u!ƒµ„†u+l:wyˆb¢ ~ {+pk}'}'m'{{wz„†ˆI„Uƒ‚€~Uu+|+w ~Ux ƒe~†},|„†u+wyÅQ~U|+wz„†ˆk{„Uƒ{l‡~UxyxĐbm,ˆk{m:l‡~U|+u+wz}'m'{}~Uxyxzm' %^

(74)  0

(75) T  # ½4¾ibwz}i°~Uum±~†{{„M},w ~U|m' |„c|+ikm#ˆk„Mbm'{0„Uƒ³|+ikm#|+um'm < ~†{gwyxyxypk{+|+u~U|m'ºwyˆ :wy¢žpbum ”=?·(hikm:„†ubm,u(„Uƒ~¦ƒµu„†ˆ¶|~UxŠl‡~U|+u+w´Ç wz{¢žwy²£m,ˆ®¼¯›º|+ikmnˆ¯pbl¼ m,u#„Uƒˆk„†ˆ7eÅm,u„ž{¼ m,xz„¾ |+ikm‡7w ~U¢£„†ˆ€~Ux^wyˆ®|+ikmnÁku{+|(}'„†xypbl:ˆ9„Uƒ|+ikm {pb‚m,u+ˆk„Mbm±~†{{„¯},w ~U|m'®¾wy|+i°|+ikm‡|+um'm¦ˆk„¯bmž· Ä~†}i9ƒ`u„†ˆ¯|~Ux³l‡~U|+u+w´Ç°wz{(7wy²Mwzbm'Iwyˆ¯|„ |â¾³„ ‚€~Uu+|{ :|+ikm %' '0 ¼bxz„¯}À½~Uxz{„.}~Uxyxzm' %+"  #-+ .2 ¼bxz„M}À ½Š¾ibwz}i9}'„†u+um'{‚„†ˆkb{|„.|+ikm ‚€~Uu+|³ƒe~†},|„†u+wyÅm'¦7pbu+wyˆb¢|+ikm-m,xywyl:wyˆ€~U|+wz„†ˆc‚bu„M}'m'{{ M~Uˆk‡|+ikm  

(76) /+ 

(77) 1 , "! ¾ibwz}ic}'„†u! um'{‚„†ˆkb{|„|+ikm{  p€~Uum-¼bxz„M}À¦pb‚k~U|m'c¾ikm,ˆ—‚bu„M}'m'{{+wyˆb¢#|+ikm-ƒµu„†ˆ¶|~Ux¸l‡~U|+u+w´Ç¸ · #-ˆk}'mv|+ikm ‚€~Uu+|+w ~Uxƒe~†},|„†u+wyÅQ~U|+wz„†ˆ wz{#b„†ˆkmž½³|+ikm±}'„†ˆ¯|+u+wy¼bpb|+wz„†ˆ ¼bxz„M}À®wz{#‚€~†{{m'°|„ |+ikmc‚€~Uum,ˆ¶|:ˆk„Mbmž· ­ ikm,ˆ‡}'„†ˆ¯|+u+wy¼bpb|+wz„†ˆk{ăµu„†l ~Uxyx€}ibwyxz7um,ˆc~Uum-~'²ž~Uwyx ~U¼bxzm„†ˆn|+ikm‚€~Uum,ˆ¶|½¯|+ikm,›#}~Uˆ‡¼m~†{{m,l# ¼bxzm' <     {+pbl:lnm'¿¾wy|+i°|+ikmn²ž~Uxypkm'{}'„†ˆ¯|~Uwyˆkm'®wyˆ®|+ikm:ƒµu„†ˆ¶|~Ux³l‡~U|+u+w´Ç®„Uƒ|+ikmn‚€~Uum,ˆ¶-| =?· hikm(m,xywyl:wyˆ€~U|+wz„†ˆ¿~Uxy¢£„†u+wy|+ibl wz{~:‚ „ž{+|„†ubm,u|+u~²£m,u{~Ux < ¾³m(b„‡ˆk„†|-‚bu„¯}'m'{{-‚€~Uum,ˆ¶|vˆk„Mbm'{ ¼ m?ƒ „†um:|+ikm,wyu}ibwyxz7um,ˆ =„Uƒ|+ikm¦~†{{m,l(¼bxy›š|+um'm ?y”,Ÿ Aâ· 8E|0pk{m'{|+ibum'm¦~Uum~†{„Uƒ{+|„†u~U¢£m‡wyˆ®~ }'„†ˆ¯|+wy¢žpk„†pk{vlnm,ln„†u+› {+‚€~†}'mž½€„†ˆkmgƒµ„†u-|+ikmƒ ~†},|„†u{Q½€„†ˆkmg|„¦{+|~†}À—|+ikm(}'„†ˆ¶|+u+wy¼bpb|+wz„†ˆš¼bxz„M}À7{½ å%$IiSå&.

(78) “. 

(79)  

(80)   

(81) 

(82)  

(83) !"$#&%'

(84) (!)*+" , %-.

(85) /0

(86) 1!)32. ~Uˆk9~Uˆk„†|+ikm,un„†ˆkm‡ƒµ„†u|+ikm¦},pbu+um,ˆ¯|#ƒ`u„†ˆ¯|~Uxl‡~U|+u+w´Çd?ӜBAâ· vpbu+wyˆb¢ |+ikm¦|+um'm±|+u~'²£m,u{~Uxe½|+ikm lnm,ln„†u+›¿{‚€~†}'m:um  pbwyum'I¼¶›š|+ikm:ƒe~†},|„†u{(~Uxy¾~'›7{¢žu„Q¾

(87) {¾ibwyxzm‡|+ikm‡{+|~†}Àšlnm,ln„†u+› < }'„†ˆ7 |~Uwyˆbwyˆb¢#|+ikmv}'„†ˆ¶|+u+wy¼bpb|+wz„†ˆ—¼bxz„M}À7{-=²†~Uu+wzm'{^bm,‚m,ˆk7wyˆb¢:„†ˆ±|+ikm-„†‚ m,u~U|+wz„†ˆk{l‡~†bmK¾ikm,ˆ±|+ikm ‚€~Uu+|+w ~UxOƒ ~†},|„†u+wyÅQ~U|+wz„†ˆ@„Uƒ~(ƒµu„†ˆ¶|~UxOl‡~U|+u+w´Ç—wz{‚bu„M}'m'{{m'¸½k~#}'„†ˆ¯|+u+wy¼bpb|+wz„†ˆ.¼bxz„M}À±wz{

(88) {+|~†}À£m' ¾ibwz}i.wyˆk},um~†{m'{|+ikm({+wyÅm„UƒŠ|+ikm{|~†}À „†ˆ.|+ikm„†|+ikm,uvi€~Uˆk¸½€¾ikm,ˆ.|+ikm0ƒµu„†ˆ¶|~Uxl‡~U|+u+w´Ç wz{ ƒ „†u+lnm'c~Uˆkc~†{{m,l¼bxzm'¸½M|+ikm}'„†ˆ¶|+u+wy¼bpb|+wz„†ˆ—¼bxz„M}À7{„UƒS|+ikm-}ibwyxz7um,ˆ—ˆk„¯bm'{~Uum-‚„†‚b‚ m'±„†pb| „Uƒ4|+ikmv{|~†}À¦~Uˆk±wy|{{+wyÅmvbm'},um~†{m'{·Šhikm{+|~†}À‡lnm,ln„†u+›‡wz{^|+i¯pk{²£m,u+›¦bm,‚ m,ˆkbm,ˆ¶|

(89) „†ˆ±|+ikm ~†{{m,l¼bxy›±|+um'm0|„†‚ „†xz„†¢ž›£·  _aUdS `F_ ZbZ_a`. L&ML. |„¿¼m—|+ikm lnm,ln„†u+› um pbwyum,lnm,ˆ¯|‡ƒ „†u swy²£m,ˆ ~šˆk„Mbm—wyˆ |+ikm@|+um'mž½³¾^m@bm?Ákˆkm |+ikm(ƒe~†},i|„†u{g„Uƒ i ½S~Uˆk cb |„c¼m(|+ikm:{+|„†u~U¢£fmactor ƒ „†uv|+ikm#}'„†ˆ¯|+u+wy¼bpb|+wz„†ˆI¼bxz„M}Àš„U ƒ i · store = um,‚bum'{m,ˆ¶|{-|+ikm0{+wyÅmg„UƒK|+ikm0}'„†l:‚bxzm,|m0ƒµu„†ˆ¶|~UxOl‡~U|+u+w´Ç „UƒKˆk„¯bm i · f actor + cb ­ m:~Uxz{„cpk{m SubT |„cbm,ˆk„†|m#|+ikm:{+pb¼b|+um'm#u„M„†|m'¿~U| i ½Owyˆk},xypk7wyˆb¢ i ½S~Uˆk f |„—bm,ˆk„†|m |+ikmg~Uln„†pbˆ¯|-„Uƒƒ ~†},|„†u{-}'„†u+um'{+‚ „†ˆk7wyˆb¢n|„ SubT ½7|+i€~U|

(90) wz{  i. i. i. i. i. i. i. i. fi =. X. f actorj. j∈SubTi. Sm,| A < um'{+‚m'},|+wy²£m,xy› T =¼ m|+ikmÄ{|„†u~U¢£mŠƒ „†u~†},|+wy²£m³lnm,ln„†u+› < um'{+‚m'},|+wy²£m,xy›~†},|+wy²£m³lnm,ln„†u+› ~Uˆk‡ƒe~†},|„†u{-=Šum pbwyum'±|„‚bu„M}'m'{{^|+ikmv{+pb¼b|+um'm u„¯„†|m'c~U| i ~Uˆk n ¼ m|+ikm-ˆ¶pbl(¼m,u „UƒŠ}ibwyxz7um,ˆ „Uƒ i ·Šhik„ž{mg}ibwyxz7um,ˆš~Uumgbm,ˆk„†|m'—SubT ¼¶› (c ) · i. i. i. i. i,j j=1,...,ni.  Zb  UdS/Ud_ ] V

(91)  b aS. L&M. ¯¹ wyˆk}'m¦|+ikm‡ƒ ~†},|„†u{#~Uum‡ˆk„†|#~†}'}'m'{{m'9~U¢¶~Uwyˆ9~=ƒµ|m,u|+ikm,›°~Uum‡}'„†l:‚bpb|m'¸½Š¾^m¦}~Uˆ9{+pb‚b‚ „ž{m |+i€~U|~¼€~†{+wz}-„†pb|!â„Uƒ`â}'„†um{}ikm,lnmv¾^„†pbxz—{+|„†um|+ikm,lr„†ˆc7wz{+À½M„†u|+i€~U|³|+ikmv{+›7{+|m,l ‚€~U¢žwyˆb¢ lnm'}i€~Uˆbwz{+l ¾wyxyxk‚bpb|Š|+ikm,l‘|„07wz{+À· 8Eˆ:{+pk}i‡}~†{m'{½žwy|Šwz{Kwyˆ¶|m,um'{|+wyˆb¢|„0}'„†ˆk{+wzbm,u|+ikm{|~†}À lnm,ln„†u+›c‚ m~UÀ· 8Eˆ¿|+ikm‡lpbxy|+w´ƒµu„†ˆ¶|~Uxlnm,|+ik„¯¸½|+ikm‡‚€~Uum,ˆ¶|ˆk„¯bm‡wz{gpk{+p€~Uxyxy›I~†},|+wy²ž~U|m'®~=ƒµ|m,u~Uxyx³wy|{(}ibwyxz {pb¼b|+um'm'{Äi€~²£m-¼ m'm,ˆ±‚bu„¯}'m'{{m'¦~Uˆk¦|+ikm-{+|„†u~U¢£m-um  pbwyum'‡|„~†{{m,l¼bxzmv|+ikmv}'„†ˆ¶|+u+wy¼bpb|+wz„†ˆk{ „UƒK|+ikm0}ibwyxz7um,ˆ.wyˆ¶|„‡|+ikm0‚€~Uum,ˆ¶|

(92) ˆk„Mbm i wz{ storei +. ni X j=1. æ. ii jk3l.m-n. cbi ..

(93)  G. + 

(94) +1!). 

(95)        . kpbu+|+ikm,u+ln„†umž½£¾ikm,ˆ#‚bu„¯}'m'{{+wyˆb¢~-{+pb¼b|+um'm^u„M„†|m'#~U|Š~-}ibwyxz c ½†|+ikm³}'„†ˆ¯|+u+wy¼bpb|+wz„†ˆn¼bxz„M}À7{ „Uƒ~Uxyx¸‚bum,²Mwz„†pk{+xy›c‚bu„¯}'m'{{m'—}ibwyxz7um,ˆ@i€~²£mg|„#¼ m0{+|„†um'¸½bxzm~†7wyˆb¢n|„¦~#{+|„†u~U¢£mgm  p€~UxO|„ :. i,j. Aci,j +. j−1 X. cbci,k .. h ikm,um?ƒ „†umž½7|+ikmvl‡~=ÇMwyl(pbl {+|„†u~U¢£mvum  pbwyum'c|„(‚bu„M}'m'{{|+ikm}'„†l:‚bxzm,|mg{+pb¼b|+um'mvu„M„†|m' ~U| ˆk„Mbm i wz{¢žwy²£m,ˆ@¼¶›  X X < ” = cb ), store + cb ), A = max( max (A + ~†{{m'm,ˆ—wyˆ ?y”B Aâ· < ov„†|m|+i€~U|ƒ „†uxzm~=ƒˆk„Mbm'{

(96) ¾wy|+i—ˆk„n}ibwyxz7um,ˆO½ A = store · = ov„†|m^|+i€~U|{„†lnmwyl:‚bxzm,lnm,ˆ¶|~U|+wz„†ˆk{^„Uƒ|+ikml(pbxy|+w´ƒ`u„†ˆ¯|~Ux~Uxy¢£„†u+wy|+ibl ~Uxyxz„Q¾ |+ikm}'„†ˆ¶|+u+wy¼bpb|+wz„†ˆ „Uƒ

(97) |+ikm±x ~†{+|#}ibwyxz°|„.¼m±~†{{m,l(¼bxzm'®wyˆ7e‚bx ~†}'m±wyˆ¯|„.|+ikm±‚€~Uum,ˆ¯|ˆk„Mbmž½Kwyˆ°¾ibwz}i }~†{m¦|+ikm }'„†ˆ¯|+u+wy¼bpb|+wz„†ˆ®¼bxz„M}À.ƒµu„†l |+ikm#x ~†{+|g}ibwyxzIwz{0ˆk„†|}'„†pbˆ¯|m'ºwyˆ¿|+ikm:x ~†{+|~†{{m,l¼bxy›º{+|m,‚O· 8Eˆ |+i€~U|-}~†{mže½ :k„†u+lpbx ~ < ” =^¼ m'}'„†lnm'{  X X <œ = A = max( max (A + cb ), store + cb ), k=1. j−1. i. j=1,ni. ni. ci,k. ci,j. i. ci,j. j=1. k=1. i. j−1. 0 i. j=1,ni. ci,j. i. ni −1. ci,k. i. k=1. ci,j. j=1. P _ Zb X U S U _a] V b S. L&M. :€„†ug|+ikm:wyˆ7â}'„†um‡}~†{mž½O¾³m±~Uum:wyˆ¶|m,um'{|m'¿wyˆ¿|+ikm:‚ m~UÀº„Uƒ^|„†|~UxÄlnm,ln„†u+›£· ^„†l:‚€~Uum'®|„ |+ikmlnm,ln„†u+› pk{~U¢£mƒµ„†u-|+ikm~†},|+wy²£m < {+|~†}Àe=

(98) lnm,ln„†u+›£½¾³m(|~UÀ£m|+ikm(lnm,ln„†u+› pk{m' ƒµ„†u-|+ikm {|„†u~U¢£m-„UƒO|+ikmƒ ~†},|„†u{^wyˆ¯|„#~†}'}'„†pbˆ¯|{+wyˆk}'m|+ikm,›‡lpk{+|^¼mÀ£m,‚b|wyˆ‡lnm,ln„†u+›±~=ƒµ|m,u^|+ikm,›‡~Uum }'„†l:‚bpb|m'¸· v{+wyˆb¢:|+ikmg{~Ulnm0ˆk„†|~U|+wz„†ˆk{~†{~U¼ „²£mž½€|+ikm~Uln„†pbˆ¶|„Uƒ|„†|~UxSlnm,ln„†u+›—ˆkm'm'bm' |„#‚bu„M}'m'{{~:ˆk„¯bm i < ~Uˆk—wy|{{+pb¼b|+um'm =wz{|+ikm,ˆ ¢žwy²£m,ˆ@¼¶›  . Ti = max( max (Tci,j + j=1,ni. L&M . . j−1 X. (cbci,k + fci,k )), storei +. k=1. ni X. (cbci,j + fci,j )).  . < =. j=1. X _ ] [0Z QFU  Z_ fTS )] Sb BS@Z QFS UdS/Ud_ ] V

(99)  b aS.  Z [aS UdS/Ud_ ] V b S. h ikm@|+um'm@|+u~²£m,u{~Uxi€~†{±~I{+|+u„†ˆb¢°wyl:‚€~†},|±„†ˆ |+ikm@‚ m~UÀ „Uƒglnm,ln„†u+›£· :k„†ucm?Çk~Ul:‚bxzmž½~ bm'm,‚cpbˆ¯¼€~Ux ~Uˆk}'m'c|+um'm-¾wyxyx xzm~†±|„:~g¼ m,|+|m,ulnm,ln„†u+›‡pk{~U¢£m-w´ƒS‚bu„M}'m'{{m'±pk{+wyˆb¢:~bm,‚b|+i7 Áku{|^‚ „ž{+|„†ubm,u|+u~²£m,u{~Ux4~Uˆk±|+ikmˆ¶pbl(¼m,u„UƒK{+wylpbxy|~Uˆkm'„†pk{-}'„†ˆ¶|+u+wy¼bpb|+wz„†ˆ.¼bxz„¯}ÀM{¾wyxyx¸¼ m {l‡~Uxyxzm,uÄw´ƒO|+ikm

(100) |+u~²£m,u{~Ux¸{+|~Uu+|{ă`u„†l |+ikmbm'm,‚ m'{+|Äxzm~²£m'{·hibwz{³wz{wyxyxypk{+|+u~U|m'±wyˆ :wy¢žpbumœ7· å%$IiSå&.

(101) •. 

(102)  

(103)   

(104) 

(105)  

(106) !"$#&%'

(107) (!)*+" , %-.

(108) /0

(109) 1!)32. Best. Worst. :wy¢žpbumœ &8El:‚ „†u+|~Uˆk}'mg„Uƒ|+ikm|+um'mg|+u~'²£m,u{~Uxe·³hikm|+um'm0wz{‚bu„M}'m'{{m'cpk{+wyˆb¢‡~:bm,‚b|+i7 Áku{| ‚ „ž{+|„†ubm,u

(110) |+u~²£m,u{~UxS{+|~Uu+|+wyˆb¢nƒ`u„†l |+ikmxzm?ƒµ|!ei€~UˆkMâ{+wzbmž· 8Eˆ ?y”œ Aâ½b¸wyp@{pb¢ž¢£m'{+|{

(111) ~Uˆ „†‚b|+wyl‡~UxO|+um'm0|+u~²£m,u{~Ux¸ƒ „†u

(112) m,xywyl:wyˆ€~U|+wz„†ˆ@|+um'm'{¾ikm,ˆ—|+ikm~†{{m,l# ¼bxy› „Uƒ|+ikm(x ~†{+|}ibwyxzš}~Uˆ ¼m(b„†ˆkmwyˆ7e‚bx ~†}'mž·hibwz{~Uxy¢£„†u+wy|+ibl }~Uˆšm~†{+wyxy› ¼ mg¢£m,ˆkm,u~UxywyÅm' |„~†{{m,l(¼bxy›:|+um'm'{^~Uˆk¦}'„†ˆk{+wz{+|{½M~U|³m~†}i±xzm,²£m,x„UƒO|+ikm|+um'mž½¶wyˆ±{„†u+|+wyˆb¢|+ikm-}ibwyxz7um,ˆcˆk„Mbm'{ wyˆ—bm'},um~†{+wyˆb¢:„†ubm,u

(113) „Uƒ max(A , store ) − cb ~Uˆk±|+ikm,ˆcpk{m0~(bm,‚b|+i7 Áku{+|‚„ž{|„†ubm,u |+u~²£m,u{~Ux³„Uƒ|+ikm:um'„†ubm,um'®|+um'mž·‡hibwz{um'{+pbxy|gwz{0¼€~†{m'I„†ˆ¿|+ikm:|+ikm'„†um,l ¼ m,xz„¾g½4‚bu„²£m' wyˆ ?y”

(114) œ Aâ½k~Uˆk—|+i€~U|

(115) ¾³mg¾wyxyxOum?ƒµm,u|„¦~†{

(116) ¸wypOÂÃ{|+ikm'„†um,l wyˆ—|+ikmg„†|+ikm,u

(117) ‚€~Uu+|{„UƒK|+ibwz{‚€~U‚ m,u   y§^¨  É¯Í © É     E#'1 ^% B  +-# (x , y ), i = 1..n  !)F 

(118) B

(119) +ec^% max x + ci,j. ci,j. #c 1 0

(120) .2  (#'

(121) 1"$!)c#'

(122) 1+e1 i  . yi  j=1  x1 − y 1 ≥ x 2 − y 2 ≥ . . . ≥ x n − y n. Pi−1. #. i. i. (xi , yi ). g2  1  . #"

(123) ^2" ^%. xi − yi.  i=1,n !) . vo „¾ w´ƒ|+ikm—~†{{m,l(¼bxy›I„Uƒ|+ikm¦x ~†{+|#}ibwyxz9}~Uˆbˆk„†|:¼m¦b„†ˆkm±wyˆ7e‚bx ~†}'mž½c?y” A{+ik„¾r|+i€~U||+ikm }ibwyxz7um,ˆ {+ik„†pbxz |+ikm,ˆ ¼ m—{„†u+|m' wyˆ bm'},um~†{+wyˆb¢°„†ubm,u¦„Uƒ A − cb ·^›°b„†wyˆb¢°{„b½ |+ikm(²ž~Uxypkm(„Uƒ A ~†{-bm?Ákˆkm' ¼¯› < ”=ƒ „†uv|+ikm(}'„†l:‚bxzm,|m|+um'mu„¯„†|m'š~U|vwy|{u„M„†|-ˆk„Mbm(wz{ l:wyˆbwyl:wyÅm'¸F· 8∱|+ikmum'{|„UƒO|+ibwz{‚€~U‚ m,u^¾³mv¾wyxyx pk{m|+ikm-|m,u+ln{ B

(124) 

(125) /T^%  +  #   -!  |„#um?ƒ m,u

(126) |„#|+ibwz{-„†ubm,u· Cv„¾^m,²£m,u½Om'{+‚ m'},w ~Uxyxy› ƒ „†u²£m,u+›@¾wzbm#|+um'm'{ < ¾ikm,um P l‡~›@¼ mx ~Uu+¢£m}'„†l:‚€~Uum' |„ store =?½ A }~Uˆ®{+|+wyxyxŠ¼m#x ~Uu+¢£mž· :k„†ug|+i€~U|gum~†{„†ˆO½4¸cbwyp®~Uxz{„—m?Ç7‚ m,u+wylnm,ˆ¶|{gwyˆ ?y”Bœ A^~ {|+u~U|m,¢ž›.}'„†ˆk{+wz{+|+wyˆb¢—wyˆš‚bum?E~Uxyxz„M}~U|+wyˆb¢ |+ikm:{+|+u+pk},|+pbum:„Uƒ^|+ikm#‚€~Uum,ˆ¶|0¼ m?ƒµ„†um#}ibwyxz7um,ˆ°~Uum ƒ „†u+lnm'¸· Ä~†}i±}'„†ˆ¯|+u+wy¼bpb|+wz„†ˆ—¼bxz„¯}À#ƒ`u„†lrm~†}i±ˆkm,¾ }ibwyxz‡wz{³|+ikm,ˆ—~†{{m,l¼bxzm'±7wyum'},|+xy›nwyˆ¯|„ |+ikm³{+|+u+pk},|+pbumĄUƒk|+ikmÀ~Uum,ˆ¶|½†|+i¯pk{~²£„†wz7wyˆb¢0~‚„†|m,ˆ¯|+w ~Uxyxy›gx ~Uu+¢£m³}'„†xyxzm'},|+wz„†ˆ#„Uƒk}'„†ˆ¶|+u+wy¼bpb|+wz„†ˆ ¼bxz„M}À7{-wyˆ {+|~†}À—lnm,ln„†u+›£·-hikm{+|„†u~U¢£m„†¼b|~Uwyˆkm'.ƒµ„†u-|+ibwz{~U‚b‚bu„£~†}išwz{|+ikm,ˆ um'},pbu{+wy²£m,xy› bm?Ákˆkm' ¼¶›  < "Ÿ = A = store + max (A ) ci,j. root. j=ni j=1. i. root. i. æ. ii jk3l.m-n. i. j=1,ni. ci,j. ci,j. ci,j. i.

(127)  h. + 

(128) +1!). 

(129)        . -Ž ¢¶~UwyˆO½4wy|~U‚b‚ m~Uum'¿|+i€~U||+ibwz{{+|+u~U|m,¢ž›º¾~†{~Uxz{„@{„†lnm,¾i€~U|7wz{~U‚b‚„†wyˆ¯|+wyˆb¢k½K¼m'}~Upk{m±~ }i€~Uwyˆ:„Uƒk‚€~Uum,ˆ¶|Šˆk„Mbm'{ < ~U|Km~†}i#xzm,²£m,xb„Uƒk|+ikmÄ|+um'm=Si€~†(|„¼ m^{+|„†um'¸½†‚„ž{{+wy¼bxy›~Uxz{„-xzm~†7wyˆb¢ |„n~n{+wy¢žˆbw´Á€}~Uˆ¶|-lnm,ln„†u+›±pk{~U¢£mž· ­ m#{+ik„†pbxz ˆk„†|+wz}'m#ikm,um|+i€~U|u~U|+ikm,u0|+i€~Uˆš‚bum?E~Uxyxz„¯}~U|+wyˆb¢ |+ikm‚€~Uum,ˆ¯|½¸|+ikm#{}ikm,lnmƒµu„†l Owyp®}~Uˆ¿¼ mn{+xywy¢ži¯|+xy›ºwyl:‚bu„²£m'®¼¶›¿~Uxyxz„¯}~U|+wyˆb¢ |+ikm:‚€~Uum,ˆ¯|‹!pk{+|~=ƒµ|m,u|+ikm#Áku{+|(}ibwyxz®i€~†{ ¼ m'm,ˆ°‚bu„¯}'m'{{m'¸· v{+wyˆb¢@|+ikm¦ˆk„†|~U|+wz„†ˆk{n~U¼„Q²£mž½K|+ikm‡lnm,ln„†u+›®pk{~U¢£m¦wz{|+ikm,ˆ°um'},pbu{+wy²£m,xy› bm?Ákˆkm' ¼¶›  <“ = A = max(A , store + cb , store + max (A )) w´ƒK|+ikmg~†{{m,l(¼bxy›cwyˆ¶|„:|+ikm0‚€~Uum,ˆ¯|wz{ˆk„†|‚ m,u!ƒ „†u+lnm'—wyˆ7e‚bx ~†}'mž½~Uˆk . i. ci,1. i. ci,1. i. j=2,ni. ci,j. <G =. Ai = max(Aci,1 , storei + max (Aci,j )). w´ƒK|+ikmg~†{{m,l(¼bxy›cwyˆ¶|„:|+ikm0‚€~Uum,ˆ¯|-}~Uˆ ¼ m0b„†ˆkm0wyˆ7e‚bx ~†}'mž· ov„†|m|+i€~U|ļ „†|+i#ƒ „†u+lpbx ~†{ < “ =Š~Uˆk < G =ˆkm'}'m'{{~Uu+wyxy›(xzm~†:|„(~²†~Uxypkm„Uƒ A {+l‡~Uxyxzm,uĄ†um  p€~Ux |„v|+i€~U|„Uƒ :k„†u+lpbx ~†{ < "Ÿ ="†ƒµpbu+|+ikm,u+ln„†um|+ikm,›(¼ „†|+i#‚bu„²Mwzbm~Uˆ:„†‚b|+wyl‡~Uxk²†~Uxypkm„Uƒ|+ikm

(130) ~†},|+wy²£m lnm,ln„†u+› A ¾ikm,ˆ:|+ikm³}ibwyxz#¾wy|+i|+ikm^x ~Uu+¢£m'{+|K‚ m~UÀgwz{|+um~U|m'(Áku{+| < |+ikm^„†ubm,uKwyˆ|+ikm^|m,u+l i€~†{Kˆk„vwyˆ kpkm,ˆk}'m

(131) ~Uˆk(ln„Q²¯wyˆb¢0|+ikm³Áku{+|Š}ibwyxz#¾wy|+ix ~Uu+¢£m'{|‚ m~UÀ max |„ |+i€~U|{m,(A|}~UˆI+store „†ˆbxy›šwyˆk)},um~†{m‡|+ikm:‚ m~UeÀ =?· 8∺|+ikmnƒµ„†xyxz„Q¾wyˆb¢k½S|+ikm'{m‡|â¾^„@{+|+u~U|m,¢žwzm'{¾wyxyx ¼ mum?ƒµm,u+um'¿|„ ~†{ 

(132)  1 ) O 2 e

(133) .1/D 3 B 

(134)  ~Uˆk 

(135) / 1 ) O.2 )

(136)  / ' B 

(137)     ,   ½bum'{+‚ m'},|+wy²£m,xy›£· j=2,ni. i. i. j=2,ni. ci,j. i. P _ Zb X U S U _a] V b S. ¹¯wyl:wyx ~Uu+xy›:|„g|+ikm

(138) }~†{m

(139) „UƒO~†},|+wy²£m

(140) lnm,ln„†u+›#‚ m~UÀ½¯{„†u+|+wyˆb¢g|+ikm

(141) }ibwyxz7um,ˆ¦ˆk„Mbm'{wyˆ‡bm'},um~†{wyˆb¢ †„ ubm,uK„Uƒ T −(cb +f ) ¢žwy²£m'{~Uˆ„†‚b|+wyl‡~UxM|+um'mÄ|+u~'²£m,u{~Ux7wyˆ(|m,u+ln{4„Uƒkl:wyˆbwyl:wyÅQ~U|+wz„†ˆ:„Uƒ |+ikm|„†|~Ux€lnm,ln„†u+›:‚ m~UÀ:bm?Ákˆkm'‡¼¯› <   =?·Šhibwz{Äwz{Äm?Ç7‚bx ~Uwyˆkm'‡wyˆ¦ln„†umbm,|~Uwyxz{³wyˆ ?Ӗ A⽶¾ikm,um m?Ç7‚ m,u+wylnm,ˆ¶|~Uxum'{+pbxy|{{+ik„Q¾&|+i€~U|-|+ibwz{ < „†‚b|+wyl‡~Ux=„†ubm,u¼bu+wyˆb¢£{v{+xywy¢ži¶|-¢¶~Uwyˆk{-„Q²£m,u|+ikm(„†ˆkm ƒµu„†l|+ikm‡‚bum,²Mwz„†pk{‚€~Uu~U¢žu~U‚bi < A − cb =0¾ikm,ˆ°|+ikm±|„†|~Ux³lnm,ln„†u+›®wz{(}'„†ˆk{+wzbm,um'¸· < 8âƒg|+ikm.¢žu~U‚bi „Uƒ|+ikm.l‡~U|+u+w´Ç wz{¦ˆk„†|±}'„†ˆbˆkm'},|m' < um'7pk},wy¼bxzm l‡~U|+u+w´Ç/=?½

(142) „†ˆkmš{+ik„†pbxz ~Uxz{„ „†ubm,u|+ikmvu„M„†|^ˆk„Mbm'{}'„†u+um'{+‚ „†ˆk7wyˆb¢:|„|+ikmv7 w ¸m,um,ˆ¶||+um'm'{wyˆ—bm'},um~†{+wyˆb¢n„†ubm,u

(143) „Uƒ4|+ikm,wyu ½¶~†{Š|+ibwz{¾wyxyx€bm'},um~†{m|+ikm|„†|~Uxblnm,ln„†u+› max T + P f ž|+ibwz{Šwz{³~7wyum'},| T −f }'„†ˆk{m  pkm,ˆk}'m(„UƒÄ¸wypOÂÃ{|+ikm'„†um,l },wy|m'@m~Uu+xywzm,u· =g8∠|+ikmgƒµ„†xyxz„Q¾wyˆb¢k½¾³m}~Uxyx4|+ibwz{v~Uxy¢£„†u+wy|+ibl ' , #-#- 

(144) a0 0

(145)  

(146)   

(147)  · :wyˆ€~Uxyxy›£½³wyˆ |+ikm—}~†{m—„Uƒ-|+ikmc|„†|~Uxlnm,ln„†u+›®wy|:wz{n~Uxz{„š‚ „ž{{+wy¼bxzmc|„¿~†},|+wy²ž~U|mc|+ikmc‚€~Uum,ˆ¶| ˆk„Mbm-~=ƒµ|m,uÄ|+ikmÁku{+|^}ibwyxz¸½Mxzm~†7wyˆb¢(|„|+ikm /' ) O.2D)

(148)  /Y ' B 

(149)  ~Uˆk

(150) / 1 ) O.2 e

(151) .1/ '  T     ²†~Uu+w ~Uˆ¯|{½€~Uˆk@~lnm,ln„†u+›cpk{~U¢£mm  p€~Ux¸|„ ci,j. ci,j. ci,j. ci,j. i. i. ci,j. i∈roots. i. i−1 j=1 j. å%$IiSå&.

(152) –. 

(153)  

(154)   

(155) 

(156)  

(157) !"$#&%'

(158) (!)*+" , %-.

(159) /0

(160) 1!)32. Ti = max(Tci,1 , fci,1 + cbci,1 + storei , storei + max (Tci,j +. j−1 X. fci,k )). w´ƒ|+ikm‡~†{{m,l¼bxy›º„Uƒ^|+ikm‡}'„†ˆ¶|+u+wy¼bpb|+wz„†ˆ®ƒ`u„†l |+ikm#Áku{|}ibwyxzIwyˆ¶|„—|+ikm:‚€~Uum,ˆ¯|wz{0ˆk„†|(b„†ˆkm wyˆ7e‚bx ~†}'mž½~Uˆk j=2,ni. Ti = max(Tci,1 , fci,1 + storei , storei + max (Tci,j +. k=1. j−1 X. fci,k )). w´ƒ|+ikm:~†{{m,l¼bxy›@„UƒÄ|+ikm}'„†ˆ¶|+u+wy¼bpb|+wz„†ˆ¿ƒ`u„†l |+ikm(Áku{+|v}ibwyxz wyˆ¶|„±|+ikm‚€~Uum,ˆ¯|g}~Uˆ.¼ mb„†ˆkm wyˆ7e‚bx ~†}'mž· :€„†u³|+ibwz{~Uˆ¶|+wz},wy‚€~U|m'c‚€~Uum,ˆ¯|~†},|+wy²ž~U|+wz„†ˆO½7¼„†|+i‡ƒµ„†u³|+ikmwyˆ7e‚bx ~†}'m~Uˆk¦ˆk„†ˆ¦wyˆ7e‚bx ~†}'mv}~†{mž½Mwy| l‡~UÀ£m'{³{m,ˆk{m|„g|+u+›|„„†ubm,u|+ikm}ibwyxz7um,ˆ‡ˆk„Mbm'{wyˆ‡bm'},um~†{+wyˆb¢„†ubm,u³„Uƒ |+ikm,wyu T − f ~†{

(161) |+ibwz{¾wyxyx4~U|xzm~†{+|

(162) l:wyˆbwyl:wyÅm|+ikm|m,u+l max (T + P f ) ·

(163) $€ 

(164) %@"( c 6S

(165) *   ®

(166) 

(167) ¦€Š 

(168) 

(169) 

(170)  Š- Žv{nm?Ç7‚bx ~Uwyˆkm' wyˆ |+ikm—‚bum,²Mwz„†pk{‡{m'},|+wz„†ˆO½Äˆkm,wy|+ikm,u¦|+ikm {+›7{+|m,l‡~U|+wz}—‚bum?E~Uxyxz„M}~U|+wz„†ˆ „Uƒ|+ikm ‚€~Uum,ˆ¯|nˆk„Mbmž½ˆk„†u:wy|{n~Uxyxz„M}~U|+wz„†ˆ ~=ƒ`|m,u:|+ikmcx ~†{+|:}ibwyxz < },x ~†{{+wz}~Uxl(pbxy|+w´ƒ`u„†ˆ¯|~Uxlnm,|+ik„M/ = ‚bu„Q²¯wzbm ~Uˆ „†‚b|+wyl‡~Ux-lnm,ln„†u+› pk{~U¢£mž· C-„Q¾³m,²£m,u±wy|nwz{n‚ „ž{{+wy¼bxzm—|„°~†},|+wy²†~U|m@|+ikm—‚€~Uum,ˆ¶| ˆk„Mbmg~U|-~Uˆ@~Uu+¼bwy|+u~Uu+›c‚ „ž{+wy|+wz„†ˆO½€~=ƒµ|m,u-~(Áku{+|

(171) {m,|

(172) „UƒŠ}ibwyxz@ˆk„¯bm'{i€~†{

(173) ¼m'm,ˆ |+um~U|m'¸· j=2,ni. k=1. ci,j. j. . . ci,j. ci,j. j−1 k=1 ci,k. . ... S1. ... S2 P. wy¢žpbum  K Çk~Ul:‚bxzmg„Uƒ~‚€~Uum,ˆ¶|ˆk„Mbm~Uˆkcwy|{

(174) }ibwyxz7um,ˆO· :. swy²£m,ˆ&~I‚€~Uum,ˆ¯| ˆk„Mbmš~Uˆk wy|{—{m,| „Uƒ}ibwyxz7um,ˆ&ˆk„¯bm'{cwyˆ|+ikmº~†{{m,l(¼bxy› +| um'mž½v¾^mšpk{m |+ikm‡ƒ „†xyxz„¾wyˆb¢šˆk„†|~U|+wz„†ˆk{ n{+pb‚b‚ „ž{+wyˆb¢ |+i€~U|(|+ikm¦‚€~Uum,ˆ¶|#ˆk„Mbm‡wz{#~†},|+wy²ž~U|m' < U~ xyxz„M}~U|m'°wyˆ æ. ii jk3l.m-n.

(175) ”. + 

(176) +1!). . 

(177)        . lnm,ln„†u+›=ċ!pk{+|~=ƒµ|m,u0}ibwyxz j i€~†{¼ m'm,ˆš‚bu„¯}'m'{{m'¸½ ¾³m:bm?Ákˆkm p = j · ­ m#~Uxz{„±bm?Ákˆkm S ~†{

(178) |+ikm{m,|

(179) „Uƒ}ibwyxz7um,ˆ ˆk„Mbm'{|+um~U|m' ¼ m?ƒµ„†umg|+ikm(~†},|+wy²ž~U|+wz„†ˆ.„UƒK|+ikmg‚€~Uum,ˆ¶|-ˆk„¯bm(~Uˆk S ~†{|+ikm:{m,|g„Uƒ^}ibwyxz7um,ˆIˆk„¯bm'{0|+um~U|m'¿~=ƒµ|m,u0|+ikmn~†},|+wy²ž~U|+wz„†ˆI„Uƒ³|+ikm#‚€~Uum,ˆ¯|gˆk„¯bmž·n¹M„c¾^m }'„†ˆk{wzbm,u~‚€~Uum,ˆ¯|4ˆk„Mbm i ~Uˆkwy|{S}ibwyxz7um,ˆ (c ) = S ~Uˆk (c ) ~†{ {ik„¾ˆ#wyˆ :4wy¢žpbum

(180)  7·hikm{+|„†u~U¢£m^ˆkm'm'bm'#|„v‚bu„¯}'m'{{K~v}ibwyxz#ˆk„¯bm c wz{ A · m?ƒ =„†uSm^|+ikm ~Uxyxz„M}~U|+wz„†ˆ#„Uƒk|+ikmÀ~Uum,ˆ¯|ˆk„Mbmž½†|+ikm³‚m~UÀg„Uƒ€{+|„†u~U¢£m³wz{„†¼b|~Uwyˆkm'(¼¯›(~U‚b‚bxy›Mwyˆb¢ :k„†u+l(pbx ~ < ” = < ¹Mm'},|+wz„†ˆ&œ7·Ó  =‡|„°|+ikmš}ibwyxz7um,ˆ ˆk„Mbm'{—„Uƒ · :bpbu+|+ikm,u+ln„†umž½|+ikmº~Uln„†pbˆ¯|@„Uƒlnm,ln„†u+› ˆkm'm'bm'š|„c‚bu„M}'m'{{|+ikm:}ibwyxz7um,ˆIˆk„Mbm'{0„Uƒ SS wz{g„†¼b|~Uwyˆkm'º¼¯› ~U‚b‚bxy›Mwyˆb¢ :k„†u+lpbx ~ < “ =-|„ |+ik„ž{m0}ibwyxz7um,ˆ < ~†{{pbl:wyˆb¢n|+i€~U||+ikmg~†{{m,l(¼bxywzm'{

(181) wyˆ¶|„n|+ikm‚€~Uum,ˆ¶|-~Uumˆk„†|b„†ˆkmwyˆ7e‚bx ~†}'m =?· hi¯pk{½b|+ikm~Uln„†pbˆ¯|-„Uƒlnm,ln„†u+›±ˆkm'm'bm'@|„#‚bu„¯}'m'{{|+ikmg{+pb¼b|+um'm0u„M„†|m'@~U| i wz{. 1. 2. i,j j=1,...,p. i,j j=p+1,...,ni. 1. i,j. 2. ci,j. 1. 2. Ai = max(max (Aci,j + j=1,p. j−1 X. cbci,k ),. k=1 p X. storei +. •. < = cbci,j ,. j=1. ÄÉ ¿ ª  !)(

(182) .2 *.%  A.  

(183) . storei + max (Aci,j )) j=p+1,ni. !)  !  ,2. 32 # . S2. 2 #   !

(184)   (!eYe.. !. ^%( 3 .  ^3.%  #-¼¶²Mwz„†pk{{wyˆk}'m b„Mm'{ˆk„†|bm,‚ m,ˆkº„†ˆ¿|+ikm:„†ubm,u„Uƒ (A )) |+ikm0}ibwyxz7um,ˆ.wyˆ S · store + max ÄÉ. ¿ª    1 j  R !  ,2 2    

(185) "

(186) "H0 S  % A  ##    !

(187)  max(A )  i. i. j=p+1,ni. ci,j. 2. !e. j.    $  .2R0. S2. ! 

(188) +)T 1  . #"R!) e ! 1. . j. i∈S2. i.  ^3.% -Œm,ln„Q²¯wyˆb¢~vˆk„¯bmƒµu„†l b„Mm'{Šˆk„†|wyˆk},um~†{m|+ikm

(189) ‚m~UÀwyˆ:|+ibwz{Ä{m,|·I:bpbu+|+ikm,u+ln„†um {wyˆk}'m A wz{

(190) {+l‡~Uxyxzm,u

(191) |+i€~Uˆ max(AS ) ½7|+ikmg‚m~UÀ±wyˆ S ¾wyxyxOˆk„†|

(192) wyˆk},um~†{mž·.  É¯Í © É   1 i    2   2g 1 c . . . c  # !  2

(193) .1  2 -#   - 

(194) 1. j. i∈S2. i. 2. ' 

(195) ( 

(196)  e !#( 1 0

(197) .2   -  

(198) i,1"R!)i,n T%1i

(199)    

(200)  

(201) ! 

(202) . Í ©¦ª . ÎÍ j = 1, . . . , n ¹¯|~Uu+|

(203) ¾wy|+i |+ikm0}ibwyxz7um,ˆ †„ ubm,um' wyˆ bm'},um~†{+wyˆb¢±„†ubm,u„Uƒ A  vm?Ákˆkm S = (c ) U~ ˆk@~U‚b‚bxy›c|+ikm²†~Uu+w ~Uˆ¯|-„UƒŠ¸wypOÂÃ{-~Uxy¢£„†u+wy|+ibl |„num'„†ubm,u < bm'},um~†{+wyˆb¢  = A − cb i. ci,k. 1. i,k k=1,...,j. ci,k. S1. ci,k. å%$IiSå&.

(204) ”ž”. 

(205)  

(206)   

(207) 

(208)  

(209) !"$#&%'

(210) (!)*+" , %-.

(211) /0

(212) 1!)32. vm?Ákˆkm S = (c )  ^„†l:‚bpb|m02 |+ikm0²ži,k~Uxypkk=j+1,...,n m0„Uƒ A < ƒµ„†u+l(pbx ~ < •=.= . wz{

(213) {+l‡~Uxyxzm,u

(214) |+i€~Uˆ ~Uˆ¶› A ‚bum,²Mwz„†pk{+xy›—„†¼b|~Uwyˆkm' ¨  É ¹Mm,| ~Uˆk—À£m'm,‚@|+ikmg}'„†u+um'{+‚ „†ˆk7wyˆb¢‡„†ubm,u„UƒŠ}ibwyxz7um,ˆ@ˆk„Mbm'{  É Î  p = j É Î Í ©  ^3.% -Om,| ¼ m~Uˆn„†‚b|+wyl‡~Ux„†ubm,uĄUƒ|+ikm}ibwyxz7um,ˆ¦„Uƒ ¾ikm,um wz{~†},|+wy²†~U|m'¦~=ƒ`|m,uŠ|+ikm }ibwyxzci€~†{¼m'σm,ˆ±‚bu„M}'m'{{m' < wyˆc|+ikm0„†ubm,u^¢žwy²£m,ˆ—¼¯› σ =?·Ši hikmv‚m,iu+l(pb|~U|+wz„†ˆ σ „†¼b|~Uwyˆkm'cp¼¶› {„†u+|+wyˆb¢c~Uxyx4|+ikm(}ibwyxz@ˆk„Mbm'{ j ¼ m,xz„†ˆb¢žwyˆb¢¦|„ S wyˆ bm'{}'m,ˆk7wyˆb¢c„†ubm,uv„Uƒ|+ikm,wyuvum'{+‚ m'},|+wy²£m wz{v{+|+wyxyxŠ„†‚b|+wyl‡~Ux < Sm,l:l‡~š” =?· 8⃠∃k ∈ S {„¦|+i€~U| A ≤ max(A ) ½|+ikm‚ m,u+lpb|~U|+wz„†ˆ A „†¼b|~Uwyˆkm'°¼¶›ºln„Q²¯wyˆb¢ |„ wz{{+|+wyxyx³„†‚b|+wyl‡~Ux < Om,l:l‡~ œ =?· hi¯pk{½K|+ikm,um¦m?Ç7wz{+|{~Uˆ σ „†‚b|+wyl‡~UxĂm,u+l(pb|~U|+wz„†ˆ σ k{pk}iºS|+i€~U| min(A ) > max(A ) „†¼b|~Uwyˆkm'¿¼¶› um,‚m~U|+wyˆb¢ |+ikm ‚bum,²Mwz„†pk{

(215) „†‚ m,u~U|+wz„†ˆO· Žv{g~c}'„†ˆk{m  pkm,ˆk}'mž½4~Uˆ¿„†‚b|+wyl‡~Ux‚ m,u+lpb|~U|+wz„†ˆ°„Uƒ^|+ikm:}ibwyxz7um,ˆIˆk„Mbm'{0}~Uˆ¿¼ m}'„†l:‚bpb|m' ¼¯› {„†u+|+wyˆb¢—|+ikm,l wyˆ¿bm'{}'m,ˆk7wyˆb¢@„†ubm,ug„Uƒ|+ikm,wyu0um'{+‚ m'},|+wy²£m A ~Uˆk¿bm,|m,u+l:wyˆkm:|+ikm#¼ m'{+| ‚ „ž{+wy|+wz„†ˆ(|„v~†},|+wy²ž~U|mÄ|+ikm³‚€~Uum,ˆ¶| ·hibwz{4}~Uˆ¼m³b„†ˆkmļ¯›g|+u+›Mwyˆb¢v~UxyxM|+ikmĂ „ž{{+wy¼bxzmĂ„ž{+wy|+wz„†ˆk{ ƒ „†u0|+ikmn~†},|+wy²ž~U|+wz„†ˆI„Uƒ^|+ikm#‚€~Uum,ˆ¯p|(~Uˆk¿{m,xzm'},|+wyˆb¢@|+ikm#l:wyˆbwyl‡~UxĄ†ˆkmž·:o-„†|m#|+i€~U|gƒµ„†ugm~†}i wy|m,u~U|+wz„†ˆ±¾³m-um'}'„†l:‚bpb|mv|+ikm

(216) ¼ m'{+|Ą†ubm,u„†ˆ ·Khikmvbm,|m,u+l:wyˆ€~U|+wz„†ˆc„UƒO|+ikm¼m'{|‚„ž{wy|+wz„†ˆ ƒ „†u|+ikm~†},|+wy²ž~U|+wz„†ˆ@„UƒK|+ikm0‚€~Uum,ˆ¶|

(217) wz{

(218) |+ikm,ˆ@b„†Sˆkm0wyˆ.~l‡~=Ç7wylpbl „Uƒ n {+|m,‚k{· ­ m^{+ik„†pbxz(um,l‡~Uu+Àgikm,um³|+i€~U|4ƒµu„†l ~Uˆ(wyl:‚bxzm,lnm,ˆ¯|~U|+wz„†ˆ:‚„†wyˆ¯|K„Uƒk²Mwzm,¾g½Uwy|4wz{ˆk„†|ˆkm'}'m'{{~Uu+› |„ {„†u+||+ikm±ˆk„¯bm'{wyˆ S ~U|#m~†}i9wy|m,u~U|+wz„†ˆO· 8Eˆ¿ƒe~†},|½Š|+ikm‡ƒµ„†xyxz„Q¾wyˆb¢I~Uxy¢£„†u+wy|+ibl }~Uˆ9¼ m ~U‚b‚bxywzm'¸½¾ikm,um ~I‚bum?â}'„†l:‚bpb|m' ‚ m,u+lpb|~U|+wz„†ˆ ~Uxyxz„¾

(219) {c|„°{m,xzm'},|cm 7‡},wzm,ˆ¶|+xy› |+ikm.ˆk„Mbm'{ ¾wy|+i.{+l‡~Uxyxzm'{+| A wyˆ S  ¹Mm,| S = (c ) ½ S = ∅ ~Uˆk p = n  ¹M„†u+| wyˆ bm'},um~†{+wyˆb¢¦„†ubm,u„Uƒ A − cb ~Uˆk }'„†l:‚bpb|m A pk{+wyˆb¢ :k„†u+l(pbx ~ < • =  © É ÄÉ Sª7¨ :4wyˆk {pk}i |+i€~U|  ¹Mm,| Sc = S \ c ½ SA = S= ∪minc ½k~Uˆk Ap = p − 1  ^„†l:‚bpb|m A   A ≤ A ¨  É 0m'm,‚ |+ikm0²ž~Uxypkm0„Uƒ ½ ~Uˆk S ~Uˆk@{m,| A = A  p S É Î   ¨   p == 1 „†u A > A hikmÄ|m,u+l:wyˆ€~U|+wz„†ˆ:}'„†ˆk7wy|+wz„†ˆ wz{4m?Ç7‚bx ~Uwyˆkm'¼¯›0|+ikmƒe~†},|S|+i€~U|¾ikm,ˆ|+ikm³¢žxz„†¼€~UxM‚m~UÀ wyˆk},um~†{m'{½€|+ibwz{vwz{-ˆkm'}'m'{{~Uu+wyxyA› ¼ >m'}~UApk{m(„Uƒ^~:|m,u+l }'„†u+um'{+‚ „†ˆk7wyˆb¢±|„ S ·v¹¯wyˆk}'mm,xzm,lnm,ˆ¶|{ . i. i. Ai. i. th. 0. 2. j. k. 1. 00. j. j∈S2. 2. n. k∈S1. k. k. k∈S2. j. 1. i. 1. ci,k. 1. i,k k=1,...,ni. 1. i. 2. ci,k. 1. i,j. ci,j. 1. 0 i. 1 0 i. i,j. ci,k ∈S1. 2. 2. ci,k. i. ci,k. i,j. i. 1. 0 i. 2. i. 0 i. i. 0 i. i. 2. æ. ii jk3l.m-n.

(220) ”œ. . + 

(221) +1!). 

(222)       . ¾wyxyx„†ˆbxy›š¼ mn~†bbm'º|„ S ½¸|+ikm:‚ m~UÀš}~Uˆ¿„†ˆbxy› wyˆk},um~†{m‡~Uˆkšwy|gwz{0ˆk„†|0¾^„†u+|+iI|+u+›Mwyˆb¢ |„ ~†b ln„†umgm,xzm,lnm,ˆ¶|{|„ · :wyˆ€~Uxyxy›£½€~U‚b‚bxy›Mwyˆb¢n|+ikm0‚bSum,²Mwz„†pk{|+ikm'„†um,l „†ˆ—|+ikmg}'„†l:‚bxzm,|mg|+um'm0xzm~†b{

(223) |„nŽxy¢£„†u+wy|+ibl»”ž· ¤  Í ©  ¨  #-‚b|+wyl‡~UxO|+um'mgum'„†ubm,u+wyˆb¢‡|„#l:wyˆbwyl:wyÅmg|+ikmg‚m~UÀc„UƒK{|~†}À±lnm,ln„†u+›£· . © É¶É ¦É¯Í ©UÎ É © < T = ¬ #É  Í ©‡ª   i wyˆ@|+ikm0{m,|„Uƒ4u„M„†|

(224) ˆk„Mbm'{ Î Í ‰Šu„¯}'m'{{ ³ibwyxz < i=  É Î Í © Î © Í « É §§   `Î < i= ¬ #É   wz{~#xzm~=ƒ ¨  É i  A store É `§ É Í © j = 1 |„ n Î Í ‰Šu„¯}'m'{{ ³ibwyxz < c =  É Î Í © vm,|m,u+l:wyˆkm|+ikmv‚ „ž{+wy|+wz„†ˆ p ¾ikm,um-|+ikm‚€~Uum,ˆ¯|

(225) {+ik„†pbxz±¼m0~†},|+wy²ž~U|m'—~Uˆkc|+ikmv„†ubm,u „UƒK}ibwyxz7um,ˆ@pk{+wyˆb¢‡hikm'„†um,l»”  ^„†l:‚bpb|m A pk{+wyˆb¢ :k„†u+lpbx ~ < • =  É Î  Î 2. 2. . . . . . . . . . i. i. i. . i,j. i. .  

(226)   

(227)  (   8Eˆ |+ibwz{‡{m'},|+wz„†ˆ ¾³m@~Uum wyˆ¶|m,um'{+|m' wyˆ |+ikm—l:wyˆbwyl:wyÅQ~U|+wz„†ˆ „Uƒv|+ikm—|„†|~Uxlnm,ln„†u+›£½¾ikm,um ¼ „†|+i |+ikm{|~†}À ~Uˆk.|+ikm(ƒ ~†},|„†u{0~Uum|~UÀ£m,ˆšwyˆ¶|„ ~†}'}'„†pbˆ¯|·(8∠|+ikm#},x ~†{{+wz}~Uxlpbxy|+w´ƒµu„†ˆ¶|~Ux lnm,|+ik„M¸½¯|+ikm

(228) |„†|~Ux€lnm,ln„†u+›:wz{Ä¢žwy²£m,ˆ¦¼¯› <   =?½7~U‚b‚bxywzm'n|„g|+ikmu„¯„†|³ˆk„¯bm„UƒO|+ikm|+um'mv~Uˆk‡wy| wz{‚„ž{{wy¼bxzm|„gbm,|m,u+l:wyˆkm-~Uˆ¦„†‚b|+wyl‡~Ux€|+um'm|+u~²£m,u{~Ux€wyˆn|+i€~U|^}'„†ˆ¶|m?Ç7|&· C-„Q¾³m,²£m,u½M{+wyl:wyx ~Uu+xy› |„.¾i€~U|i€~†{(¼ m'm,ˆ9b„†ˆkm‡wyˆ ¹Mm'},|+wz„†ˆ  —ƒ „†u(|+ikm±{+|~†}À < ~†},|+wy²£m =lnm,ln„†u+›£½Š¾³mc}~Uˆ9bm'},wzbm |„.~†},|+wy²†~U|m‡|+ikm:‚€~Uum,ˆ¶|ˆk„¯bm¦~U|~Uˆ®~Uu+¼bwy|+u~Uu+›º‚ „ž{+wy|+wz„†ˆO½4¼m?ƒ „†um‡~UxyxÄ}ibwyxz7um,ˆ®i€~²£m‡¼ m'm,ˆ ‚bu„M}'m'{{m'¸· Sm,| i ¼ m:~nˆk„Mbmwyˆ |+ikm:~†{{m,l¼bxy›@|+um'mž· ­ mpk{m|+ikm#{~Ulnm#bm?Ákˆbwy|+wz„†ˆ®~†{v¼ m?ƒµ„†um(ƒ „†u-|+ikm {m,|{ S ½ S ~Uˆk p ·Šhikm‚ m~UÀ±„UƒK{+|„†u~U¢£mvƒ „†u S ½7wyˆk},xypk7wyˆb¢n|+ikmg~Uxyxz„¯}~U|+wz„†ˆ.„Uƒ4|+ikm‚€~Uum,ˆ¶|. 1. 2. 1. å%$IiSå&. .

(229) ” . 

(230)  

(231)   

(232) 

(233)  

(234) !"$#&%'

(235) (!)*+" , %-.

(236) /0

(237) 1!)32. ˆk„Mbmž½Owz{(„†¼b|~Uwyˆkm'¿¼¯›I~U‚b‚bxy›¯wyˆb¢ €: „†u+lpbx ~ <   =|„—|+ikmn}ibwyxz7um,ˆ®ˆk„Mbm'{g¼ m,xz„†ˆb¢žwyˆb¢@|„ |+ibwz{ {m,|  P1 = max(max (Tci,j + j=1,p. j−1 X. (cbci,k + fci,k )),. k=1 p X. <h =. (cbci,j + fci,j )). storei +. j=1. kpbu+|+ikm,u+ln„†umž½k|+ikm~Uln„†pbˆ¯|-„UƒKlnm,ln„†u+›±ˆkm'm'bm' |„:‚bu„¯}'m'{{|+ikm0}ibwyxz7um,ˆ.ˆk„¯bm'{-„Uƒ S wz{  :. 2. P2 = storei +. p X. fci,j + max (Tci,j + j=p+1,ni. j−1 X. –. < =. fci,k ). Eˆkbm'm'¸½7¾ikm,ˆ—|+um~U|+wyˆb¢‡~(ˆk„¯bmž½7|+ikmvlnm,ln„†u+›±¾wyxyxO}'„†ˆ¶|~Uwyˆ—|+ikmvƒe~†},|„†u{}'„†u+um'{+‚ „†ˆk7wyˆb¢n|„ ~UxyxÄ~Uxyum~†7› ‚bu„M}'m'{{m' ¼bu„†|+ikm,u{· 8Eˆ |+ikmƒ „†u+lpbx ~†{(~U¼„Q²£mž½Oˆk„†|mn~Uxz{„c|+i€~U| T wyˆk},xypkbm'{ {„#|+i€~U|

(238) |+ikmvƒe~†},|„†u{ƒ „†u|+ikmx ~†{+|}ibwyxz@~Uumgm ¸m'},|+wy²£m,xy›c|~UÀ£m,ˆ wyˆ¶|„‡~†}'}'„†pbˆ¯|· f :wyˆ€~Uxyxy›£½ |+ikm:~Uln„†pbˆ¶|g„Uƒ³lnm,ln„†u+›@ˆkm'm'bm'š|„±‚bu„¯}'m'{{-|+ikm#{+pb¼b|+um'mu„M„†|m'º~U|-|+ikm‚€~Uum,ˆ¶| ˆk„Mbm i wz{  < ” = T = max(P , P ) ÄÉ. ¿ª  +3 ) #' ! (!e$) #

(239)  p 0 ' B O !) )

(240)  /%$!) #'1$^% !  ,2

(241)   j=1. k=p+1. 8. ci,j. ci,j. i. 1. 2. 32 -#   2 !) #'1F.% !" 2

(242)    2 -# 

(243)  c 1   !) I#1" !) !  ,2. S2

(244)  2 !)!  ,2

(245)  H 32"-#> 32 -#> S1  2" 1 . #-"g

(246) .2  ^% S1 Tci,j − (cbci,j + fci,j ) 2  1. #-"g

(247) .2 *^%  ^  32"-#*

(248)  - 

(249) "e. ! .%  

(250) 

(251) 

(252)  2S  2 S1 

(253) . ! ^%> 

(254) T ci,j

(255)  − fc i,j S2.  ^3.% D:k„†u ½^{m'm—|+ikm@m,ˆk „Uƒ(¹Mm'},|+wz„†ˆ œ7·Ó“7· :€„†u ½^|+ibwz{:um'{+pbxy|{:ƒµu„†l |+ikm—|+ikm'„†um,l ƒµu„†l Owyp@¾ibSwz}iš{+|~U|m'{|+i€~U| x + P y wz{

(256) l:wyˆbwyl‡S~Ux¾ikm,ˆ@|+ikm < x ½ y =

(257) ~Uum{„†u+|m'.wyˆ bm'},um~†{wyˆb¢¦„†ubm,u-„Uƒ x − y ·Ähikm|+ikm'„†um,l wz{-~U‚b‚bxywzm'.„†ˆ :k„†u+l(pbx ~ < – =^¾wy|+i x = T ~Uˆk y = f · ÄÉ. ¿ª  + )

(258) #'$! T!e(   P  #( 10  2E%'

(259)  !  2 j  p + 1 ≤ j ≤ n     !) 

(260) .2 # >#+3 ) #'$!   !)  *2"T  P = P (j ) . 1. 2. k−1 j=1 j. k. k. k. ci,k. k. 2. æ. k. ci,k. 2. ii jk3l.m-n. k. k. 2. 0. 0. 0. i.

(261) ”,Ÿ. + 

(262) +1!). . P2 (j0 ) = storei +. p X. jX 0 −1. fci,j + Tci,j0 +. j=1. . < =. fci,k + Tci,j0. k=1. !e c

(263)   

(264)  I

(265) +". 

(266)  #+!H!  (S1 , S2 ) ! F #> 

(267) " 

(268) $

(269) 1+)  0R!) B

(270) +e. ci,j0 . P2 (j0 ). S2. jX 0 −1. . ”ž”. fci,k. k=p+1. = storei +. . 

(271)       .   ," # 0 . ! # $e. ! '% 

(272) !) #31. S2.  ^3.%  < w =&8âƒO¾³m-ln„²£mvˆk„Mbm'{ ƒ`u„†l |„ |+i€~U|~Uum¼ m?ƒµ„†um < |+i€~U|^wz{½ =?½ |+ikm-{m'}'„†ˆk¦xywyˆkmwyˆ < ”ž”=Š¾wyxyxˆkc„†|^}i€~Uˆb¢£mžS· < wyw=T8âSƒ¸¾³m-ln„²£m-ˆk„¯bm'{ c c ƒµu„†l S |„ j S<|+ji€~U| =?½¯|+ikm{m'}'„†ˆk±xywyˆkm-wyˆ < ”ž”=¾wyxyxwyˆk},um~†{m < {+wyˆk}'m ~†bb{ ~Uum~=ƒµ|m,u < |+i€~U|wz{½ f pb‚—|„n|+ikmgc{+pbl =?½k~Uˆk—|+i¯jpk{

(273) >|+ikjm‚ m~UÀ±„†ˆ S · ÄÉ. ¿ª     #31 S Ye#'1" R  / c 0 S 

(274)  2  1 . #3 !) . ! 

(275)  $ %  2"T  P 0  !) e. ! ^%00

(276) I 

(277) g ' +e2

(278) "g!)

(279)    

(280)  S i,j1. 2. i,j0. 1. i,j1. i,j0. 1. 1. 2. 0. 1. ci,j1. 0. 2. . 1 0 1. .%(!) )

(281)  1/Y+ #". . ^3.%. ¹Mm'm>:k„†u+l(pbx ~. S10 = S1 ∪ ci,j0. ?·. <h =.  ¯É Í © É    cb.c i,k!. . . (! . P10 ≥ P1. i,j0. . 1.  >

(282) 2  W #'' .%  32 #  ! 1!

(283) . 'O #11# i,k )k=1,ni i,k 

(284)  $ # 2  F.2 . - 1  !)Y%'

(285)  ,  (c "R

(286)  

(287) !   . 32 -#  -  T aci,k 00

(288) f c. %1(!) )

(289)  /Y 32  . . Ti. i. ¹Mm,| S = ∅ ½ S = (c ) ~Uˆk  ¹M„†u+| S ~†}'}'„†u7wyˆb¢n|„nOm,l:l‡~¦  p = 0 ^„†l:‚bpb|m T = P ~†}'}'„†u7wyˆb¢‡|„ :k„†u+l(pbx ~ < – =  © É ÄÉ ª7¨ :4wyˆk {+pk}i |+i€~U| P = store + P f + T ∈S c ¹Mm,| S = S ∪ c ½ S = S \ c ½€~Uˆk p = p + 1  ¹M„†u+||+ikm0ˆk„¯bm'{

(290) wyˆ ~Uˆk ~†}'}'„†u7wyˆb¢‡|„:Om,l:l‡~¦  ^„†l:‚bpb|m P ½ P ½k~USˆk T =S max(P , P )  ¨  É  T ≤T 0m'm,‚ |+ikm0²ž~Uxypkm'{

(291) „Uƒ ½ ~Uˆk S ~Uˆk {m,| T = T  p S É Î   ¨   p = n „†u P ≥ P 1. i,k k=1,...,ni. 2. 2. i. 2. i,j0. 2. 1. 1. i,j0. 2. 2. 1. 1. 0 i. j0 k=1 ci,k. i. 2. ci,j0. k„†u+lpbx ~ < ž” ”= = . <:. i,j0. 2. 0 i. 2. 1. 2. i. 1. i. 1. 2. i. 0 i. 2. å%$IiSå&.

(292) ”“. 

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(300) 1!)32.  ^3.%  ­ m^Áku{+|um,l‡~Uu+À#|+i€~U||+ikm„†ubm,uwyˆ ~Uˆk wz{Šwyl:‚ „ž{m'#¼¶›:Om,l:l‡~( 7·K¹¯|~Uu+|+wyˆb¢ ƒµu„†l ~n}'„†ˆ7Ák¢žpbu~U|+wz„†ˆ ¾ikm,um P > P ½€wy|uSm'{+pbxy|{ƒ`Su„†l Om,l:l‡~‡Ÿ‡|+i€~U||+ikm(„†ˆbxy›—¾^~› |„ bm'},um~†{m:|+ikmn‚ m~UÀ.wz{¼¶›šln„²Mwyˆb¢ c ƒ`u„†l S |„ S ·±hi¶pk{Q½~U|m~†}iIwy|m,u~U|+wz„†ˆ°m,wy|+ikm,u ¾^mvi€~'²£m0„†¼b|~Uwyˆkm'±|+ikm„†‚b|+wyl‡~Ux¸‚m~UÀ T ½M„†u|+ikmv{„†xypb|+wz„†ˆ—¾wy|+ic|+ikm-„†‚b|+wyl‡~Ux¸‚ m~UÀnwz{{+pk}i |+i€~U| c < ¾ibwz}i®¾~†{um'{+‚ „†ˆk{+wy¼bxzm¦„Uƒ|+ikm‡‚m~UÀšwyˆ S =v¼m,xz„†ˆb¢£{(|„ S ·@¹¯wyˆk}'m‡¾³m±{+|~Uu+| ¾wy|+i S = ∅ ½¾³m(~Uum{+pbum|„num~†}i |+ikm„†‚b|+wyl‡~Ux}'„†ˆ7Ák¢žpbu~U|+wz„†ˆ¿~=ƒµ|m,uv~:l‡~=Ç7wylpbl „Uƒ n wy|m,u~U|+wz„†ˆk{Q· < Ž|m~†}i@wy|m,u~U|+wz„†ˆO½€Om,l:l‡~¦ wz{~U‚b‚bxywzm'¸· = :€„†uS|+ikm|m,u+l:wyˆ€~U|+wz„†ˆ:},u+wy|m,u+wz„†ˆO½†¾^mÀMˆk„¾°|+i€~U|S|+ikmĄ†‚b|+wyl‡~UxM‚ m~UÀvi€~†{4¼ m'm,ˆ(„†¼b|~Uwyˆkm'(¾ikm,ˆ ¼ m'}'„†lnm'{x ~Uu+¢£m,u„†um p€~UxÄ|+i€~Uˆ ½{+wyˆk}'m:wyˆI|+i€~U|(}~†{m:|+ikmnlnm,ln„†u+›º‚m~UÀ T = P P ¾wyxyx4„†ˆbxy›¦wyˆk},um~†{mgw´ƒ|+ikm ~Uxy¢£„†u+wy|+ibl Pwz{‚bpbu{+pkm'cƒµpbu+|+ikm,u < Om,l:l‡~¦“ =?· :wyˆ€~Uxyxy›£½£{wyl:wyx ~Uu+xy›(|„0Ž-xy¢£„†u+wy|+ibl ”ž½¶~U‚b‚bxy›Mwyˆb¢0|+ikm‚bum,²¯wz„†pk{Š|+ikm'„†um,l‘|„-|+ikm}'„†l:‚bxzm,|m

(301) |+um'm xzm~†b{

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