Memory-based scheduling for a parallel multifrontal solver
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Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Memory-based scheduling for a parallel multifrontal solver Abdou Guermouche (ENS Lyon) — Jean-Yves L’Excellent (INRIA). N° 5162 April 2004. ISSN 0249-6399. ISRN INRIA/RR--5162--FR+ENG. THÈME 1. apport de recherche.
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(14) }mon- wTmo aiNgº±}on?ylzL v wTn-gn¿moTn-¥4»wf|!v¼ vgwTil wTx½£©n-|g,im³wTya|¼vg|gyl¸¥À!!io»|!| n%wT Niav a©¾»¦yln-n%xv º Á¿C~¼Â,° ÃÅÄÆÇ!ÈÉ =Ê¢ ± wT!v!n¬mqwT|gg©%n%vPºam³ya|g¸¥À!»| wTmon-|gil±Ëºv!!iln%7yaalºmon-mog}. ÕÙëWÕÙÎÐÏ×æÌÒÍTìxÎÐÏRí?í¤Ñdî ÒEÓPïÀÑRð-ð!ÎÐÏ4ñ-ñ-Ô¤ñÕÐÏ×ò_Ö.ó!ôTÔ!õxؤö Ô¤÷¤ÎÐÕÙø%Ô¤ù!Ú=ô ÕÐÒ]òåú-Ô û%Ï0ð ü=Ô]ý!ÛdþÒ+ÿÏ×ÒgÔ!ÛdÜEÍÛdÒWÝÖ¤Û×Ñ4Ö¤ÞaÑdÍTҹ߱ԤÚÖ¤Û¯Ô!ÑdÖ?ÎàÛdÒ¼á=ÒÕâ ãåä=Þ[Ö¤ÛdæÔ ÑdÎÙç%èTÒáPè¬éaÔ Û¯Ô Õàê. Unité de recherche INRIA Rhône-Alpes 655, avenue de l’Europe, 38334 Montbonnot Saint Ismier (France) Téléphone : +33 4 76 61 52 00 — Télécopie +33 4 76 61 52 52.
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(56) | aT!T¥ L¥%wTn%v!NwTv!|g|!¿avivg®dyaw §av|gwo!»n%| n%±vx° 0n- TwT¥|g!©nxlw7vxJ°}Å:E| awvg|gª7wTvwT}lºw µ|g]g©n%lv{%n-|!g9a!ailIL|gv!iln nÅvgaw%¦} n|gv iln-n©n% a-|g!©L% n%v!vgv!|g !wTv{|!n-©n%}v!v+®d¥À©wa|¯}Ln%¨ nÅ|giwTl±a|!n%v!vxn-¸º¼¥gºnxw½!i|gi W_mqv!wmov!|!nCn- X awT!»±ya%n%wTv gv!|E} |g|g%ililln¬nv!mqv!|g w%wwT¦v!n%»|!vn-|!xvx©°°Jv:Ew{vjw§awT7wTl|g©%n%N(ºL|gwilvn¿ ,v!n-v!©v!n%-§a|g©©n
(57) {½©|gv¼ia|gillnn¿½]vgyl!gi{ª±©|giwwT(|]T|g¥4iln¿|gilwTn¬mo%yag!±n%|jvg T¥Nl½]7gaªoo| w¦vgn-ª{b|! Master (P0). Master (P0). Slave 1 (P2). Slave 2(P1). Slave 1 (P2) Slave 2(P1) Slave 3 (P3). Unsymmetric. Symmetric. Oya!n$4Zh¹}± nflLan%v§a©± ª±aq½|giÅ|giln¬an?¥×wTya|
(58) vg|g wT|!n-}°. jh g!iln-n(ya§a©wT±¹ ª±¥Àa¹¶|gilylnv!vgn%}L£m:¥Àmoon-|g|ggil©.n(x|¯w}Lv! ,n$nWÀv ´n%n lO4Lan%yavq!n"©vq4[!X?n-ºqyawT!qa¥×n-¼o|!¬|gil§nCwTyawTllvg%}Lnm:|gilmonn-½¼|gg©(gª:x§wn-v!n|¯½¼wTn%ln- v>!w%n ¦nfvgilvgPya½¼§an%| wv!ªL vx° ?pe4BA0|giwT|¿|giln w-|g¦n mon-mog}ŧn-iw%¦L©ya¬T¥KCEDECF-Gq©v¿l|
(59) ¦n-g}(±L¥× wTwT m:wT©
(60) ©n-v!]!n?ilz7n%n%7-yaya|ga©lvgvP|g° wThj|!i±n-yl©vfn%v¼½¼|!nqf am:!n% av!n-!P»¦|n.|gil|gn.ilmon:n-lmon?z7|³g}ov!n%§,-n-|gi©wxl¦±v©v!ya]moTn:¥Ë|gmoiln.n-mov!g¦}±n-®Exw%;° ½]> wT!nnb·l7!}±vg|® ã$# `Rã&%.
(61)
(62) "!#$&%'(#)*+,- ./ ! 0.1#2 3. amo!n-n%±v n-|!v»| |!:w{|gmoilnn-movg|g gwT}±|!®dn-§w}v!° n%O4 vgwTw%¦}nºlv!½¼n-nf©n%-|g¦©nIwvgl|g n-wT½&|!n-7}L}°wThjm:il©n-¬I| w½¼vgªnqav! n%ilv!n%-7gya§,nqav!qmo|gin:wT|¿m:wT amo!xv¿¦wTn?|® m: a!P¦±a|gilnmon-mog}b§,n-iwx¦±©ya¿T9¥ CDEC-FGa° , . !¹/ !R* ! a*j* " :E |gia©vfv!n%-|g© ½]n:an%v!-g§now{ln-½ mon-mog}»®d§wv n% v!!iln%7yaawT a a!w!iI|giwT| wTmov³wT| !wTn%7ylg-|gaiaCm |gil|gnoiwTvg|{= <xilnon-T l¥v{|gilw no av!!| ±w%!ªn%v!v moqn-mo|!Ig}´ il7±yav gnÅa(|!v{|gil%noT®dn?½¼zan%g-ªyan-|g!©vbN° :¯vg|¬wx©¦v n%wv{¥_½ya|g}´i 7mo}Ln-mowTm:g©} %nxwl i´v!|g awT!±»%|!n%vxv °v!hj¿iav ©v|gwTiwT|g|g|gilianm xwT!n%¾±aya! an%!vfL%7n%¥×v!v!gmq wTW_|gmq©w vg|!wTn-§,¬Tya¥¼||E}±|gil non:moon-lmoLaYng}X
(63) xLwT% -ya! ilwTL|g©v!n ¶T|!v¥ v!w%¦n%v.w%%!7a|!:|giln-mon-mog}b©wË°J:E|©v¦n-ŧn-©x½ Z ÉÄ å n-mog}»®d§wv n%Åvgw%¦n³v!n-©n%-|g©(vg|g wT|!n-}° ! a n T 0 ¥ g | l i ¬ n a y g ! n » | À ¥ ! » | T w 4 q m T w g | g ¸ z ÂLg||gilnj ,|!n-±|gwT vgwx¦nj a!±%n%v!v !v:!x½a³!an-¼T¥ |giln-mon-mog}±%-ya wT|g© ·llI|gilnq§an%vg| º4±yam³§n-:T¥ a!±%n%v!v !vxºv!Å|giwT| ©v vgÉmq wT©n-|gi|!wT Å|gilÊ nfÉ vgya+¥dw%n¬T¥0|giln¥_!±| wTNmqwT|gg¸z Ä 4Ê ¦n É !x½v
(64) T¥|gilnfmqwT|gg¸zb|! a!L%n%v!v! »wÄv!yavgm³ R§ n-|gilnT¥ !n-mqvgw%wT¦an%v aq !P½v.n%¾±ya| wT§a}b|!:nxw!i( a!L%n%v!v! °° 4Ê h|!jil!nilwTLv!ng|g|gilianm vgmq ilwT±©n%v v!n%|
(65) v:v!n-vg|¿w%¦Tnb¥ a a!!±L%%n%n%v!v!v v!!!v.vo|gwi%wT%|¿!7©a§´wT|!}Å §|gwTiln-wTl%mon%v¿n-mo|giln³g}¶mon-©mowËg° }(:E½||g|!iln-layav| v!lw%-¦!n%nxv¬wv!|¬a·l!|gv!il|fnv!-yag|!gv¬!n-|g»il|¿no ,nx|gwTilªn-º !v¬wv¿Ivgilx!½x½a^CO!ayan-!n³Tl¥]° |g>«iln-ilfn- mow:n-mo a!Lg}%n%©v!v!wË°oiwhjv
(66) il|!n-q il|±7©v v+n ® |g|ggiln
(67) §ayamo|!n-n%mov¼|gilgn}!wTPl½v¼ aT!¥Ën%|gv!iln-gn¦¥_n%!v]±|gil| wTn
(68) -mqyagwT!|gn-g±¸zq|j ,nxwT|gª ilnW_¸ a¥R! L%v!n%vgv!v!§a©nY!vX?°0bhjilwn ½]p wx
(69) }o®d|g§ai©wT±|¹ ª±§a|giq a§!wTL7wTyll%%n%n% v §±v!i}wT|g ,iln{nfwwTv³ gO4|giam)ya!¥ÀLn 4(§n?z7|g%in- a|gil| n¬|givg}LwTm:|mo|giln-nq|gg»©fyamwTl§,bn-:yaTl¥vg}±!m:x½mov n-|g9g©³nxwx!wi¶v!n% vjwT½g|gN|g©iw%9¦n½|gil¼n³§v n:wTmo¥dwTn mo!nfg!n-yawTx° . . .
(70). . nf ront . i X. i. (M EM [i] − M EM [j]). j=1. .
(71). j=1 i (M EM [i] − M EM [j])/nf ront .
(72). . j. .
(73). `9` #ä acbed(fg. . 1 i. .
(74) S. c =
(75) EV = .. Current memory peak P0 P0 slave selection. P1. P2. P3. Oya!n¬cZn-mog}±®d§wv!n%(vgwx¦nfv!n-©n%-|g©(v!|g wT|!n-}°. ×¥©
(76) vmo©x½g!|gnfaia³%m m:wTp(g a|!©©vn?v{z½¼an½n%vg½|giÅlmon%%«l!¥×%n¬n-q»g|g!§ n-wT|g|!yai«n
(77) wT|goil§an|g©ilLvgn!}Lª7m:yavxl°movg}Ln-m:|gg©mo n-wT|ggl©«xyawlv!vgn}Lvgm:lmo%nn-|g|ggil©n vgx}±wm:v!n%movx° n-|gg:¯©«.|gillnn ¿L%-|!yan wT|gi|g©wT|bo|gTil¥,nC|gmoiln-n%¹ i awT!a±©%vgn%m v!v yl!v!v0n%©¨v0|g|!il¶n]§a!§7gwn%a-|xwTvg¥|bw
(78) ¥×7¥Àg|ggilmq%wTm:|g©a ¬!|!n-n%!wTia|ga¦©nxwT|! ¶!n-|g ,ilng|xmo°;:En-mo|giag©} v mohji±n%yl ivxwTº4a|gil©vgnomź|gnxilwn-!!iIv³x awT!L%wn%%v!v!-ya¬m§aya!wTw|!nqax|gwilv!no|!vfl|g-il!n:n-mo¦wTn-g»|!wTvx|gº©|gIiwTT|¥jx|!wTv¬Imo§,n-nmo gvg} |g½¦ilnon- fl|³n-L»%wT-|gya¦!nvxº° mq| v wTn-»wªv!|fn|!n-§±|gfil}´n- am|g!il±n:%±n% a|!v !Åv!±%w!n%%v³v %v!wT!yaxn»°_|ªL:El x½w|gail7n-wf|gvf©v!¾±Nyaw%º¦©!n:wbªLv!mo} n-©n%n%w!-v¬i|gwT© aN©v!v!°ovgm §a±©yln-n!li¶§»vgya} wq!n%|gmoilv³n:n%|g iiwT|gwT|filan-|g©!ilvgvm nqv! b©ilv|gli©%n%wTn%%|fvn%|gv!ailv¤wTn-gl} }(nox¥×§»wT} nxw iÅ a!L%n%v!v!|!:iwx¦n³w:%iln-!n-»|wTl(wvya 7®d|!T®¯lwT|!n³¦±©n-½ wv v!vg§a©n¬T¥0|gilnfvg}7vg|!n-mÅ° ¿gv0} an%7v!¥×-ggmq§n%wT|g©wT§xn?¦zan!º±i.wTan%go|g§±ia}m)|gilpjn il a±!±v!%n%n%v0v v!|gil!njvx°vghjwx¦ian%©v¹v0§xwTwqv!n%x:wTylv!n.|gv!iln]molnvg-| gwT»|g| ©xwTwTln%vgyl|gyv¹wTmo|g©n-mlv® wwx½]vwTvg!iln:P½T¥¹9|gil n Oln?z7ya|
(79) !| nwv!ª ZoWÀ v½Xil|gn-i£wT|w(½ a!0±%§,n%nv v!|g!:nxwT!|!iln%L´v!n%Cv:|gwTillnvg|gilwxn-¦on&° l:¯nbw|givwTvg|¿wx·l¦nyaº!n|ºm³|gilyln vg|ln-§½ n | wTwlv!Cª³|g|gia!nx©vwT|!xn%wT:C§±©}nxw|giln|!bvgwxwT¦Cnj al!-±!%nxn%wv v v!nT¥¹xwT|gil:n§, n]nx¦wTn-ªÅg}T¥¹%movg|gn-mo} :g}|!°fn-g±moylv!i´T¥ vgmo|gyn-wTmo|g©gl}v±i%w%-ya¦n wT§|gn%©n- | v wTn%§l-ªLv |gn-©ag¿¦Nn%° »|!f:w a%!%n-yam:»|wT7g¥×} gn?mqz7 ,wT|gn-©gomowTn-±§,|!v ya|0W_l!nx|w7!n-}f | wgv!|!ªLn%vx:°4ilhjn-ia!Yn ©v0X4©|gvi|gwTil|n¼vgil§7x½]gn%n%-:|T|gil¥n]|gillnn%%¥×n%v!©vgx½|E} aT ¥ ã$# `Rã&%.
(80)
(81) "!#$&%'(#)*+,- ./ ! 0.1#2 3 Master. T. Slave. Processing of a new task Problem with the coherence of memory informations. Time. Time Memory information. Oya!nÅZ. 0 l z T w : m a © n l y g v g | T w g | a I g | l i b n l % n % % n ! v g v ¯ | 9 } T ¥ a ! % n 7 © g | a | g l i Å n w g | ¦ T w g | © « T
(82) ¥ ! x n w 7 } | wv!ªLvx° 6 ' /
(83) * *, * ¬
(84) !
(85) l*j* 4 " ! :E |gia©v¬v!n%-|g©´½¼n a!n%v n-»|fv!monm: a!P¦n-mon-»|!v|!b|giln 7}LwTm:©mon-mog}±®d§wv!n% vgwx¦n v wT!m:iln%7aya wTa|Åvg|gg n%wT-|!|gn-a} av!| !wTn%|gv!©n-»|!7n% ¥×g mq|gwTil|g©n 9 a!n-9¦L©|gilyln{vf7v!n%}L-|gwT©m:N©° b> v!!n iln% a7!yan%v!n-a±´|¬|!|¯C½]bmqmowTªn%nb!i| wTamo©vgmo!nv n 7q-©n-±|x_° :EIwv!n%%l´ wTg|xºR½¼n: a!n%v n-»| w{mon-mog}±®Ew%½jwT!nq| wv!ªv!n-©n%-|g©¶v!!iln-mono|giwT| %l%n-glv.|giln¬lLan%v.wv!vgln%vg| wT|g©xwT}b|!qw a!L%n%v!v!x° . Slave designation. .
(86)
(87) ! "# $ % & '( '( #)*,+.-/ "01 324 5+6 !
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(89) °(|!n-±¦yl©n-n% :vi m a| !ylw±v!vg%ªL|n%v v | v!wTwTª!xnn ° h|g4il(n%v n-nblvg| yaw!vgn{ªLvw§n?L¥×±! nbmo|giln-n-mo} gwT}´!nb§,|gn-!inxwxwT¦±|!©n%Ëya° xº0> ½]nno7m³©vgyl|gv!|³a aya!©xvg¦Li£©a|gniln|gil| nowv!ªL7v¥×|ggimqwTwT|q|g©wT! n!n-lv!wT©|ganb¦n|gil|!n lv!yaL§aa|gn%! v n%X?n%° v,W_|E}± n{pflLan%v X.wTlC|giln| wvgªLv¿T¥|gilnya a ,n-¿w%}n-!v¬T¥|giln|g!n%n W_|E}± n{pºÐq&4 `9` #ä acbed(fg.
(90) px. c =
(91) EV = .. 4Rx Ä»Ä hmojiln-mon vggya}±§a®d|g§!wn%v!n%n%vx7wT}± wT§m:n©¦]n-vgg|g}« wT-|!gn-|g}©xm³wT¬ylvg |4§,|!n¼n-gw%mo½]vwT!Tnj¥T¥amo|gn-ilmon0¥dwg-}¨|R±|gi%-wTya|0 wjwT a|g!©± %n%v ?pxv!NA¯°©vNhj|g!i»nxylwT|gvxºaw wwavg9ya§a|!|g !n%|gnil|!nb ,w%nx¦wTª9©ÅTw¥¿mo|giln-nmov!yag§a}|g!Ln%%n-º¹yal wT|o|g©!nxw lil-n%!£nxw}v!n-n | W_X?|g° iln¿¿mo|!n-nbmo|gigwT}|oT|g¥iaw ©v:vg½]wx¦n¿ya©| £wvg§ªonb½¼moya©vg | §ln-l|!n%n?9·- wT?¸px¥¹%A¯°¶m³hj§aillnbn%mo n%|! biwowTavg©| vgwTm|g© ½]vgnÅ aa|gn%|gvga T©v¥¹vg§yaw§av!|gn%! n%n%v¿ ½mo|gin%v!v¤wowTwTn%gveZn³moln-%monÅwg} a!%±%vgn%|xv ºv!wv v!hj| iawT©gv.|!vq!ilw´©vg%yan §aWÀ|gv !n-n%ln7ºja|oo§a|g!ilnwamoxn-wmov!|!vog}b|gil ,nÅnx%wTªbvg\| T¥W_|gmoiln-n¬mov!yag§a} |g! ,n%Yn nxX]wTcª ©vXmoT¥|g|gil¦n(wT|!n%%l§»%}{n-g|gliln%n ¥dwv!ya-|§a|g|g!in%wTn| ° |g§iln-n©aw-|ga¦ nÅ|!mo n-vgmoya§a|gg!}£n%n%xvowT«wT!in{w%!¦n-nCwT|g!¦nxn-wT|q} ¦vgwTmqgwTwT|gj©¶lvq|!n-lgmovg©av:nT¥.w´ avg!ya±§a%|gn%!v n%vgn awTl|g£mo|gni°´wT|hj|gi»ilylnvo| nxwwvg ª7i v a7!¥×L%gn%mqv!v!wT|g¬©iÅw½vw{N¦±§©nn-½ ylv!Tn%¥]|g|!il:nov %n-©n%vg-|f|T|g¥¼il|gn¬ilvgnqw%vg¦yan%§av|g¥À!n%n%v¬|E}±§ n-n³a l|gL!nxawTn%|!vxn%° ´|gilno|giln-!vx°q±yl i. Ä . a = É Ä x Ä±Ä |gTµ]i¥¼wTwol| %ln-½±gaajna°.v!{±hj|gil¶ilnfn¬vgmo| yawT ag |g|:n-¦wÅwT |gmq©wTwgC|
(92) vg|!©Tn-v
(93) ¥0 |g|g| iilwwTnfvg|
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(102) ln%v %|g-nilgn¼w
(103) §,| wTn%wgvgªn%ilvgn-©|4v!n| ww-©v!|gvª¬¦|gwTi¦|!wTn%n-Ë| ºxwT|g il njO§, anj!yaw±-!%no|gn%v!¦' v wTW_|!§ 0n%X?%Ë° °l>
(104) %n-n:glxn?zlwTn%qwTm:v!v!n-n% aln©nja|gv0Ti¥l|gwTil|g|¬iln]|gn¹liln-nm:½£ 7%w¥×-vg|g|0mqT|g¥lwTi|g|gwTil©|n¼ %mowTgn%!§!n%ivgya ,wT|¬a©l|gvgilam nv v!l|g%ywTm:|g©aN °¼mqhjwilvgn³|!n-7¥À| wgvgmqª7wTv |g©©v lilv
(105) n- 7¦¥_yan-j(|!§± }{w%|g¦ilnf©£ a!wTn-¶¦±©lyl-v
(106) !nxmowv!n%nb!iTwT¥.a|g©vgilmon{v
(107) mo|!n-:mo7¥×g}Igm nx|gwTilª nf9|gil|gn-ia!©vv . . P0. P0. ã$# `Rã&%.
(108) pp.
(109) "!#$&%'(#)*+,- ./ ! 0.1#2 3. Memory. Memory. Memory Memory peak. Memory peak. Memory peak. Without predictions. Slave selection for P1 P0. P2. Initial memory. Activation of a master task P0. Memory sent by P1. P2. P0. P2. Ô ÎàÑdÍ=Ö?è=ÑNÝPÛdÒgá=ÎÐÜWÑdÎÐÖ¤ä. Memory cost of Master task. . Memory. Memory. Memory Memory peak. Memory peak. Memory peak. With predictions. Slave selection for P1 P0. P2. Activation of a master task P0. P2. P0. P2. Ú ÎàÑdÍ¿Ý=ÛdÒ+á=ÎÐÜWÑdÎÐÖ?ä Oya!n³Z]µ¼g|g©xwTRvg|gywT|g©Å½|giCw lLanfT¥0|giln¬ya a n- wTg|.T¥|gilnf|g!n%n° Initial memory. Memory sent by P1. Memory cost of Master task. . `9` #ä acbed(fg. Estimation of the memory of incoming master task.
(110) xp c =
(111) EV = . n½xwTw§iav!© ya}Åi|:½]|gilwnÅ
(112) v|gvgilyan¿§ag|g|g!lian%vgn%m| vwT±wTp| l° wT9l:¯ln%|gailyln%nvn%Ëlmoº±n-¸a¥mon%½]vqn gT}¥
(113) !ºan-§±|g ail}nw|g%ilyann a|g vgilyan-n³:m mow%Tn-}¥|gn-g|g!©ilvnylTv!¥¿ln%vg|g(| ilwTnb¥×»|
(114) |gwT!l|gn%ilnÅn%nxylvgwTv. wxmo¦§n n-nbmov n-©n%gg}-n%|gº7-©|!|giln%N n º yawT±mo a|g !±n-ya7j±yl|¿w%%}Tn%n-{¥!mov§±}:n-| wmo|gv!ilª,n.gºL}l|giln-l½«n¬n%n%wTmoan%n%(!gi§»|gwT}iaam)©|gvgilmo½nfvx-°Ëya:¯gv!!n-n-©|gn%±il-|¿n.|vg!|gyan%il§avgn|]|g!vgTn%¥Nw%n ¦|gwTn%illvnÅwT l|gwTbil n³n-!¼n%mov!½] ,n-n.mon%-½|gwT}Å yl%|gv!iln.vg|.n
(115) T%¥l|gilvgg|gnf|g iawTlmn?»z7|!pv| |!:an-l|!n.g|giam p½iln-!n¬½¼n¬ylv n¬|gilnmon-|gg©³an%v!-g§n%ÅwT§x¦n° . '( '( #)*,+ 3 ) 1/
(116) ! 4 1) 9#*. :E£|gia©vv!n%-|g©. ½ ] Å n a ! % n ! v n ± b | w o m n o m g ± } E ® % w j ½ T w ! n | w g v ª ! v n © % n g | © ¨ ! v g | T w ! | n } R> n(§,n- §»} ° | !an-wn%m:v!v ª£-gl|go§ailn|gai(|¯wT}L|¹|g ,ilvgn¶now%¦§p nn-i| w%w¦Lvg|EªL©}±v¼ yawTn´!nT ¥ wmq-C|gDEw¦C-vg|!wTFn-|!Gbn%{| Iwwvgv|!ª7n-v!vbgLmovgqv| wTTw|g©¥vx|g| wTilwvgn-}¨ª7}vfwwTv v!!vg nil!n%ln%7%n%yan-¨¦a|!n%Il:°L|g§»il8
(117) }nn-|g!il an:n!L½] %ilnon%v!mov!v!n-nxq!wTv vg wx½]§»¦} nn a!L%n%v!v!x°hjiln-(½¼n¬ a!n%v!n-±|
(118) yaln-½ | wv!ªmqwTwTn-mon-±|¿v! iln-mon° . L. L. ... L. L. L : Leaf node (Type 1) T1 : Type 1 node inside a sequential subtree T2 : Type 2 node (Master task). . . . T1 T2. zawTm: a©nfT¥w ±NT¥0| wvgª7vx° CDECF-G¶iwv(w %'eT¥!nxw7}¨| wvgªLvPº.½iln-!n nxw i | wvgª¨%g!n%vg ,lavÅ|!£w¶l±an´T¥ |giln mqw|:v wTv!%n-m³wT±§a| n%wT}lvq|g½!n%|gnai¨° }¶w´:E||gv!il©| vnbw!!m:ª9nx ,wmo7}gn%| !wTi| ±wwT|fvgaª7©|!vvg{mÅvgl| ° wT|g|!:E©nxwT|g|giil}wTn(|¬wa|gv!iln?vgn¥dwTl ,yan%L|q90vg|!|g© vwT|g©|!±iln-xnbwT} aºj!|!|gLbil%nÅnxn%wv!| v! wi vgxª9°¶ a!|ghj±i%ilwTn%|onbv v!½ ,Lf.wTjl§© nv v§ n-n%©%n%-mo|!n%n% v¿©!v nxw|gil7nb} W×wTlnb0|! v!|giailn{©7|!!n- ´Ti¥.wx¦|gnilnb§,n%vg| n-w ª, a°!L:E%9n%v!v!wn%a0 7X|g©|¿N©v¿º¹ a½ylilvgn-il£n% w(l L|gailn{nT|!¥. |gilTn{¥¼|g|g!iln%nn ½v!wT| ia|gwil©! ªi:ya°hji{n-ial|g©n-v]ia ©wTv!v]m:©}v al©m:n.|mo|!wTvn%½] |giilwxwTnj}LavwT©movgwTm) a ayan-±©lx|¹vgwTyaT§a!¥ ©n%n¿|!vjn-m:qw |g-ilan
(119) n-wT a g|g}³wTi7 ®×mowT·l!n-©n-vgmo|,¿xgw|g} v nwxl¦K° n%n-:¯n%!aav wTn%|gËwTT§»¥R}} |gº±il|g|gilniln¼n.w¥×v! wv!-Ln-|!m³ §agT=<P¥Ë}qwT| |g|gw©!vgn%ª7Nnv ºº |g%nxiw± wTiC|j| wT|gvgillyanv¬§a|g©wT!nxn%wxRn³¦|gn%ilylvnfvgTa©¥0nxw=w ¥wvglya¦L§awTa|gg!n%wTn%v¿»n¬T|wT¥!T|gnf¥0il|gwTnfil|gn!n%wTn³w v!| %gwT|g|g±ia©|gxm wTyl§»}C} ylwv.v!?pxvgv!Bn-A |&X?l°jn%W×wT( l|!b!on-m: a|gil!n%anfv!n-m: a±!=| L<xwTn%|gn%©|gv!ilv!´n¿xTmoº,¥0v!n-|gmoilgnf|!n%g ,C}{Lv!T ¥ ©Tv]¥|gil¦nn- a!LL%On%v!v!ya¿!n ©v
(120) L°ya
(121) lwT|!|!n¿n% |giwwTv¿|jv ¥×±¼ |gilwnv½]wo| gwª±vg©ªÅw§Ln%®d§%wmov!n%n%v¿7!}Lnxw7wTm:} ©W_¬d°ànv! °.il©n%v¿7yalv!an-gl|!ºan%|gCiln¿©|gwil n ya Ll) wTX0|!½n%iln-½!ilnxwn-v]Å¥×|gil¼nf|gil| wn
(122) vgmoªqn-©v.mow-g|g}»¦®d§wTw|!n%v n%{W+7° d}L°àn½wTm:iln-©(v!!|.il%n%7lyavgyaamon%|gvilmon
(123) n-momon-mogc}gX?}q° 7¥×gmqwT|g©b©v Subtree 1. Oya!n -Z. Subtree 2. . . . ã$# `Rã&%.
(124) pe4.
(125) "!#$&%'(#)*+,- ./ ! 0.1#2 3 L. L. ... L. Subtree 1. L. . . . T1. Pool of ready tasks of P0. T2. Subtree 2 Task A Activation of new tasks. Memory occupation. Memory occupation Memory peak. Memory peak. Cost of A. Cost of the current subtree (Subtree 2). Activation of A (Type 2 node). Oya!n$SZ. Activation of the nodes of Subtree 2. zawTm: a©nfT¥w:-g|g©xwTRv!|gywT|g©¥À|gilnf| wvgª{v n-©n%-|g©N° Current memory occupation. . jhaLiln%nv al!|§a©| n-wTmªn½|gil|gni mo|giln-n³mo§wg}Cvg© ln%vg|gn% awTn%|! n-§±}(}(a|gn%ilv n-g| w§,vgn%ªCC»C|!Å|gilwn%% a!yan-¦L±©|¬yl¥Àv
(126) |gwTil nwTv! n-wT©n% a-i´|g©©v
(127) |gTi¥]wT||giln| l½Lilan-no!n{|giwwT a| !½L%n%¹v!v!§nfw©v-|g|g¦!nxwTwT|!|gn%Ëa° ´:Ewlavgyan%n%§a |g!¸n%¥n½]wTnl %ilwv!v©awÅn-³wT|ggiln{novg|E}±|gy wTn|g©b lLa¦non-¶|giwT |O§,n%%ya!mon n%Sv !©vnxwwT7!}nxºw|g7il}Cnf-§gw|gvg©©x³wTvgI |g WÀvgwT|!|gn-ywT}b|g©½´4n?z7|g|g ilwn³-|.©n?|g¥À |ilX?n¬°hjwTgi»yln³vl|g±ilan n%n-¦vgn-|ÅT¥¸¥|g|giail©v
(128) nflmoLn-amon½g}±w%a-yaC yawT |g©|! |gLil%nb-ya% wTvg|g|©TN¥.° |g
(129) ilnbvg|gyail§an³|g!n%|gnÅiln-wT¿li¶wT|gliaË©º,v¸¥xwT|g9ilnf©nx aw! L%|!n%´v!v!wC¿7a wTn-mqw%}7wTv
(130) |g©x|gilwTnwl--|g!nx¦wwTv!|gnb©T¥.T|g¥il|gn{iln³mo|¯n-}Lmo ,ng} l LnxwTaª:n:½ya±|g,¹l|g|¼ilnolvg-!|gnxywwTv!|gn©;° >¨§,|gn%i{%|gmoiln.n%v³ aya©n%gv! vf-v!gn¿|gT©¥4xwTw% ¦WÀvg©7|gyawTo|g©vgyl ibwvg|gil|gynowTg|g©i»N | ºLX?½¼º|gn¿ilan:n%vgmon-lmon%g}w mo½0!n§v!n:w a-ia|g©vg¦|gwT©x|!n%wT|!n%(| wT| ªLwavgªv!mon-©n-n%mo-|g©g}C %wTlvgg|g |gwTiam±|!vfWÀ.±|!Ågw|g%ia%m ya»[|xX?° ° O4:¯|
(131) !!v!il|xLº¸¥jv!n%wv a|gil!±n³%| n%wv v!vgªb|giiwwT|v v!v!| yawT§ag|g|!!n%n%n.w{|vgwya-§a|g|g¦!wTn%n|!n%wTv]l|jvg¸}7¥¹vg|g|!iln-mqn³| wTw|gv!©xªÅwT©±}x°wThj|!n%ia ©v]©v¹§|giln%n xwT|!ylv! n¿Tvgya¥¹§a|gil|g!nn%n% vj±xwT©v¿§§,nn-¦©n-ga}qn?az7b |!n-lvg|gil¦nn wTg|!|0n-gTmo¥|gvilTn¼¥|g!mon%&nn-moWÀyag|!}(vg©ylav n]wTw
(132) nvg"° ya§a:¯¥0|g!|gn%ilYn n X?ºl|gLilanjnwTwT|¿|ggil|gn³iam |! ´v!n-©Tn%¥-|!|gilvn| ,La}³R§¸¥n-©|aaLv¿n%v|!lq|g|4ilnl-ya! anx wv n-n |g|gilg©nqn%v¼mo|! n-mov!n-©gn%} -|j wTnxlwTª |giln-§l]v!n-| gw¦vgn%ª: §±}ov!l%¦±nqa|gilnq ag§,©n-g|¯a}a|!a³ |gilTn.¥l|gil±noan%¥dv]w-§|!n-©g=a<PwT|ga©N|!°
(133) |gil|giln.n-vggya½§a©|gv !nn%ºn%vP| º |gg}La{|!o!n%vg n%-|
(134) |gilnan- a|gi7®×·l!vg||g w%¦n-!v wTwv
(135) m³yl i´wv
(136) v!vg§a©n$° W a{|!L:¥dwT.¥À!m . `9` #ä acbed(fg.
(137) -p c =
(138) EV = . |g|gil!n%n¿nawn-- a|g|g¦i7nf®×·lvg!vgm³|+®¯yav!nx| wTwT!l n%iÅyl|g v!w%}¦n-wT!lv¤wTÅN%l-ya!©nxqwv!|!n¬n-l|g{iln¬|! ©l§-wT!nxNwmov!n¿n-mo|giln.g}b±yaylm³v wT§n-n° X T¥R§a wTl!iln%vT¥4|giln ÉÄ å n-mog}»®Ewx½]wT!n³| wvgª{v n-©n%-|g©CwTg|giamÅ° | gÄiln¬ Rl ±an¬|gilTn¬¥0l|gilLn¬anf|!T Å¥0|gTil¥0n|gil|!n¬ ( T±¥0Ë|gil©vn¬ ,lLv!©a n³ wvgya§a|g!n%n Ä Ä _ ÄÉ N|giln¬ ±NT¥0!nxw7}{| wvgªLv WÀvg| wTg|ga:¥À!m |giln¬|! (T¥0|giln ,L) X Ê É g}{ moÄ n- nxmo RwT ªg} §l v %n-gvg¦Y| n%W_() Xvg l%-nyag|g!iln-n¬±|§n-mon-amoaag} qW_T¥0l|g-ilyln7¥dwa-o|! gnx=<PwTwTª|g©T¥ v!ya §a|gÄ !n%Yn X mon-m® Ä _ Ä Ë§,n-©av.|!ow:vgya§a|g!n%n Ä ÄÄ 4Ê 4 Ê Ä R Ä Ä4 Ê R É |giln¬lLanfT¥0|giln|! (T¥0|giln¬ ,L 4Ä Ê 4Ê .
(139). . .
(140).
(141). . . . .
(142). ≤.
(143). . .
(144).
(145). .
(146). .
(147). .
(148).
(149).
(150). . .
(151). '0
(152) * +* > n³vgilya©{·l!vg|mon-±|g©(|giwT||gilnfan%v!(T¥|gilnwTg|giamov
(153) T¥¹ÂLn%-|g©lvlº7°pfwTlC7°Ð v!h4wTil!n{x½vgwT|gyla!7nxo}:w|g7|gil} iln-n¿|gilm:n© !§n%wvgwT-ya|j4nT|!¥4Tn%|g¥
(154) -il|!w(n
(155) vx° 7wT}LgwTnm:wT©movg|gya ±wT||!n-T¥.©n%n?vjz7 a !n-g ,mov n-n%»q| wT|g{©|g iln¿wT al! n-¦±½]©noyl¥×vjLv!-n%yl-v³|g©iln-l!vxnºL½] n |g|gn?ililz7nbn
(156) n-~ s.g¿yamos.|giln-¬»n-|:+ ¥À !ya L:® %jvg|gL ©n-n%wT-|!a|gn-©© %n%
(157) v°0©n%hj¶-il|gv!©n.n-¦|!n%n-vg ? YA|!wTv¹¯jºi a|gw%!il¦nbn
(158) §a©%§,n-n%mon-©n%v-|g ,WÀv!©n-n%+n{¥×h4¥_g!mowT§am n%©qn Á
(159) Yp aX|g¦n?ilz7n-n !|gvg :wj|¯-}¶|!n%TI¥ÂL~¶¥ÀO!©vgm²}7gvg©l|!n-n-w m|g qiln-T¥ : s :g¹%m: v!n% T¥ v!n-¦n- wTLl±an%vT¥ 4. a!±%n%v v!!v4nxw iN° ¹w!ilLan¹©vn%¾»ya a n%½|gi w m:amyam T¥T T¥4mon-mog}vgiwT!n%(wTmoaq|!vI4:~4P½¼n-gW+pºH4 8 <BX¼ a!±%n%v!v !vx° "' ìxìxã_ä=í-í-ÏÀí í Ñdî î ÎàÑdïÀïÀè=ð-ð-ÑRð ð ñ?ñ?á=ñ-ñ-è)ññ(0òò©î!ë+ ¤?Ø-û õ¤ÒWóa %ÕÙÖ¤÷-òÝTx÷Ýú% $# ÷7Ò+æò_ò%óÒW&äxP#Ñ,ð ò©ÒWô= Ñ4 ù- ð!á=î Ò+ ?-+Ï ûõ-`R xð%Ò+õ?õgÏ×î!ù-Ï× ¤÷ Ö?û=èPõ¤Ûdó%Ü+ðÒ+ÏNÒ+äãåä=Þ[Ö¤ÛdæÔ ÑdÎÙç%èTÒ+*PÜ+ÎÐÒ+ä%ÑdÎ-,ç%è=Ò. ã$# `Rã&%.
(160) px.
(161) "!#$&%'(#)*+,- ./ ! 0.1#2 3 "#%$'&)(+*-,. GHJI K*6 Y9V J( 9]J^ I ,+43436 I (J_)Z #$' +J JJ_ H NL#(+*V9 H J6.
(162) . /101232!4 .812!0!6 /O.M5303836 .Z.@2!Z30 835!7343636 .Z!412!534 .0!5L.736. 35 6373836!038 /812!4L.M435 .436!Z30363737 /O.43630!0L. 53735373Z!03Z .Z3Z@/Z!Z@/ .3.6!734383Z35. j_ JZ. .532/8@/. 630383838!030. 93: 93: 93: 93: H 9 H 9 H 9. ! *+; !<= ! >1? @A BCDE <=
(163) F N OM@ !P3 @<Q<Q P <R TSE*VU*-W X %M
(164) L ;<[ \
(165) ! 91B % ;O ; *`+aJb BC@<= ! cCF d O M<=BO
(166) *`+aJb BC@<= ! cCF d O M<=BO
(167) e I !C@P ! @Df6! ;F ! !;O
(168) hg# >1 #P1O id
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