Determining the Idle Time of a Tiling: New Results
Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Determining the Idle Time of a Tiling: New Results Fr´ed´eric Desprez , Jack Dongarra , Fabrice Rastello , Yves Robert. N ˚ 3272 Octobre 1997 ` THEME 1. ISSN 0249-6399. apport de recherche.
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(115) o [ ]efsR]Tp vkmtp ]TlV[¤ wzk [ o evLpgEem]tE¨p© }emkf]uªt«g o km] o tgEw7uz¬7] ofo gEw?kf]®_
(116) |T¯? ]k- [ o + &#' [ B # ]k ] em] gikk ]k s k ]km] gikf wemsRgEv]£gEw7u( ) . )*&+#&+, B/ # ]k ] ef] gik°k ]. J p kmk p ^ p k [ ]°km] o gikf p wemsRgEv]t¤ V]]¬¥th o ]z|° p}o giwHh7efk o gEkm p w¥¤. a) rt h. 1.5h. w. Upper iteration-space boundary. b). h rbh. h. -h. A tile tile boundary. Lower iteration-space boundary. ¬¥th o ]z| 84 [ gEsR] p 1k [ ]°km] o gikm p w£efsRgivT]07k [ ] o ef]es p efkft]-w§´¸g}·giw7uw7]T?gikft]-w§´ ·¤. öªá þ. $ $.
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(143) ]e jV] v]hOsOkmk p uOwª¤ ]a w [ Z ]g w ]]°w7uO_tiwOv p w7k u7[ ]°kf s p ofw£p vT´W«]em·¥efwOe gap km]Tv p [ w7w7]°vLv gip Jh7v p^ wOw u7p km1 p kmw]´ eH gief emt´fw7_(]T!u£k ! p i·-k ?zV·w k p[ k gi[ k] emo k0gi kmpt]To e¹u7k eL[ Ogiwk¹vLw gEp ws ^ omp p vt]T] eme p p}w o k p kfew7]jVkv p h7^Hwk [7p h7k gEkmw7° p}o giwVluRgEkºgk p ]zv p ^H^h7w7vLgEkm]uª¤ & ( *+-,/. 0, wk [ eef]vkm p w£]zs ofp s p ef]wO] s ompVp ¸e gEw7u]jVkm]wOuQk [ ] p}omp ®_L|T¯m¤ Tile stacks 10 9 8. processors. 7 6. active. 5 4. P. 3. 4. idle. M=11 p =7 r =-1. 2 1. P. 3. 0 1. P. 2. 2. 3. P. 4. 1. 5 6. ¬ª´¶}hOh emo k ]¹o gi{ kf8wO-vTkmk [ t]°]¹gEp}w7o u°^h7u7¶g]1s?7omp
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(145) ¸kf] oomp kºgEkm p wª¤ time-steps.
(146) .
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(170) l áªá 4Ù K ÷ ' ø ' .
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(184) *· ^ hO*w7&, vL'ªgikf}]u kvLgigi pwwOv p wVk [km] wVh7[O]-pto k [ ]-p wVs kºomgip vgi]efjVemew7 e-p giª o ]LkmgEeu7v l£p g
(185) h7t^ gEw±g ´¶ :¤ ]t]t¤¤ika[ k ] [ o] ]o ^£gEem]wOwOk- ^H h7ef!§k¹´ ? gikLB¤ _
(186) Z*· p sZof[ p ] vTo ]]0em ezp}o ´W]tgiJw7ku[ v]p ^ p w7^th7]efwOkv
(187) sRgigikmk ]/[ uRgiwHk0gak [ om]p u7^£]·¥sJ]k [w7]u7] wOo veflQk´ ? o giBs [_
(188) ·[ kºgEgEee ]e w7p tk [k [ ] [7pto ( p wVkºgERgijVekºg ]e°´ ? B_L·T´m_9!·?;*&+, '¤ ´ ? B _L·V^gij7´ B ' _ !$· *&, ' !M´ H !M´ ? B _L*· i?· *&+, ' $ Z[ e p w7}]emksJofgip k [ e ptoo ]s o ]em]Tw?kf]u§w£¬¥th o ]z 7¤ Ots v]eme ? B§_^h7efk gEk ´ ? B_
(189) *· *&, ' km^ ]0:emkf]s7e/ pto s omp v]efe p}o °k p v p ^ s7]kf]kme o emk ´ ?CB _L· kºgEe eL¤ Z[ ]wQs ofp vT]eme pto ? BF_°^hOemkgik-giw p k [ ] o ´ ?CB _L·T´m_:! T·$ *&, ' km^ ]0:emkf]s7e) pto ]jV]vTh7kmw7 ßçiÔºÞTß1Õ:çißfÕ:ÖßmÜãWçiåæàíià¸åæÙiÚÔÕfØéèiÛWßmçißmÙ
(190) àWåæÞßRàWåæéíiõæÔã¸åæØTÙ¹èiÛWØÚÛWÔé°ï0à¸ßfß = Eü >¶ÿ $ öªá $ þ . . . . . . 9;:#,. < = )>. . .9. !. #". ' '(. . . )". *. +" ,. .-. 0/21 #" 435(. #?A@. . . $&%. . ,. 5". B. DC. FE. G". #?. ,. FE. FE. HE. IKJ. )1. )" 76. 8(.
(191) Tile stacks 10 9 8 7 6 5 4 3. M=11 p =7 r =-1. 2 1 0 1 2 3 4 5 6. ¬¥th o ]z 8 ! p w7}]emkasRgik [ [ ] w. ´¸giwOuEH ?> > · Processors. ¬¥th o ]z~ 8 ! p w7t]TemksRgEk [ [ ] w. ´¸gEw7u%H ?B> > ·. kmgik]kme°^Hgiw7]0¡uefkmv ]p s^ ^ h7w7´ ? vLgEB km_
(192) w7·°´f_"uRgE!kºg"!#gi p w7·$*k&[, ] 'V[7vLptgio w( p s wVofpkºgEvª]emu7e ]ptsJo ]w7uO ]w7? v]B!sR_-gEk ef[ kºg k o [ k-gik k-[ ]°]LgE]jVu7]e-vkh7p kf p kmw e p o emkakfe+kºgEH e ¤kºgE-e wOeLJlgik w7[ uH]wªk w£¬¥Rw tp h k o ]z ]a~7¤ ¸h o k [ ] o uO]¶g
(193) lt]uHu7h o w7k [ es ofp vT]emefwOR¤ Z[ ]- p w7}]emksRgEk [ wHk [ ]°u7]sJ]w7uO]w7vl o gis [ e o ]s o ]Tem]wVkm]Tu ¦§]°emh7^H^£g o ] Rp k [ vLgEem]eak [ k [ ]zemw7}] p}o ^h7¶g ® H ! ´ ?CB _L·T´m"_ !/:!· 1 ¯ ;*&+, ' $ Z[ ]a p}o ^h7¶ga p}o eu7] o t]uH ofp ^ k [ ]°] VhRgEkm p %w ?( ! Vk [ * - H ?+ *&, ' ¤ o Á §7 omp < ¦§]zem]]o k [ gikak [ ] o ]Temh7kme[ wA®o_
(194) |T¯mìgEe po ]s pto kmp ]u§w Z g ]H_tg o ]wRgE[ vvh [ o gikm]}¤@ ef^£pVp gE o em]u p ]e°w p k sk [ ]]£tv]pwV^ k)^hO^ w7vLg giVkf hRp gEwVu :k p gE:kmv vzp ^u7s7hO]°kºkmgikf^ p 0] w Vk o gi]zkf sp]p vemk ][ ]£v k0w7g u7o tkm] kwg oezv [ _ kf!#]v"kmh !/o ] [ gEe77k p s7¶v g
(195) lµ^g g omp ]e¹]}¤ wFug²ef]w7empt] o u¥]vL gi[ hO]emw ] p_9k ! [ [] o! [[ gEw7 uªV7}k [ [ ]T]owop "_ em]a! ompe!#e p efp}^£ o gEH´¶´" [ v [ Bz´me_9!gi·fg
(196) ·lVk e¹[ k gio kh7]°gi7km[ ]]w v +p hO^ Bw7_
(197) e·vLVgEk w[ ]° k ]ap sk0giompvu7]ef]-em]kfu²^ wO]-u7 ]omsJp ]w7eu7]VwVhRkfgilVu ¤o giakfwv
(198) gik [ l] k k ]wVh7^ ] s v]efe eL¤ p Á §p 7 [ D p Z[ ]giefemh7^H[ s7km p w H ´ ? [ Bµ_L· om>p >iep}w7o ]]u7]Tu£k p ]Tw7[ emh o o ]zk [ ]-tgiu7k:l p ªk [ ]a p}o ^[ h7¶gie
(199) o ¤ V]]z¬¥th o ] 8 ´ w7] ·k ]° w7}]emksJgik et}]w l k ]°s v]Teme V ] ] H ´ B _
(200) · > > ?gEw7uk ]°sRg gi]kf^H]e ® H ! ´ CB _L·T´m"_ !/:! T· 1 ¯ ;*&+, ' $ p km]°k [ gik ^£g
(201) l ]°^h7v [ em^£gE] o k [ gE%w ? wk [ evLgEem]t¤ Z[ ]°uO]°kf^H] ]Tv p ^H]e ? p´ k [ >] B o _L·Tem´m] "_ !:!$· *&, ' "_ !/!# GE. . . GE . . . . áªá 4Ù K ÷ ' ø '. . .
(202) ´th o ].
(203) . Processor Q. 8 gEw7uEH ?B> >. ! *%
(204) ! % 2
(205) % °iÅÅWWÅ D ½º½¾@,3W»/, ¿¾'-A"t¼)( , .%:½/* ,'7¾4(H¼3W¾L»-, 3W¼)" H½* , .0¾H"2,°½¼.-
(206) ¾+H
(207) )? / 1#".'3+(t¼½'3W»¼)1A 3¸¼)" A6 , 8(»¼½¾T½ !) /21'" 3 3 " 6, D ( //3*"' 6$9F: , H ´ ?CB _L·´/> > !A> ) > · @ 3*7¾ ® H ! ´ ? ! B _L·T´m´m_:! ) !#· 1 !/ B ) · 1 ¯ *&+, ' $ p°ivÅp [ÅDhOC ^ !w o ]¿k ¤ ´zEom·_p ! H ) p}o!! 7 pt´ om B 7 p )k ·1[ ] ptwµo gE v p h7 ^ ? w7eaB vL_ giw ]- k ][ s]om p ptv]omefem ]p uAgEu w7p u7ª]s sJom]p w7vu7]Tem]Te w?p}kfo l?¿¤ }kZ[ [ gi]kk eLp }kºkgE[ ]-kmk ^Hp kº]gE7e°¹t]T} ][ w k p kl[ k] o ]¶efg ]t¤ }]emkas v]Teme omp ptgEou8/ p ´ t · H ) pgE[Ow7ptu1o p ´ p?CB _L· H p ! ´ ?CZ[ B _L·T´ B p ) · e-kmeef ^H-o em_"k±g ! 0o $!k´)pB !#k )[ ·°gEk-kf]TwQ7 e 7k gEgE[ kz]skms ^ ofpVv] p ]¡ efefekmp ] s9e°n omemp sJ s ]pw77 emuQ kme p h7wµ^ k]°_tk [¤ u7] wA ]- w7kf]^ ]uO]-O eJuOk h7s p ]zomp k v]giefkze p}h7oHwVE kf/wV kºkmgi ^ v ]0E :^ em kf]^s9?ChO w7B vL_tgiJkf´mvL_gEw7!weL¤ ef kº·"g o;k°]°* &+s u7, om' pemvvh7 ]efef]0emem ptw7o w] sh7ofwVp kmvT]emkmef^ wOQ] ¡efk km[ ]] s9o ]Temk p ´m¹_kf! e-:v p ! hOi^ · wªO* k &[ , gE' ke
(208) ]0R p}k [o ]] emo k0]g^o kfgiw7w7Qw7k p ´pttom· ! ¤ ! )w Rkmp k][ eL¤ vL gE em] e
(209) ¹ s 7 ompJvs ]omefp e vpt]To eme Qp}o kf] o ^H[ gEwJe-gik kmp ]Tezk gi[ ] k . . . . . . .
(210) . ]jV]vhOkm p w p 1kmev p h7^ w gEkkm^ ] ¡efkm]s ®´m_"! ! ) ·" ! H ! ¿´" B( ) ·W¯ *&+, ' $ pt o ]sJ pt]o w7u7p w7? [ h7sB!o p _tw²¤ k [ ]zeftw p -´m[ _"!o ! !)·9p !Mp ´!o B!)º· _"!#"!/[07k [ Je#p ?hJgip}wVokmk:le^£gEj?^h7^ ]Tk [ ] o pto -k t]k ] 7]gEemef]^ ]°k ] ]Temh7kme h vLgEem]gEwRgilVemew£k ]zg t]a ^h7¶g7¤ [7pofp uO-eLV}gik pt[ wªo ] }p ¹[ w7]alzs pu7wV]k kmpo h7^Hk]¹k k[ [ giptgikomk k o[ p ]]a^£v gEp wOw7u7e kme p u7wQh7p]a_ p ! k p !omk );[ [ !]- hOw °gi¶gie¹w7kvT[p ]]u ]p}lom k p p ^HgEu£w7´ ^ k T[ wOg°k o uOgisJ]a]'kmp p ^ uJ[ ] gi8 J pªkfk] [o gEekm vp p wwOefu7sRkmgip vTp ]tw sk [ ]l vT]emw7e vh e o ·¤ g
(211) t]u7] ]wVk gEu7eº·T hOkªw }] ]
(212) giuzeu7h7]/k uJgikºguO]sR]Tw7u7]w7vT]Te-´¸gEw7uzk k ]v ^ ^h7wOv
(213) gikm w7e.
(214) . 6 !
(215) % 6
(216) ! F 2
(217) % °iÅÅWWÅ G ½º½¾@,; ,}»/,2¾#" t»-,$&%:½/* ,'7¾'(¼03¸¾
(218) »/, 3¸¼)" ½* , .0¾H"2,a½ ¼ -L¾ H
(219) ? !//.A? .¼.G(}¼½'3¸»T¼)1 3W¼)" 76 , 8(H»¼½¾ 9;:#, H ´ ?CB _L· @43(O* ¾9 'H*&, ' ¼ ,°_"!/! ·T´! '? B _L ·9! H ¯ *&, ' ® m ´ " _ ! ®´m_"! ¼¼ ,°,°_"_"!/!/! ·T´ !!?CB _
(220) ·9 /! ' HH 'H ¯ ´m´m_"_"* &/!/! , ' ::!! ·-·-?? ' . . . . . . . . öªá þ. $ $.
(221) M. M+(P-1)r. M+(2P-1)r (P-1)(1+c) 0. 1. 2. P(1+c). M+(bP-1)r. P(1+c). P-1. 2P-1. bP-1. -(2P-1)r. -(P-1)r. -(bP-1)r. ¥¬ th o ] 84 ]Tkmv [Qp k [ ]s ompVp k [ ) 7¤ °iÅÅ -s ofp vT]eme pt[o e [ g
(222) o t]ak [ ]ao eºgEp ^ ]-[ pto of p giu p 'Hp *&, ' ¤ ¹_ ! ! O?gEJkm]-v p h7^ wOevLgiw ]°s ofp v]emef]u w7u7] sJ]_w7!uO]wVkm! l?J gi w7uH7k s ] omp v]efefk-e ptsRog ? k B k _] ezgE]emh7g
(223) k)lV e-k [ ]eL¶gi¤ km]Temk p w7]k p km] o ^ wRgEkm]]jV]vh7kf p wª¤¦§]u7emvTh7emezk [ ]k: p vL?CgEemB ]e _ /| ? B µ_ giwO$%$%u $ '? B §_L·TefV]sRvLgig w o gikf]]°l?u7¤ ]v p ^ /s p ef]uQ7gike[ ]Qp p ptomea!´¶emp ]T] s ¬¥omptvh ]oef]ze ptto ·'8 ? B_§´¶ [ v [ e giefemtw7]Tu v p h7^Hw7e ' ' ' ^£gij7´f_"! ' B i·T´ ? B _
(224) · , » ,9 , 3W¼ , 3 ,),2 * t»¼ 3 ,2 , t¼½ , . 3¸¼ )3¸¼¾½ 0¾' , }¼½z¼%. ?CB _ ! ) ) ^gij7´m´f_ !#/· ? ' H B ? i· »m, ¾ E,t3W¼ ¼7¾0¼ ½ º¾3W ¼ ¾½°, ¼ %}¼½z. ¼ %. ? QB ´ _ !M_
(225) ·/?CB _ !H !M´ '? BF_
(226) *· ¬ p}»mo ¾ +,t ¼ 770¼ V k [ 3W]z¼ eº¾gE½°^ ¼ %]a.u7] v p ^ s p emkf p w]Lgiu7ek p ´¶ef]]¬¥th o ] _t·8 ´m"_ !:!·´ ?CB _L· ½'3 ,t»'3 53¸¼ ¾ ! ) ) & ^gij7´f´m_ !# ! i-· ? ' H · 3W¼ ¾T½0¼ %. ! ? ! H 3W¼ ¾T½0¼ %. ! ´ B _L-· ? o p [ [ p o p pto p p [ o pto [ ]jVs o kz]emkmefh p wQw7e eh7tgik°k ugE ptkLo J]Tp vLk gi[ h7ef] :_ ! gEw7u ! 7O¤ Zk [ ]ek:u7 o ]Tv]kmjVl s ]T]LemgEemu7 ew7k ezp k [ ] o ]efv h7kLw7¤ vu7 ] 8°w k ] u7eLk ]¶giefk . DC. " , "' D("2, 4" . . A D(. 1 #" B. A. (. 1. . #" "$B. D". . '" '" '" . . .-A". '" . . 1
(227) %. % K. . . ). '" #. . '" #.
(228) . Z[. ! *%
(229) ! F ]°k o gisJ]' p uJgiªe [ gisJ]uVvlVvvzu7emk o h7kf p w ek [ ]^. áªá 4Ù K ÷ ' ø '. . . . . % p efk¹u7 vhOkvLgEem]t¤/¦§].
(230) . [ g
(231) t]°k [ ]a p p w7 o ]emh7k8.
(232) M. M M M. P(1+c). {. Pr P(1+c) (P-1)(1+c). {. Pr. {. Pr. ¬¥th o ] _ 84 ]kmv [p k [ ]s ompVp k [ ) O¤ °iÅ Å W WÅ I ½º½ ¾ ,E3W»/, ¿¾ t¼ , :½ * , 7¾ £¼03¸¾
(233) »/, 3¸¼ Q½ , .0¾ ½¼
(234) ¾@H
(235) '? -. .'¼. t¼½'3W»¼ 3W¼ E, Q»¼½¾T½ !) 3 3 , ! /3 H ´ ? B _L·* E´?> > !A> ) > · 43(*O¾9 ® 'H ¼ !z_") ! ) & ) !? ´ ! B( !!) ·W¯ *&) , ' ) ) ) ?!/ i´' ? B _L·f·"´ &B(!)·W¯ *&, ' ® '. H ! ´ ' ¼ªz´ L´_"t! · & ! ?C' !B _L· ) ¼ z"_ ! ' ! *O¾L»f¾ L´t· ®´f_"!· ! ) ) & ^gij7´f´m_ !#/· ? ' H B$? ) ! ´ ! ?·´ B ) ·m9· ! H ! ´ !M´ B _L·-?z·* ¯ *&, ' [°gEieÅk Å ) [ "]a¶g o }_]em!/k ) pt!om p [ gEuo 7Jegis omp kmv]]zefe v ptp o hO^ -w7 e vL gEw ) gE]zw7s uomp s vof]Tp emvTem]]Temu§e pto wOu7? ][sJB]_w7u7p ]k wV[ kf][ o lV¤ p ef ]twQ¤p k Z[ [e-]avL gEemp}]tom k p [ gi]zu s p omp sv]ofefp e vTpt]o eme k pt[ o gE k e ) B ´ _}! ¤ Z?z[ ·ei ]
(236) gi] u7e] k p ´ k [ ! "] ?zo em·k-CsJg H o k p!§1´ k [ ! ] o ]T?emh7·´9k 8/ B !) · ek )
(237) 7]-k []]^ gik jV^ªvhO^ h7^ e-w giv [ ! ]}]?uH pto s omp v]T?eme p}Bo _t ´¸gEw7u )) ) ´ ?z· H ! ) ) & ) ?$´!) )B)0·m·O [ ]- !" )
(238) Vk [ ]^£gijV^h7^ e-giv [ ]t]Tu pto s ofp vT]eme pto ?CB!_ ´¸gEw7u 1_.!)) !3´ ? B7
(239) "_ kª! eª^ ?pt· o ] u7 ' H vh7!Mk´ k p &u7) ] kf] ?o ^H! wO E]¹´ ? k [ B ] _
(240) p ·mw7·}"´ ]FemkB sR!gE)ºk ·f[ ·¤¤ ) OL]h7em]k [ ]eºgE^ ]u7]v p ^ s p efkm p w giLe´ EwH· k [ ]z^s giomj7pV´mp "_ !p nB omp s )0"·p emkm p w£{k p u7]v p ^ s p ef]°k [ ]z ptofp s ofp vT]eme pto 7 ?CB _ 8 ' , » ,9 , 3W¼ , 3 ,),2 * t»¼ 3 ,2 , t¼½ , ) & . 3¸¼ )3¸¼¾½ 0¾' , }¼½z¼%. ! ) ^gij7´m´f_ !#/· ? ' H B ? !) !M´ ! ?z·T´ !B)0·m· »m, ¾ E,t3W¼ ¼7¾0¼ ½ º¾3W ¼ ¾½°, ¼ %}¼½z. ¼ %. ! ? ! ´ ! _
(241) ·/? !H !M´ !M´ B _L-· ?z*· ! ¦§»m] ¾ [ g
(242) ,tt¼ ]a70¼ k p kº3Wg ¼ ¾]°½°k ¼ [ %]z. ^£ gEj?^h7^ ! t"´ gi 4Bh7] _
(243) p /· 1? k [ ]em]VhRgEwVkmkm]ek ppt k0giw k [ ]°sRg o gi]]jV]vh7kf p wQkf^H] 8 ª´ L´ t· ' ?CB _L· $ p M !) OVk [ ]ze0gi^H]-u7]v p ^ s p efkm p w]LgEu7ek p k [ ]z]jVs o ]emef p wµ´ pto s omp v]efe p}o t'· 8 0. . . . /21'". " 6. . D(. G/ *"' 6$9F: ,. 1. '-A" )(. 2. %. P-1. 2P-1. ' '(. )". *. bP-1. +" ,. .-. 0/ A?. !(. )1A )" A6. D(. @. . . ,. . . . . . ,. . . ,. . B. . . . 9. DC. . . . . . . . . . . . A D( A. . . . . . " , "' D("2, 4" . . . . (. 1. #" "$B. D". . . . .-A". . '" . . . '" . 1 #" B. . '" . . '" . . öªá þ. $ $.
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