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Determining the Idle Time of a Tiling: New Results

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(1)Determining the Idle Time of a Tiling: New Results Frédéric Desprez, Jack Dongarra, Fabrice Rastello, Yves Robert. To cite this version: Frédéric Desprez, Jack Dongarra, Fabrice Rastello, Yves Robert. Determining the Idle Time of a Tiling: New Results. [Research Report] RR-3272, LIP RR-1997-35, INRIA, LIP. 1997. �inria00073417�. HAL Id: inria-00073417 https://hal.inria.fr/inria-00073417 Submitted on 24 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. 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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(113) ¤  ‘ ”k [ ]a›™kf] o gEkm› p wu p ^£gi›˜wz›˜eg°sRg o gE—™—˜]— p ‡ o gE^œš0—˜]k    ]k [ ]aem— p sJ] p ”k [ ]k p s gEw7u Rp kmk p ^ Jp hOw7uRg o ›˜]eL¤ ‘ wk [ ›˜evLgEem]tšO¨’p© ‡temkf]uªšJ«g o kf] o šgEw7u£¬O] omo giwVkm]H®™_

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(194) |T¯mšìgEe po ]s pto kmp ]u§›˜w Z g  —™]H_tšg o ]’›˜wRgE[ vvh [ o gikm]}¤@–  ef^£pVp gE—™— o ›˜em]u  p ]e°w p k sk [ ]]£tv]pwV^ k)”^ŠhO^ w7›˜vLg giVkf› hRp gEwVu Ÿ:k p gEŸ:kmv ›˜vzp ^’›˜u7s7—˜hO]°kºkmgi›˜kf^ › p 0] w Vk o gi]zkf› spœ]p v”ž›˜emk ][ ]£v k0w7g u7o ‡t›˜km]› kwg o›˜ezv [ _ ›˜kf!#]v"kmh !/o ] [  gEeŠ7š7k p• s7›™—¶v g

(195) lµ^g g omp ]—™e¹]}¤ ‡ ‘ wFug²ef]•w7empt] o u¥š]vL• gi[ hO]emw ] p_9k ! [ []  o! [[ gEw7 uªšV7•š}k [ [ ]T]ow­op ›˜"_ em]a! om›˜pe!#e p efp}^£ o gEH—™—´¶´"•  [  ›˜v [ Bz›™´me_9!gi—˜•·fg

(196) ·lVk e¹[ k gio kh7]°gi—˜•—7km[ ›˜]—˜]w v +p —™ hO^ Bƒw7_

(197) e·vLšVgEk w[ ]° k ]ap sk0giomp—v›˜u7]ef—˜]-em]kfu²›˜^ ›™wO]-u7‡ ]omsJp ]•w7eu7]VwVhRkfgi—™lVu ¤o giakf›™wŠv

(198) gik —˜[ —˜l] •›˜k k ]žwVh7^ ] ”s v]efe eL¤ Šp Á §p Ž7 [ D p Z[ ]žgiefemh7^H[ s7km› p w H   ´ ? [ Bµ_L· om>p >i›˜ep}w7o ]]u7]Tu£k p ]Tw7[ emh o o ]zk [ ]-tgi—˜›˜u7›˜k:l p ”ªk [ ]a” p}o ^Š[ h7—¶gie

(199) o ¤ V]]z¬¥›˜‡th o ]  8 ´ w7] ”·k ]°— w7‡}]emksJgik ›˜e‡t›˜}]w l k ]°s v]Teme  ŠšV• ] ] H  ´ B _

(200) · > > š?gEw7uk ]°sRg gi—˜—˜]—kf›™^H]›˜e    ® H ! ´ CB _L·T´m"_ !/:! T· 1 ¯ ;*&+, ' $  p km]°k [ gik ^£g

(201) l  ]°^Šh7v [ em^£gE—˜—™] o k [ gE%w ? ›˜wk [ ›˜evLgEem]t¤ Z[ ]°›™uO—™]°kf›™^H]  ]Tv p ^H]e    ?Š p´ k  [ >] B o •_L›˜·Tem´m] "_ !:!$·  *&, ' › ”"_ !/!#  GE. . . GE . . . . áªá 4Ù K ÷ ' ø '.  . .

(202) ¬¥›˜‡th o ].

(203) . Processor Q.  8   ŠgEw7uEH  ?B> >.  ! *%  

(204)  !   %    2

(205)   % °iÅÅ‹WŒWÅ D  ½º½¾@,3W»/, ¿¾'-A"t¼)( , .%:½/* ,'7¾4(H¼3W¾L»-, 3W¼)" H½* , .0¾H"2,°½¼.-

(206) ¾+H

(207) )? / 1#".'3+(t¼˜½'3W»¼)1A 3¸¼)" A6 , 8(»¼˜½¾T½ !) /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 [ÅD—™hOC ^ !w o ]¿k ¤ Š‘ ´”zEom·_p  ! H ) p}o!! 7 pt´ om – B 7 p š )k ·1[ ]” ptwµo gE —™— v p  —˜h7 ^ ? w7eaœB vL_ giw  ]- k ]Š[ s]om• p ptv]om–efem— ]p uAgEu ›˜w7p u7”ª]s sJom]p w7vu7]Tem]Te w?p}kfo —˜l?¿¤ š}kZ[ [ gi]’k›˜k eLp š}kºkgE[ —]-km›˜k ^Hp kº]žgE—7›˜e°•¹‡t]T›˜›™}‡ ][ w k p kl[ k]‘ o •]›™—¶efg ]t¤ ‡}]emkas v]Teme • omp — ptgEou8/  Šp ´ t · H › ”     ) špŠgE[Ow7ptu1o  p  Š´ 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 š7k gEgE[ —™kz]’—skms ›˜^ ofpVv] p ]Ÿ¡” efefekmp ]” s9e°n  omemp sJ s ]pw77 emuQ›˜š kme› p h7wµ^ k†]°_tk [¤ ›˜u7]‘ wA—˜” ]- w7kf›˜]^ ]uO]-O eŠšJuOk h7s p ]zomp •k v]gief›˜kze p}h7oH›wVE kf›™š—/wV kºkmgi›˜ ^ —v ]0E Ÿ:^ em kf]^Šs9?ChO  w7B ›˜vL_tgišJkf´m›vL_gEw7!wœeL¤ ef kº·"g o;k°]°* &+s u7, om›˜' pemvvh7 ]efef]0emem” ››˜ptw7o w‡] sh7ofwVp kmvT›˜—]emkmef›˜›™^ wO‡Q] Ÿ¡efk km[ ]] s9o  ]Temk p ´m”¹_›˜kf! e-:v p ! —™hOi^ ·  wªšO* k &[ , gE' kž›˜e

(208) ]0šR” p}k [o ]] emo k0]g^o kfgi›˜›˜w7w7‡Q›˜w7k ‡ p Š• ´pttom· – ! ¤ !‘ )w  Rkm›˜p —˜k][ eL¤ vL‘ gE” em] e

(209) š¹ s 7 ompšJvs ]omefp e vpt]To eme Qp}o kf]  o ^H[ gE›™wJe-gik kmp ]Tez•k gi[ ›˜] k . . . . . . .

(210) . ]jV]vhOkm› p w p ”1›˜kmev p —˜h7^ w gEkkm›˜^ ] Ÿ¡efkm]s  ®˜´m_"! ! ) ·" ! H ! ¿´"  B( ) ·W¯  *&+, ' $ pt o ]sJ” pt]o w7u7p ›˜ w7‡Š? [ h7sB!o p _tw²¤ k [ ]zef›˜‡tw  p ”-´m[ _"!o  ! !)·9p !Mp ´!o B!)º· _"!#"!/[0š7k [ J›˜e#p ?hJgip}wVokm›˜k:lŠ›˜e^£gEj?›˜^Šh7^ ]T›™k [ ] o ” pto    -—˜k ‡t]k ] š7•]ƒgEemef]^ —˜]°k ] ]Temh7—˜kme ” h vLgEem]’gEwRgi—˜lVem›˜e›˜w£k ]zg t]a” ^Šh7—¶g7¤ [7pofp —™uO-eL‡Vš}gik pt›˜[ wªo ] š}p •¹[ w7]a—˜lzs p›˜u7›™wV—˜]k kmpo ›˜h7^Hk]¹k k[ [ giptgikomk– k o[ p ]]a^£v gEp ›˜wOw7u7e ›˜km›˜e› p u7wQh7p]a_ p ! k p !omk );[ [ !]- hOw  °gi—¶›˜gie¹w7kvT[p ]]uŠ– • ]p}lŠom– k p— p ^HgEu£›˜w7´ ›˜•^ ›™k › T›[ ™wOg°‡k ›™o uOgi—™sJ]a]'kmp ›˜p ^ ›™uJ[ ] gi8 —J‘ ›™p”ªkfk] [o gE›˜ekm› vp p wŠwOefu7sR›˜kmgip› vTp ]tw š sk [ ]l vT]›˜emw7e vh e o ·¤ g

(211) t]u7›] ]wVk• — gEu7eº·Tš hOkªw }] ]

(212) giuz›˜eu7h7]/k uJgikºguO]sR]Tw7u7]w7vT›™]Te-´¸gEw7uzk k ]v ^ ^Šh7wO›™v

(213) gikm› w7e.

(214) . 6   !

(215) %  6

(216)  !  F    2

(217)   % °iÅÅ‹WŒWÅ G  ½º½¾@,; ,}»/,2™¾#" t»-,$&%:½/* ,'7¾'(’¼03¸¾

(218) »/, 3¸¼)" Š½* , .0¾H"2,a½ ¼ -L¾ H

(219) ? !//.A? .™¼.G(}¼˜½'3¸»T¼)1 3W¼)" 76 , 8(H»¼˜½¾ 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 ]ƒ‚ 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_ !  !   Oš?gE—™—Jkm›˜—˜]-v p —˜h7^ wOevLgiw  ]°s ofp v]emef]u ›˜w7u7]‘ sJ”ƒ]_w7!uO]wVkm! —˜l?šJ gi w7uH7k šs ] omp v]efefk-e ptsRog ? k B ”k _] ›™ezgE]em—™•h7—˜g

(223) k)lV” e-—˜k — [ ]Š•eL—¶gi¤ km]Temk p w7]Šk p km] o ^ ›˜wRgEkm]]jV]vh7kf› p wª¤Š¦§]u7›˜emvTh7emezk [ ]Šk:• p vL?CgEemB ]e _ /| ?  B µ_ giwO$%$%u $  '?  B §_L·TefšV]sRvLgig w o gikf]]°—˜l?u7¤ ]v ‘p ” ^ /s p  ef]uQ7šgike[ ” ]Qp —˜—• p pt•omea–!´¶emp ]T”ƒ] s ¬¥omp›˜‡tvh ]oef]ze pt‚to ·'8 ? B_§´¶• [ ›˜v [ ›˜e giefem›˜‡tw7]Tu v p —˜h7^Hw7e ' ' '   ^£gij7´f_"! ' B i·T´ ? B _

(224) · , » ,9 , 3W¼   , 3 ,),2   * t»¼  3 ,2 , t¼˜½ ,  . 3¸¼ )3¸¼˜¾½ 0¾' , }¼˜½z¼%.  ?CB _ ! ) )   ^gij7´m´f_ !#/· ? ' H B ? i· »m, ¾  E,t3W¼ ¼7˜¾0¼ ½ º¾3W ¼ ˜¾½°, ¼ %}¼˜½z. ¼ %.   ? QB ´ _ !M_

(225) ·/?CB _ !H !M´ '? BF_

(226) *·  ¬ p}»mo ¾ +,t ¼ 770¼ šV k [ 3W]z¼ ˜eº¾gE½°^ ¼ %]a.u7] v p ^  s p em›˜kf› p wŠ—˜]Lgiu7ek p ´¶ef]]’¬¥›˜‡th o ] _t·8    ´m"_ !:!·´ ?CB _L· ½'3 ,t»'3 53¸¼ ¾ ! ) ) & ^gij7´f´m_ !# ! i-· ? ' H · 3W¼ ™¾T½ž0¼ %.  ! ? ! H 3W¼ ™¾T½ž0¼ %.  ! ´ B _L-· ? ‘ o p [  [ p o p pto p p [ o pto [ ]jVs o kz]emkmefh › p wQw7e ›˜eh7tgik°—˜›˜k ugE” ptkLoš J]Tp vLk gi[ h7ef]œ :_ ’! gEw7u !     7O¤ šZk [ ›˜]Šek:u7• › o ]Tv]kmjV—˜l s —˜]T]LemgEemu7› ew7k ezp ” k [ ]  o ]efv h7—˜›˜kLw7¤ v›˜u7 ] 8°›˜w k ] • u7eLšk ]Š—¶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 ›™uJgi—ªe [ gisJ]uVvlVv—˜›˜vzu7›˜emk o ›  h7kf› p w ›˜ek [ ]ž^. áªá 4Ù K ÷ ' ø '. . . . . % p efk¹u7›  vhO—™kvLgEem]t¤/¦§].

(230)   . [ g

(231) t]°k [ ]a” p —˜— p •›˜w7‡ o ]emh7—˜k8.

(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‹ WŒ WÅ I ½º½ ¾ ,E3W»/, ¿¾ t¼ ,  :½ * , 7¾ £¼03¸¾

(233) »/, 3¸¼ Q½ , .0¾ †½¼

(234) ¾@H

(235) '? -. .'˜¼. t¼˜½'3W»¼ 3W¼  E,  Q»¼˜½¾T½ !) 3 3 ,  ! /3 H  ´ ? B _L·* E´?>  > !A> ) > · 43(*O¾9 ® 'H ¼ !z_") ! ) & )  !?Š ´ ! 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·*  ¯  *&, ' [°gEieÅk Å ) [ "]a—¶g‘ ”žo ‡}_]em!/k• ) pt!om– — p [ gEu’o 7›˜šJegis —˜om—p kmv›˜]—˜]zefe v ptp o —™hOƒ^ › ”-w7 e vL gEw  )  gE]zw7s uŠomp s vof]Tp emvTem]]Temu§e pt›™o wOu7? ][sJBœ]_w7u7p ]k wV[ kf][ —™o lV•¤ p ›™ef‘ ]twQ¤p k Z[ [›˜e-]avL• gEemp}]tomš– —k p [ gi]zu s p om”p sv]ofefp e vTpt]o eme k pt[ o gE k ›˜e  )   Š B ´  _}! ¤ Z?z[ ·›˜eši• —™]

(236) gi] u7e] k Šp ´ k  [ ! "] ?zo em·k-CsJg H o k p!§”1´ k  [ ! ] o ]T?†emh7·´—˜9k 8/ B › ”  !)  · ›˜ek )

(237) š7]-k [•]ž]›˜^‡ gik jV›˜^’”ªvhO^ —˜h7›˜^ e-w giv  [ ! ›˜]}]?ŠuHš ” pto s omp  v]T?eme p}Bœo _t š ´¸gEw7u )) )  Š´ ?z·  H ! ) ) & )  ?$´!) )B)0·m·šO• [ ›˜—˜]-› ” !" )

(238) šVk [ ]†^£gijV›˜^h7^ ›˜e-giv [ ›˜]t]Tu ” pto s ofp vT]eme pto ?CB!_ ´¸gEw7u‘ ”1_.!))  !3Š´ ? B7š

(239) ›˜"_ kª! ›˜eª^ ?†pt· o ] u7 '›  H vh7!M—˜k´ k p &u7) ] kf] ?o ^H! ›™wO E]¹´ ? k [ B ]— _

(240) p ·mw7·‡}"´ ]FemkB sR!gE)ºk ·f[ ·¤¤ ‘ ” )  OšL•]h7em]k [ ]eºgE^ ]u7]v p ^ s p ef›˜km› p w giLe´ ›˜EwH· k [ ]z^s giomj7pV´mp "_ ” !p ”  nB omp s )0"·p  em›˜km› p w£{k p u7]v p ^ s p ef]°k [ ]z• ptof–Šp ”s ofp vT]eme pto 7š     ?CB _ 8 ' , » ,9 , 3W¼   , 3 ,),2   * t»¼  3 ,2 , t¼˜½ , )  & . 3¸¼ )3¸¼˜¾½ 0¾' , }¼˜½z¼%.   ! ) ^gij7´m´f_ !#/· ? ' H B ? !) !M´  ! ?z·T´ !B)0·m· »m, ¾  E,t3W¼ ¼7˜¾0¼ ½ º¾3W ¼ ˜¾½°, ¼ %}¼˜½z. ¼ %.  ! ? ! ´ ! _

(241) ·/? !H !M´  !M´ B _L-· ?z*· ! ¦§»m] ¾ [ g

(242) ,tt¼ ]a70¼ k  p kº3Wg ¼ –˜¾]°½°k ¼ [ %]z. ^£ gEj?›˜^h7^ ! t"´ gi 4—˜Bh7] _

(243) p /· ”1? k [ ]em]VhRgEwVkm›˜km›˜]ek pŠpt k0gi›˜w k [ ]°sRg o gi—˜—˜]—]jV]vh7kf› p wQkf›™^H] 8   ª´ L´ t· '  ?CB _L· $  p •M› ” !) OšVk [ ]ze0gi^H]-u7]v p ^ s p ef›˜km› p wŠ—˜]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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