Impact of Mixed–Parallelism on Parallel Implementations...

34
ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE Impact of Mixed–Parallelism on Parallel Implementations of Strassen and Winograd Matrix Multiplication Algorithms Fr´ ed´ eric Desprez, Fr´ ed´ eric Suter No 4482 June 2002 TH ` EME 1

Transcript of Impact of Mixed–Parallelism on Parallel Implementations...

Page 1: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

ISS

N 0

249-

6399

ISR

N IN

RIA

/RR

--44

82--

FR

+E

NG

ap por t de r ech er ch e

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Impact of Mixed–Parallelism on ParallelImplementations of Strassen and Winograd

Matrix Multiplication Algorithms

Frederic Desprez, Frederic Suter

No 4482

June 2002

THEME 1

Page 2: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT
Page 3: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

������������ �������������������� �!�!���!�!"�� ��#$������� �!�!����%���&�'�(�)��# ��*+�!��#&"�� -,./�0��"/"+��#$��#&�21 �'#3��4������$���*/�0���

��53����6�&�'�!�7��+�!��#98:�!4����0��+;��)"

<>=@?BA0?C=EDGFIH&JBK@L0=EJCMONP<>=@?BA0?C=EDGFRQ+S+TUJC=

VXWZY'[�\�]_^a`�bEcd\Eegfihj\!klcdmicnkoY'[�\Ecprqosutn\'k(`�\EvIegp

`(egwxwOsBqyk(zi\�qo\E{�WZ\'q�{�Wx\}| ~u�B�B�C�R^��gfZ|x\ �u�B�B�_^��x]*wZeu�B\Ec

�j���E�'���x�g�g�_� |�kdWx��crwOeuw�\'q�P\lcykdfZzxm koWx\(��[.wOeg{6k7su��kdWx\_cy��[ fZ��k�eu|x\EsgfZc>\'hiwx��sB��k�e�ko��sB|.sg��zxe�k�eU��eu|Ozk�egcd�B�CwZegqoeg����\'����cd[:sg| �Ckdq�egcocd\'|¡eu|Zz£¢¤��|Zsg�gq�egzR[¥e�kdqo��hR[ fx��kd��wx����{Ee�ko��sB|¡eg���Bsgqo��koWx[¥c'¦.¢§\¥wxqd\@cy\E|Ckkn�PsR[���hi\EzC�¨wZeuq�eu����\E���[�wx��\E[�\'|Ckoeukd��sg|ZcE¦©VXWx\3�ªsgqo[.\Eq �ªsB����s��(c kdWx\&wxWZeBcy\@c sg�(kdWx\&sgqo���g��|Zeu�Xeu���gsg«qo��kdWx[¥c_��Wx����\.kdWZ\.��e�kdkd\Eq_WOegc_¬O\E\'|¡zi\Ecd���g|x\@zRegc­koWx\�qd\@cyfx��k*su�Pe3����cyk�cd{�Wx\@zifx����|Z�&eg���Bsgqo��koWx[�¦�¢£\�B��®B\ e¥koWx\'sBqd\'kd��{'eg��{'sg[�wZeuqo��cysB|�¯O��|°ko\'qo[�c_sg�r[.\E[�sgqom&fOcdeg�g\ eg|Zz°\!hi\@{!fiko��sB|Ikd��[�\g¯�¬�\!kn�P\'\'|£sgfxqeg���Bsgqo��koWx[¥c�eg|Zz±{'��eBcdcd��{'eu�Pzxe�k�eU�¨wZeuq�eu����\E���[�wx��\E[�\'|Ckoeukd��sg|ZcE¦±VXWx��c�eu|Zeg��micd��c���c�{'sgqoqdsB¬OsBqoeukd\Ez§¬¨m\'hiwO\Eqd��[�\'|CkocE¦>²+��|Zeg����mj�P\*�g��®g\*cysB[.\*Wx��|Ckoc(eg¬OsBfikleu|&Wx\'kd\'qosg�B\'|x\EsgfZc7®g\Eqocd��sg|jsg��sBfxq(eu���gsBqd��kdWZ[�cE¦³°´Cµ�¶o· ¸ ��¹��g� vI��hi\Ez¨�CwZegqoeg����\'����cd[�¯¨[�eukdqo��h3wxqdsizifO{6kE¯��Ckdq�egcocy\E|�¯¨¢¤��|Zsg�gq�egz�¦

º�»%¼d½¿¾EÀX¼�Á�Â�½¿ÃGÄ!Å

Æ�ÇUÈÊÉrËGÌÎÍ'ËXÈÊÉ7Ï�ÐÊÉGÑ�ÏnÒ6Ï�ÈÊÐÓÏ�ÔUÐÊÌ­Ï�ÉPÏ�ÕÖÌnÉGÌyÏ�ÕØ×GÇjÕÖÌnÙuÑ6ÕÖË�Ñ�Ú�ËØÇUÌlÛiÏ�ÔuÑ6ÕGÏoËØÑ6ÈÊÕØÌ_ÜUÌ(Ð�Ý Þàß@Ú�Ñ6ÕÖá(ÏoËØÈÊâEãUÌ(ÜUã�äZÏ�ÕØÏ�ÐÊÐÊånÐÊÈÊÉGá�Ìæ!ç6ç�èZéªê6êdë6ë6ëZìàíoî@ï'ðdñoò'ódîZì�ô6õ'êoöU÷døUù

Unit e de recherche INRIA Rhone-Alpes655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN (France)

T el ephone : 04 76 61 52 00 - International : +33 4 76 61 52 00T el ecopie : 04 76 61 52 52 - International : +33 4 76 61 52 52

Page 4: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

�%���&� �r��&5 �����0���!� � �!�'"+�)� �)����+� "/53� �&��" �'����� � �)��#�+�*+�'��#3"�&���0���!�����!��" �&��" ���!4��������;&�)��" �&� �3�0�¥�35&��+"��3� �)�*/�0�!�P��" �&�

,./�0��"/"+��# �P 1 �!#&��4��0���

��� ��� � ��� eu|Zc�{!\'k0egqyko��{'��\r|ZsgfZc+b!kofZzi��sg|Zc��� ��[�wZeg{!k�zi\P�� fiko������coe�ko��sB| cd��[�fx��k�eu|xbE\Pzxf�wZeuq�eu����b'����cd[�\zx\>k��B{�Wx\Ec/\!k+zif wZeuq�eu����bE����cd[.\rzx\zisg|Z|xb'\@c/cdfxq���\Ec/eg���Bsgqo��koWx[�\Ec�zx\wxqosizifx��k+zi\r[¥e�koqd��{!\@c/zi\P�Ckdq�egcocy\E|\'kr¢¤��|Zsg�gq�egz�¦��(sBfZcrwZqdb@cy\E|Bkosg|Zc0zi\Efih���[�wx��b'[�\'|Ck�e�kd��sg|Oc��*wZeuq�eu����b'����cd[�\P[���h¨kd\g¦��/e*wZqd\E[.��Y'qo\(cdfx��k��\Ec­wxWZeBcy\@clsBqd���g��|Zeg��\@c(zi\Ec_eu���gsBqd��kdWZ[.\@c�k�eu|Zzi��c��BfZ\���ejcd\E{'sg|Zzi\�e�b!kob.{'sg|��'fx\�{'sg[�[�\ ��\ qobEcdfx��koe�kz� fx|©eg���Bsgqo��koWx[�\¥z� sBqozxsg|x|Zeg|Z{!\E[�\'|Ck�wZeuq�����cykd\B¦��lsgfZc�zisg|x|Zsg|Zc*fx|Z\j{!sB[.wOeuq�eu��cysB|IkdWZb'sgqo���BfZ\����e£�ªsg��c�\'| kd\Eqd[�\@c¥zi\I{!sB|ZcysB[�[�eukd��sg| [�b'[�sg��qd\ �Cf� \'|¤{!s"!ikjzi\°{'eg��{'fx�دP\'|Ckoqd\�|xsCc�eu���gsBqd��kdWx[�\@c\'k�zx\Ec3��[�wx��b'[�\'|Ck�e�kd��sg|Oc&zZe�koeu«GwOeuq�eu����YE��\@c�{!��egcocy���Cfx\EcE¦$#7\'kyko\§eg|Zeu��micy\£\Ecyk&{!sBqdqosg¬�sgqob'\°wOeuq�zi\Ec\'hiwObEqd��[�\'|Ckoeukd��sg|ZcE¦&%r|('O|�¯+|ZsgfZc�zxsg|x|xsB|Zc)�Cfx\E�*�Cfx\@c�cyfZ�g�g\@cnko��sB|Zc_wOsBfxq�fx|Z\�®g\Eqocd��sB|RWxb'kdbEqdsB�gY'|Z\zx\_|ZsBc�eu���gsBqd��kdWZ[.\@c'¦+ ¸ �@� ¶ �-, � � p>egqoeg����b'����cd[�\­[���h¨kd\g¯ZwxqdsizifZ��klzi\�[¥eukdqo��{'\EcE¯i�¨kdq�egcocy\E|�¯i¢¤��|Zsg�gq�egz�¦

Page 5: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

������������ �������������������� �!�"���#�%$&� �

' ��#�/�0�¥�&53�+�!��#

p>egqoeg����\'�/co{!��\'|Cko� 'O{ euwxwx����{'e�ko��sB|Zc({Eeu|°¬�\�zi��®¨��zi\@z���|Ikn�7s¥[¥e�tnsBql{!��egcocd\Ec)(�����*����eg|Zz+*� $�, ���-����� �!�.��egwxwx����{Ee�ko��sB|Zc'¦£VXWx\��ªsBqd[�\'q�{!sB|Zcd��cykoc ��|±egwxwx��m¨��|x�RkdWx\&cdeg[�\jsgw�\'q�e�ko��sB| ��|±wZeuq�eu����\E�sB|±zi�./�\Eqd\E|Bk\E��\E[�\'|Ckoc�su�­e£zxe�k�e£cd\!k@¯r��Wx����\3kdWZ\3��e�kdkd\'q.��c.zx\ 'Z|x\@z kos£¬�\&{'sg|Z{'fxqoqd\E|Bk.{!sg[�wxfxkoe�ko��sB|Zc�sg| zx���à«�ª\Eqd\E|Ck¥zZe�koe cd\!k�c'¦-VXWZ\Ecd\�kn�7s {!��egcocd\Ec�{'eg| ¬�\R{!sB[ ¬x��|x\@z©kos �g\!k�e cd��[�fx��koeu|Z\'sgfOc�\!hiwx��sg��koe�ko��sB|sg�zxe�k�eU�°eu|Zz�k�egcd�B�CwZegqoeg����\'����cd[�¯Ocds3{Eeu����\@z0�1�����2� ���-����� �!�"���#�%$&�.¦ � |£[.��hi\EzC�¨wZeuq�eu����\'�+egwxwx����{Ee�ko��sB|Zc'¯cd\'®B\'q�eu�lzZe�koe��CwZegqoeg����\'�({'sg[�wxfik�e�ko��sB|Zcj{Eeu|¤¬�\R\!hi\@{!fiko\Ez-{'sg|Z{'fxqdqo\'|Cko��m ��|�e koegcd�B�CwZegqoeg����\'���XeUmg¦vI��hi\Ez¨�CwZegqoeg����\'����cd[ wxqdsB�gq�eu[�[���|Z�¥\'[�wx��s�m¨c_e&v&�i�¨p7v �43 v°fx��kd��wx��\��¨pv �65 cnknm¨��\ ��Wx��{�WR��c­kdWx\{'sg[�¬x��|Oe�kd��sg|�sg�0¬�sukoW�koeBcy�B�CwOeuq�eu����\E����cy[ 3 v°pv ��5 eu|Zz°zZe�koeu«GwOeuq�eu����\E����cy[ 3 �¨pv ��5 ¦%�ifZ{�WIeu|�\!h¨«wZ��sB��k�e�kd��sg|¥sg��[���hi\Ez¨�CwZegqoeg����\'����cd[ WZegc[�eg|¨m.eBzi®�eu|Ckoeg�g\EcE¦87­|Z\lsu�%koWx\'[ ��c>koWx\*eu¬x������knm�kds ��|Z{'qd\@egcd\co{'eg��eg¬x������knm�¬�\E{EeufZcd\l��k�eu����s��(crkdWx\_fZcd\lsu�+[.sBqd\lwZeuq�eu����\E����cd[ ��Wx\E|�koWx\­[¥euh¨��[¥eu��eu[�sgfZ|BkPsu�/zxeukoe��sBqjk�egcd�B�CwZegqoeg����\'����cd[ kdWZeuk�{Eeu|�¬�\¡\!hiwx��sg��kd\@z-��c3qo\EeB{�Wx\Ez�¦:9 �BsCsiz�s�®g\'qo®¨��\E�:su��koWx��c&kosgwx��{£��c�B��®B\'|3��|<; � =G¦rvIsBcykPsg�/kdWZ\�qd\@cy\@euq�{�Wx\Ec7eu¬�sgfikXkdWZ\�cy��[ fx��koeg|x\'sBfZcP\!hiwx��sg��koeukd��sg|jsg��zxeukoe��¥eg|Zz�k�egcd�B�wOeuq�eu����\E����cy[�WZeU®g\�¬�\'\E|Rzisg|Z\���|�koWx\�egqd\@e�sg�>wxqdsB�gq�eu[�[���|Z�.��eu|Z�gfZeg�g\EcPkos��g��®g\�cd��[.wZ��\�WZ���BW&��\'®B\'�eB{'{'\Ecocy\@c/kos*[.sBqd\XwZegqoeg����\'����cy[ eg|Zz���|.kdWZ\(euqo\Ee_su��{!sg[�wx����\'q�c'¯u��WZ\'qo\PwZqdsB¬x��\E[¥c>cyfZ{�W¥eBc>cd{�Wx\@zifx����|Z�eg|Zz¥eu����s¨{Ee�ko��sB|.sg��{!sg|O{!fxqoqd\E|Ckzxe�k�eU�CwOeuq�eu����\E�¨koegcd�icreuqo\(cykdfZzi��\Ez>;�]@?�=G¦ � |>;�]E��=¿¯¨`(eu[¥egcd�Xeu[ m ��|Ckdqosu«zxfZ{!\@c7e�cnkoqdfZ{!kdfxqo\(kos�zi\@cd{'qd��¬�\l[���hi\@zC�CwZegqoeg����\'�Owxqosg�gq�eu[¥c)(%kdWZ\_vIeB{!qos � e�koe�AZs��CB_q�euwxW 3 v � B 5 ¯ejzi��qo\E{!k_eB{!mi{!����{ �gq�euwxW���Wx\Eqd\�|xsizi\Ec­qo\'wxqo\Ecd\'|Ck­cd\��Cfx\E|Bko��eg�/sBq­zxeukoeU�¨wZeuq�eu����\'�/{'sg[�wxfik�e�kd��sg|Ocleu|Oz\@zi�g\@c�qd\Ewxqd\@cy\E|Ck�wxqo\E{'\Ezi\E|Z{!\�{'sg|Zcykdq�eu��|CkocE¯�����koW kn�Ps�zi��cykd��|x�Bfx��cdWx\@z |xsizi\@c'¯gsg|x\Xwxqo\E{'\Ezi��|x�*eu|Zz�sg|x\cdfZ{E{!\'\@zi��|x��eu���OsgkdWx\Eqr|xsizi\@c'¦D7­|Z{'\�kdWx\_v � B ��c\!h¨kdq�eg{!kd\Ez��ªqosg[ kdWZ\l{!sizi\B¯Ce*kn�Ps cykd\Ew�eu���gsgqo��kdWx[��c�euwZwx����\Ez3kos.wZ��eB{!\�eu|Zz�co{�Wx\Ezxfx��\_k�egcd�¨cXsB|jkoWx\�{!sB[�wxfikd��|x��qo\Ecdsgfxq�{!\@c'¦

7­| e cd\E{'sg|Zz WOeu|Zz�[�eukdqo��h [�fx��ko��wZ����{'eukd��sg|���c kdWx\ �g\Eqd|Z\'�&su�R[¥eu|¨m co{!��\'|Ckd� 'O{ euwxwx����{'e�«ko��sB|Zc6;�])Ei¯Z�x]&=/eu|Zz&cd\'®g\Eqoeg��wZeuq�eu����\E����[�wx��\'[�\'|Ck�e�kd��sg|OcXWZeU®g\_¬O\E\'|&wxqosgw�sBcd\Ez%¯¨[�sBcyk�su��kdWx\E[ fZcd��|x�zZe�koe��CwZegqoeg����\'����cy[�¦�¨sB[.\�eu���gsgqo��kdWx[¥c�cdfx¬Zcykd��kdfiko\�[ fx��kd��wx����{Ee�ko��sB|Zc�¬¨m3eBzxzi��kd��sg|Zc­eu|Zz3kdW¨fZc(qd\@zifZ{!\koWx\j|¨fx[�¬O\Eq sg�X[�fx��kd��wx����{'eukd��sg|Zc.{!sB[�wxfikd\@z%¦R�CkoqoeBcdcd\'|F;Ê�g�@=�eu|Oz¡¢¤��|xsB�gq�egzG;�]@�@=�euqo\�cyfO{�W±eu���gsg«qo��kdWx[¥c.kdWOe�kjeuqo\�¬O\@cnkjcdfx��kd\@z±�ªsBq�e§wxqoeB{6ko��{Eeu����[.wZ��\E[.\E|Ckoe�ko��sB|�¦-VXWx\Em©WZeU®g\�¬�\'\E| \'h¨kd\'|Ocy��®g\E��mcykdfOzi��\@zjsB|j[�sg|Zsgwxqosi{!\Ecocdsgq[¥eB{�Wx��|Z\Ec7kds���|Z{!qo\EeBcy\lkdWx\�{'sg[�wxfik�e�kd��sg|Oeu��wO\Eqy�ªsBqd[¥eg|Z{!\@csg�+|¨fx[�\'qo��«{Eeu�ieuwZwx����{'eukd��sg|ZcH;�]u¯u�i¯i]E�x¯i]@I¨¯g�g��=¿¦0�¨\E®g\Eqoeg�BwZegqoeg����\'�C��[.wZ��\E[.\E|Ckoe�ko��sB|Zc0su���Ckdq�egcocy\E|�eg|Zz�¢¤��|xsg�BqoeBzeg���Bsgqo��koWx[¥c0WZeU®g\�¬�\'\E|¥wxqosgw�sBcd\Ez%¦ � |�; I�=G¯¨zxe�k�e�«¿wZeuq�eu����\'�i��[.wZ��\E[.\E|Ckoe�ko��sB|Zcrsg�/�Ckdq�egcocy\E|.eg|Zz�¢¤��|xsu«�BqoeBz*sg| e_� � v � [¥eg{�Wx��|x\Peuqo\rwxqo\Ecd\'|Ckd\@z%¦ � |J; ?�=¿¯ueufxkdWxsBqoc/wxqd\@cy\E|Ck/koeBcy�C«¿wZeuq�eu����\'�u��[�wx��\E[�\'|Ckoeukd��sg|Zcsg���¨kdq�egcocy\E|£eu���gsBqd��kdWZ[ c*sg|§qo��|x�Cc*su�Xwxqos¨{'\EcocysBqoclkdWOe�k�w�\'qd�ªsgqo[�c*¬�\!kdkd\Eq�kdWZeg|§kdWx\�{'��eBcdcd��{Eeu�>qo��|Z�eg���Bsgqo��koWx[�¦K9 [���h¨\@zjwOeuq�eu����\E�O��[�wx��\E[�\'|Ckoeukd��sg|jsg�+¢¤��|xsg�BqoeBz¥��cXwxqd\@cy\E|Ckd\Ez���|<;�]B]&=¿¦rVXWx\_{'��eBcdcd��{'eu�wOeuq�eu����\E��eg���Bsgqo��koWx[ ��cXfOcy\@z3eBcPe��g\Eqd|Z\'���ªsgq�e�koeBcy�¥wZegqoeg����\'��®B\'q�cy��sg|�sg�+¢¤��|xsg�BqoeBz¥sg|&W¨m¨wO\Eqy«¿�gqo��zZc����kdWML�N.wxqos¨{'\EcocysBqocE¦DOPsukoW°��[�wx��\'[�\E|Bk�e�ko��sB|Zc(sg�P;�]E� =�eu|Oz+; ��=�fZcd\ zxe�k�e�«¿wZeuq�eu����\'��eu���gsBqd��kdWZ[�cX����kdW\Q�{!��\'|Ck.zxeukoeu«¿zi��cykdqo��¬xfxkd��sg|ZcE¦I²���|Zeg����mg¯r`leu[¥egcd�Peg[ mR;�]E� =PfZcd\Ec��CkoqoeBcdcd\'|©eBc e°¬�\'|Z{�WZ[�egqd�¡{!sizi\�ªsBq�koWx\jp>euq�egzx���B[ wZegqoeg����\'���.S@e�ko��sB|£kos¨sg�G¦£v°��hi\Ez¡wZegqoeg����\'����cd[ ��c sg¬ik�eu��|x\Ez¡koWxqosgfx�BW©e�kn�7s£cnko\'wZcegwxwxqosBeB{�W 3 {!sB[�wxfikoeukd��sg|&su��koWx\*|¨fx[ ¬�\'qXsu��wxqos¨{'\EcocysBqoc|x\'\@zi\Ez3eu|ZzjkdWZ\'|�cd{�Wx\@zifx����|Z��sg��kdWx\*wZegqy«eg����\'��koegcd�ic 5 ¦VXWx\�zxeukoe�zi��cnkoqd��¬xfiko��sB|���c({�WZsBcd\'|�¬¨m¥kdWx\�kds¨sg�/eg|Zz&|xs¥����¬xq�euqom¥��c�fZcd\Ez3�ªsgq(��s�����\'®B\'��B\'qo|x\'��cE¦

7­fxq7[�sukd��®�e�ko��sB|���crkds�¬xfx����z¥\Q�{!��\'|Ck7wZegqoeg����\'�Oeu���gsgqo��kdWx[¥c0�ªsBq7{!����\'|Ckn�icy\Eqd®B\'qegwxwx����{Ee�kd��sg|Oc'¦�¢£\koW¨fZceBcdcdfx[�\XkdWZeuk[¥e�koqd��{!\@creuqo\(zi��cnkoqd��¬xfiko\Ez�sg|�zx��cÖtnsg��|Ck�gqo��zZc>su��wxqos¨{'\EcocysBqoc0¬�\E{EeufZcd\�su�%wxqo\'®¨��sgfZc{'sg[�wxfik�e�ko��sB|Zc'¦���\'k7{!sB[�wxfikd\(kdWx\(sBwO\Eqoeukd��sg|UTWVFXZYj¯B��WZ\'qo\ZX-eu|OzUY euqo\Pkn�7s�c��CfZeuqo\�[¥e�kdqo��{!\Ec

[\[ ß^]_&_�`&a

Page 6: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

� ����� ��$ � ����������� � *��sg�­zi��[�\'|Zcd��sB|�� zi��cykdqo��¬xfikd\@z©sB|±kn�Ps£cd\'wZegqoeukd\&c �CfZegqd\3�gqo��zxc.sg�lwxqos¨{'\EcocysBqoc su�­kdWx\�cdeg[�\&cd�.SE\g¦vIsgqo\'s�®g\Eq T ��c�zi��cykdqo��¬xfxkd\Ez�sB| kdWx\Rcoeu[�\°�Bqd��z�egc X ¦ VXWx\��y{'��eBcdcd��{Eeu���3�XeUm©kds©{'sg[�wxfiko\°kdWZ��cwZqdsizifZ{!k���c­kds°eg�����g|£[�eukdqo��{'\Ec*eu|OzIkoWx\'|¡{'eg���>eu|§\�Q�{'��\E|Bk�wZegqoeg����\'�0qosgfiko��|Z\g¯%fZcd��|Z�&e&����¬xq�euqom������g\�x{'e-�\9(p 9 #�� sgq�eIzxeukoe��CwZegqoeg����\'�0��[.wZ��\E[.\E|Ckoe�ko��sB|¡su�(�Ckdq�egcocy\E| eu���gsgqo��kdWx[�¦ 9Pk�koWx\�\E|Zz§su�XkdWx\{'sg[�wxfik�e�ko��sB|�¯¨kdWx\�qo\Ecdfx��k���c 'O|Zeu����mjqd\@zi��cykdqo��¬xfikd\@z%¦ � |&kdWx��c�wZegwO\Eq��7\*wxqosgw�sBcd\�e�[���hi\Ez3�PeUm¥koWZe�k�B\'\EwZc�eu���P[�eukdqo��{'\Ec���|©wx��eg{'\jeu|Oz¡\'®g\E|x��m§zi��cykdqo��¬xfxkd\Ec�koeBcy�ic 3 \g¦ �Z¦�¯>egzxzi��kd��sg|Oc���eg|Zz¡wxqosizifZ{6k�c�sB|[¥eukdqo��h �CfZeuqdkd\Eqoc 5 sB| kdWx\3kn�PsR�gqo��zxc'¦£¢£\&fZcd\3e§cy\ �Cfx\'|Ckd��eu�7��eg|x�gfOeu�g\j�����g\ # sgq.²xsgqdkdq�eu| ����kdWWZ���BWi«Gw�\'qd�ªsgqo[¥eu|Z{'\�����¬xqoegqd��\Ec*�����g\j�i{Ee-� 9lp 9 #�� ; E�=Peg|Zz§��k�c�egcocds¨{'��eukd\@z£{'sg[�[ fx|Z��{Ee�kd��sg|§����¬xq�euqomg¯koWx\ZO � 9 #X�%¦Bv°��hi\EzC�¨wZeuq�eu����\E����cd[ ��ceuwZwx����\Ez.kos��CkoqoeBcdcd\'|¥eu|Oz�¢¤��|xsB�gq�egz�zi\@{!sg[�w�sBcd��ko��sB|ZcE¦�/s��7\Eq��\'®B\'�>wxqosizifZ{!koc�euqo\�{!sg[�wxfxkd\Ez£fZcd��|Z�3koWx\�cykoeu|Ozxeuq�zIWx���gWi�Cw�\'qd�ªsgqo[�eg|Z{!\.��\'®g\E�r�jp O �\9­�£qosgfiko��|x\����������� ¦VXWx\§qo\'[¥eu��|Zzi\Eqjsu��koWx��cjwZegwO\Eq3��c3sgqo�Beu|Z�.SE\Ez¤egc��ªsg����s��(cE¦ �i\E{6ko��sB| � qo\E{Eeu����c&�Ckdq�egcocd\'|-eu|Oz

¢¤��|xsB�gq�egz�eg���Bsgqo��koWx[¥c'¦ � |¥�¨\@{6kd��sg|��x¯��P\Pwxqd\@cy\E|Ck+kn�Ps­sBwikd��[��.SE\Ez.zxe�k�eU�¨wZeuq�eu����\E�g��[�wx��\E[�\'|Ckoeukd��sg|Zcsg�%kdWx\@cy\_eu���gsBqd��kdWx[¥cE¦�VXWZ\'|���|3�¨\@{6kd��sg|j�Z¯¨�7\lwxqd\@cy\E|CkkdWZ\l��[.wZ��\E[.\E|Ckoe�ko��sB|3{�WxsB��{'\Ec��\@egzi��|x��fZckdskoWx\lzi�./�\'qo\'|Ckr[���hi\Ez¨�CwZegqoeg����\'�i®B\'q�cy��sg|ZcE¦��i\E{6ko��sB| I_�g��®g\Ece_kdWx\Esgqo\!kd��{'eg�x\'®�eg��fZeukd��sg|.su��kdWZ\lzi�./�\Eqd\E|Bkeg���Bsgqo��koWx[¥cX��|Ikd\'qo[¥c(sg�0[�\E[.sBqdm3fZcoeu�g\�eg|Zz�\!hi\@{!fiko��sB|3ko��[�\B¦(VXWx\'|I��|R�¨\@{6kd��sg|>E��7\��B��®B\ cysB[�\WZ��|CkocXeg¬OsBfikXeg|jWx\'kd\EqdsB�g\'|Z\'sgfOcr®g\Eqocd��sg|¥su�+sgfxqX[���hi\@zC�CwZegqoeg����\'�O��[�wx��\'[�\E|Bk�e�ko��sB|Zc'¦�%>hiw�\'qo��[�\E|Bk�eu�qo\Ecdfx��kocXkoWZe�k_{!sgqoqosg¬�sgq�e�kd\­sgfxq(kdWx\Esgqo\!kd��{'eg�/cykdfZzim�euqo\�cyWZs���|&��|R�¨\@{6kd��sg| �i¦7²���|Zeu����m�eu|Oz3¬�\!�ªsBqd\�e{'sg|Z{'��fOcy��sg|�¯¨�P\�\!hiwx��eu��|3��W¨m��P\*sg|x��m�fZcd\�sg|Z\_qo\E{'fxq�cy��sg|&cnko\'w�¦

! ,�+����"+"/��# ��#3�21 �!#&��4��0���$8:�!4��������;&�)"

� |¤]@?�E ?x¯+�CkoqoeBcdcd\'| ;Ê�g�@=P��|CkoqdsizifZ{'\Ez eg|¡eg���Bsgqo��koWx[ kos°[ fx��kd��wx��m"�$#%� [¥e�koqd��{!\@c���Wx��{�W WZegc�e��s��P\'ql{'sg[�wx��\!hi��knm&koWZeu|°koWx\�{!��egcocy��{'eg��&(')�+*�,6¦*VXWx��c­eg���Bsgqo��koWx[ eg|Zz°��koc_v � B euqo\�wxqo\Ecd\'|Ckd\@z°��|²����gfxqo\�]g¦ � k���c�¬Zegcd\Ez£sg|¡e�cd{�WZ\'[�\.�ªsgq�koWx\¥wxqos¨zxfZ{6k�su�Pkn�Ps.-/# -3[¥e�koqd��{!\@c*��Wx��{�W¡��|¨®gsg��®g\@c0�[�fx��kd��wx����{'eukd��sg|Zc�eg|Zz£]E��egzxzx��ko��sB|ZcX��|Ocnko\Eegz&sg�/kdWZ\�fZcyfOeu���.[ fx��kd��wx����{Ee�ko��sB|Zc(eu|Ozj�¥egzZzi��ko��sB|ZcE¦� ���7\_{!sg[�wxfxkd\­kdWx\­kdsgkoeu��|¨fx[�¬O\Eq7sg�+euqo��koWx[�\!ko��{­sgw�\'q�e�ko��sB|ZczisB|x\_fZcy��|x���Ckdq�egcocy\E|jeu���gsBqd��kdWZ[&¯

koWx\(�¥[ fx��kd��wx����{Ee�ko��sB|Zc(sB| �CfZeuqdkd\Eqoc�¬�\'��|x�&{!sg[�wxfxkd\EzI����koWIkdWZ\�kdq�egzx��ko��sB|Zeu��eu���gsgqo��kdWx[�¯��7\�WOeU®g\�(L�')-1')�324-5, *�6 ')�324-5,879,;:=<?>@')�32A-A,87�VB' LA2DC�,E� * :F'8<5<�2DC�,G�+7U¦�VXWx\q�e�ko��sl¬O\'kn�7\E\'|�koWx��c�{!sB[�wx��\'hi��knmeg|Zz�koWx\�{!sB[.wZ��\'h¨��knm*qd\ �Cfx��qo\Ez ¬¨m*kdWx\X{!��egcocy��{'eg�Ceg���Bsgqo��koWx[ kd\E|Zzxc�kds��Xeuq�zxc8LA2D>l��Wx\'|H�)�B\!koc+��euqo�g\g¦VXWZ��cX��[�wx����\Ec�kdWZeuk��ªsgq(cdf Q�{!��\'|Ckd��m���euqo�g\*[¥e�kdqo��{!\EcE¯CkdWx\Eqd\���c�e�kdWx\Esgqo\!ko��{Eeu�%�Beg��|&su��<I-�J K�L�¦VXWx��c�cd{�WZ\'[�\&{Eeu| ¬O\�\EeBcy����m¡egwxwx����\@z kosM-.#"-R¬x��s¨{��¡[¥e�koqd��{!\@c'¦©V+s¡{!sg[�wxfxkd\3kdWx\�wxqosizifZ{6k

T VCXZY�¯Z���^X}eu|Zz Y euqo\�zi��cnkoqd��¬xfiko\Ez&��|�c��BfOeuqo\_¬Z��si{��icPsu�>zi��[�\E|Zcy��sg|N�324-x¯i�7\*WZeU®B\�(O TQPRP TSP 7T 7 P T 7E7MT VO X0PEP X0P 7X 7 P X 7E7%T O Y0PRP YUP 7Y 7 P Y 7E7 T

�CkoqoeBcdcd\'|�eg���Bsgqo��koWx[ {'eu|�eg��cds�¬�\lqo\E{'fxqocd��®g\'��m�egwxwx����\@z�sg|jc��CfZeuqo\�[¥e�koqd��{!\@crsu�/zx��[�\'|Ocy��sg|/� V-AV�kos 'Z|Zeg����m¡sg¬ik�eu��|©eR{'sg[�wx��\!hi��knm§su�W&('X�ZY [E\I]_^R`G,1Va&('X� 7;b cEd ^I,!¦¡�¨\E®g\Eqoeg�rsukoWx\'q�®Uegqd��e�ko��sB|Zc�sg�koWx��cleu���gsBqd��kdWZ[ eu����s���kos�WZeu|Ozi��\�[¥eukdqo��{'\Ec(sg�0egqd¬Z��koqoegqdmjcd�"S'\g¯O[�sBcyklsu�0kdWZ\'[:¬O\E��|Z��qd\'�ª\'qo\'|Z{'\Ez&sgqzx\!koeg����\Ez3��|<;�]@I�=G¦e ÞàßlËGÇUÌ0Ú�Ñ6ÐÊÐÊÑEf/ÈÊß�g�h?f%Ì>ÜUÌnßUÑ�ËØÌrÌyÈ ËØÇUÌnÕ+Ï�ÜUÜUÈ ËØÈÊÑ6ßUÉ+Ñ6Õ�ÉØãUÔ@ËØÕGÏ�×ÎËØÈÊÑ6ßUÉ0Ô;i_Ï�Ü@ÜUÈ ËGÈÊÑ6ß ù

ÞkjK[�Þml

Page 7: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

������������ �������������������� �!�"���#�%$&� I

����� �+��� ������� ����\X���Y��� �x� ´��

� P�V X0PEP :RX 7E7 ���ZVFYUPEP :RY 7E7� 7 V X 7 P :RX 7E7 � ^ VFYUP 7 6 Y 7E7� * V X0PEP :RX0P 7 � c VFY 7 P 6 YUPEP��� V X 7 P 6 X0PEP ���ZVFYUPEP :RYUP 7���ZV X0P 7 6 X 7E7 � P d VFY 7 P :RY 7E7��� �x� ´ �! P�V"� P$#%��� ! �ZV&� * #HY 7E7! 7 V"� 7 #HY0PRP ! �ZV&���'#%���! * VCX PRP #%� ^ ! ^ V(� � #%� P d! � VCX 7R7 #%� c ��� �x� ´*)+ P V ! P : ! � + 7 V ! �Q6 ! ^+ * V ! * : ! P + � V ! 7 6 ! �TSPEPZV,+ P 6 + 7 TSP 7 V ! * : ! �T 7 PZV ! 7 : ! � T 7E7 V-+ * 6 +.�/ �+� � �+�&� TWV ' T V10 ,

T4

U1

T1 T6 T2

B11

A11 T7A22

T8 T3

B22

T9 T10T5

Q7Q6Q5Q4Q3Q2Q1

U2 C12 C21 U3 U4

C22

STOP

START

C11

²����gfxqo\±] (©²���q�cnk&��\E®g\E�­sg� qd\@{!fxq�cd��sB|¤su���CkoqoeBcdcd\'|�eg���Bsgqo��koWx[ 3 ��\!�àk 5 eg|Zz¤��koc�v � B qd\Ewxqo\Ecd\'|Ckoeu«ko��sB| 3 qd���gWCk 5 ¦

¢¤��|xsg�BqoeBz§®Uegqd��eu|Ck�sg�(�¨kdq�egcocy\E|R9(���gsgqo��kdWx[�¯���|CkdqosizifZ{'\Ez¡��|F;�]U�@=G¯0fZcd\Ec�kdWx\&cdeg[�\¥|CfZ[ ¬�\'q sg�[�fx��kd��wx����{'eukd��sg|Zcl¬xfiklqo\EzxfZ{!\@c�kdWZ\�|¨fx[ ¬�\'q­su�regzxzx��ko��sB|Zc��ªqosg[ ]E�.kosI]@Ix¦XVXWZ��c­eu���gsgqo��kdWx[ eu|Zz���kocv � B euqo\*wxqd\@cy\E|Ckd\Ez3��|&²����gfxqo\��i¦

2 3 �j+�E�)��47���65 �*��­���-���0���!�'��� 8 �'4����0��+;��)"

� |§koWx��c cy\@{6kd��sg|/¯/�P\¥wxqd\@cy\E|Ck�zZe�koe��CwZegqoeg����\'�0��[�wx��\E[�\'|Ckoeukd��sg|Zc�sg���¨kdq�egcocy\E|§eg|Zz§¢¤��|xsB�gq�egz§eu���gsg«qo��kdWx[¥cE¦K9­c0[�eukdqo��{'\Ec8X¤eu|Oz1Y euqo\Xzi��cykdqo��¬xfikd\@z.sg|�kn�7s*zi��cªtnsB��|Ckd\@z �Bqd��zxcE¯�kdWZ\'m�WZeU®B\kos*¬O\(eg�����g|x\@z¬�\!�ªsBqd\­{!sg[�wxfxkd��|x�*kdWx\­wxqos¨zxfZ{6kfOcy��|x� e��i{Ee-�\9(p 9 #�� wZeuq�eu����\E�xqosgfiko��|x\B¦�VXWZ��c7eu�����B|x[�\'|Ck��[.wZ����\Eckn�Ps&qd\@zi��cykdqo��¬xfikd��sg|Oc­¬�\!�ªsBqd\.koWx\�\!hi\E{'fikd��sg|§su�7kdWx\�eg���Bsgqo��koWx[�¦�²xfZqykoWx\'qo[�sgqo\g¯�e�k*koWx\�\'|Zz§su�7kdWx\{'sg[�wxfik�e�ko��sB|�¯i�7\*�Xeu|CkZT}kos�¬O\�eg�����g|x\@zj����kdW�X ¦VXWx��cX��|ZzxfZ{!\@c(eu|&sukoWx\'q�qo\Ezi��cykdqo��¬xfxkd��sg|�¦VXWx\_�ªsB����s�����|Z� ��[.wZ��\E[.\E|Ckoe�ko��sB|ZcP�ªsgqX¬�sukoW&eg���Bsgqo��koWx[¥cr�ªsi{!fOc7sB|jkoWx\*qd\@zifZ{!kd��sg|jsg��kd\E[�wOsBqoegqdm

®�egqd��eu¬x��\EcE¦+VXWZ\'m¥zis�koW¨fZc|Zsuk7\'hiWx��¬Z��kPwxWZegcd\Ec7egc7cyWxs���|¥��|j²����gfxqo\ ](¬ZfikXeg|�eu��kd\Eqd|Zeukd��sg|¥¬�\!kn�P\'\E|sBwO\Eqoeukd��sg|Zclsu�wZWZegcd\Ec�]�eg|ZzR�i¦�²����gfxqo\��3cyWxs��(c­sgfZq_zxeukoe��CwZegqoeg����\'��®B\'q�cy��sg|°sg�P�CkoqoeBcdcd\'|I[¥eukdqo��h[�fx��kd��wx����{'eukd��sg|jeg���Bsgqo��koWx[�¯g��Wx����\­²����Bfxqo\l� cdWxs��(ce�zxe�k�eU�¨wZeuq�eu����\E�i��[�wx��\'[�\E|Bk�e�ko��sB|�su��¢¤��|xsg�BqoeBz®�egqd��eu|CkE¯¨��Wx��{�W°{'eg|3¬�\*�ªsgfx|Oz3��|+;�]�I@=¿¦

[\[ ß^]_&_�`&a

Page 8: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

E ����� ��$ � ����������� � *����� � �+�&� ��� ��� ���� X ��Y

��� �Z� ´��� P�VCX 7 P : X 7R7 ���ZVFYUP 7 6 YUPRP� 7 V"� P 6 X0PEP ����VCY 7E7 6 ���� * VCX PRP 6 X 7 P � ^ VFY 7R7 6 YUP 7� � VCX P 7 6 � 7 � c VCY 7 P : ���

��� �Z� ´ �! P�V X0PEP #HY0PRP ! �ZV&� * #%� ^! 7 V X0P 7 #HY 7 P ! �ZV"� �%#HY 7R7! * V&� P #%� � ! ^ VCX 7R7 #%� c! � V&� 7 #%� � ��� �Z� ´*)+ P V ! P : ! � + 7 V,+ P : ! �+ * V-+ P : ! *TQPRP�V ! P: ! 7 TSP 7 V,+ * : ! �T 7 P�V-+ 7 : ! ^ T 7R7 V,+ 7 : ! */ �+� � �/��� T VB' T V10 ,

T1

T2

T4

T3 T5 T7

T6

T8

Q1 Q2 Q3 Q4 Q5 Q6 Q7

C11 C12 C21 C22

A11

A11

B22

A22

B22

B21A12B11

U1

U2U3

START

STOP

²����gfxqo\�� (�²���qocyk���\E®g\'�su�Xqd\@{!fxq�cy��sg|£su�PkdWx\¥¢¤��|xsB�gq�egzI®�egqd��eu|Ck 3 ��\!�àk 5 eu|Zz§��koc�v � B�qd\Ewxqo\Ecd\'|Ckoeu«ko��sB| 3 qd���gWCk 5 ¦

� �����������¥�-���0���!�!���©8:�!4 �����6+;&�)"

����� ��� ���������������������������������!

V+sj¬�\'|Z\ 'xk_�ªqdsB[:kdWx\�fZcd\�sg�r[.��hi\Ez¨«¿wZegqoeg����\'����cd[ ��|¡�Ckdq�egcocy\E|Reu���gsBqd��kdWZ[&¯OkdWZ\.cykdq�e�ko\'�gm&\'[�wx��s�mg\@z�B\'\EwZcP[�eukdqo��{'\EcX��|3wZ��eB{!\�eu|Zz&zi��cnkoqd��¬xfiko\EcXk�egcd�icPeg[�sg|x�.wxqos¨{'\EcocysBqoc7��|Zcnko\EeBzjsg�0eg�����g|x��|x�.[¥e�kdqo��{!\Ec¬�\!�ªsBqd\�{'sg[�wxfiko��|Z�Z¦ 9­c��7\�fZcd\�cykoeu|Ozxeuq�zRwOeuq�eu����\E�>|CfZ[.\Eqd��{'eg�>qdsBfikd��|x\@c*�ªqdsB[ �i{Ee-� 9lp 9 #���¯��7\|Z\'\Ez°kdsj�B\'\Ew�kdWx\�zxe�k�e�zi��cnkoqd��¬xfiko��sB|R��[�wOsCcy\@z°¬¨mjkoWx\Ecd\ �B\'qo|x\'��c 3 �ªfx����«¿¬x��si{���sgq_¬x��si{��B«Î{!mi{'����{�zi��cn«koqd��¬xfiko��sB|Zc 5 ¦ 7­fxqX�gsCeu�%��cPkdW¨fZcPkos¥qd\@zifZ{!\�{'sg[�[ fx|Z��{Ee�kd��sg|OcPeg|Zzjkds¥¬Zeg��eg|Z{!\*egcX[�fZ{�W°egcPwOsCcdcd��¬x��\{'sg[�wxfik�e�ko��sB|Zc(eu[�sB|x��kdWx\ wxqdsi{'\EcocysBqocE¦7VXWx\�kn�7s¥wxqosi{!\@cdcdsgq��Bqd��zxc 3 sgql{'sg|Ckd\'hCk�c 5 ��WZ\'qo\�[¥e�kdqo��{!\EcX eg|Zz0Y euqo\�zi��cnkoqd��¬xfiko\Ez¡euqo\�c��BfOeuqo\�eg|Zz§su��zi��[.\E|Zcd��sB|#" V - V ¯+¬xfxk sgfxq�eu���gsgqo��kdWx[¥c�euqo\�|xsuk����[.��kd\@z¥kos�cdfZ{�W�®�eu��fx\Ec�ªsgq�"/¦>VXWx��cPqd\@cnkoqd��{6ko��sB|¥��c7[¥egzi\_��|je�w�\Ezxeg�gsB�g��{'eu�OwxfxqowOsCcy\B¦+¢§\*{!sg|Ocy��zi\'qkoWx\Ecd\*{!sg|Cko\!h¨kocPeuqo\lcdfx¬i«¿�gqo��zxc7su��e�®¨��qdkdfOeu��qd\@{6koeg|x�gfZ��egqwxqosi{!\Ecocdsgqr�Bqd��z¥su��cy�"S'\$"=#F-%" 3 ��\EeBzi��|Z��kdse�kdsgkoeu��su��-&"@7 wxqdsi{'\EcocysBqoc 5 ¦­prqdsi{'\EcocysBqoc�sg�>kdWx��c_�g��sg¬Zeg�/�Bqd��zIeuqo\ qos�� [¥e�tnsgq­|¨fx[ ¬�\'qo\Ez��ªqosg[ ; �x¯��=%��|3kdWx\*fxwZwO\EqP��\!�àkl{'sgqo|x\'q7kds ; " 6 <B¯1-%" 6 <�=���|�koWx\*��s��P\'qPqd���gWCk@¦�¢§\*�������%�g\E\'wjkdWx��cX|¨fx[ ¬�\'qo��|Z��ªqosg[ kdWx\��B��sB¬Zeu�%{!sB|Ckd\!h¨kXkds¥��zi\E|Bko���ªm3wxqos¨{'\EcocysBqoc7fx|Ckd���%koWx\�\'|Zz&sg�/kdWZ��c�wOeuw�\'q@¦� k*��c_��[�wOsBqyk�eu|Ck­kosj|xsgkd\�kdWZeuk_wOeuq�eu����\E�+®B\'q�cy��sg|Zclsu�7�CkoqoeBcdcd\'|£eu���gsgqo��kdWx[ euqo\ ¬�\!kdkd\Eq­koWZeu|RkdWx\

cykoeg|Zzxegqoz [¥e�koqd��h [�fx��ko��wZ����{'eukd��sg|-eu���gsBqd��kdWZ[ ����eB{'{'fx[ fx��e�ko��sB|Zc¥sg�*[¥eukdqo��{'\Ec�{Eeu|�¬O\§{!sB[.wZfikd\@z��|�����|x\@euq�kd��[�\I����kdWxsBfikj{!sB[.[�fx|x��{'eukd��sg|ZcE¦ � �6X 3 sgq Y 5 ��cjzi��cykdqo��¬xfikd\@z �ªsg����s�����|x� e� �� �!�('�)�.�)�2,zx��cykdqo��¬Zfikd��sg|�¯�koWx\jegzZzi��ko��sB| su��kn�Ps �CfZeuqdkd\Eqoc����������|BkoqdsizifO{!\j{'sg[�[ fx|Z��{Ee�kd��sg|Oc'¦�V/s£eU®gsg��z£kdWx\@cy\

ÞkjK[�Þml

Page 9: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

������������ �������������������� �!�"���#�%$&� ������������ ÏoËGÕÖÈÊ×nÌyÉ�lRÏ�ßUÜ ÆiÌná�ÙuÑ6ÕGÏ�Õki(Ò6Ï�ÕÖÈÓÏ�ÔUÐÊÌnÉ���� h��Qh�� h�����h����9h����� �����! "���# $%��&'������()�+*,��� -".0/����+*0�+1#&2*43

56(��7*,(�8�.0 [�ÌyÜ@ÈÊÉØËØÕØÈÊÔUãEËGÌ:9<;=�[/ÌnÜUÈÊÉÖËGÕÖÈÊÔUã@ËØÌ%>?;@�

� � A � ���CB � �D� EGF� � A � �D�IH � �J� EGK� � A � �IL � � M�N� � A � ���IH � �D� E��� � A � ��L � �D� MO�� �D� A � �6B � � BQPRF� � A � ���CB � �D� EGS��� A �T�D� L ��� M F� ��� A � �CH � � � ���� � A � �D�IH � �D� ER�� � A � �D�IH � �D� EGN�C�D� A �O� L �Q� M �� �D� A � �D�CH � �D�� �D� A � �D��H � � PU�� � A � �J�QB � �D� E�V� � A � �D��L � � M���I�D� A �I�D� H ��� ���D�� � A � �D�IH � �J� EG�� � A � ��L � �D� MOW� �J� A � �CHYX�� � �J��C�D� A �C�D� B �Z�� � A � �J�QB � �D� E�W� � A � ���IH � �D� ER�\[� � A � �IL � � MOV� �D� A � �D��H � � � �D�

[�ÌnÜUÈÊÉØËØÕØÈÊÔ@ã@ËGÌ%�];_^` ���!����� ^ba?cd^Ue fhg

²+���gfZqd\��-( � e�k�eU�¨wZeuq�eu����\E���¨kdq�egcocy\E|3[¥e�koqd��hj[�fx��ko��wZ����{'eukd��sg|�eu���gsgqo��kdWx[�¦

{'sg[�[ fZ|x��{Ee�ko��sB|Zc'¯U\@eg{�W wZqdsi{!\@cdcdsgq+��|¨®Bsg��®g\Ez���|�kdWx\�{'sg[�wxfik�e�kd��sg|�[ fZcyk>s���| e­wZeuqdk�sg�O\@eg{�W [¥eukdqo��h�CfZegqyko\'q�c'¦IVXWZ\j¬x��zi��[.\E|Zcd��sB|Zeu� )��.�@�2, ' �ji������ ��zi��cykdqo��¬xfikd��sg|±��c kdWx\3[�sBcyk.egzZeuwiko\Ez§kds£sg¬ik�eu��|©cyfO{�WwZqdsBwO\EqyknmB¯i¬xfik(koWx\�¬x��si{��jcy�"S'\�wOeuq�eu[�\!ko\'q�WZeBcXkds¥¬�\ {'egqd\'�ªfx����mj{�WZsBcd\'|�¦ � |Ozi\'\@z°e.kds¨s¥����kyko��\�¬Z��si{��cd�"S'\�����������|ZzifO{!\�\'hCkoqoe.[�\'[�sBqdm&eg{'{'\Ecocy\@cXeu|Zz&�������+��|Z{'qd\@egcd\�{!sg[�[�fx|x��{'e�ko��sB|&{'sBcyklegc��P\��������/cd\'\��|Rcd\E{!kd��sg|MIx¦ �x¦lk VB�32A-%"I��clkdWx\�[�euhi��[�fx[:¬x��si{��°cd�.SE\�kdWZeuk*eg����s��(cXkoWx\���si{'eg�+{!sB[�wxfikoeukd��sg|Isg�eg���0egzZzi��ko��sB|ZcE¦l²����Bfxqd\ I¥cdWxs��(cleg|I\!hxeu[�wx��\ sg�>kdWx\.{�WxsBcd\'|IzZe�koe3zi��cnkoqd��¬xfiko��sB|I��Wx\'| ">V -i¦_²xsgq\@eg{�W�¬x��si{���¯ucdfx¬Zco{!qo��wik0�g��®g\@c���k�c0[�eukdqo��h �CfZegqyko\'q0eu|Zz.cdfxwO\Eqoco{!qo��wik0{'sgqoqd\@cyw�sg|Ozxc�koslkoWx\X¬x��s¨{��C«Î{!mi{!����{zx��cykdqo��¬Zfikd��sg|�¦

[\[ ß^]_&_�`&a

Page 10: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

� ����� ��$ � ����������� � *������������� ÏoËGÕÖÈÊ×nÌyÉ�lRÏ�ßUÜ ÆiÌná�ÙuÑ6ÕGÏ�Õki(Ò6Ï�ÕÖÈÓÏ�ÔUÐÊÌnÉ���� h��Qh�� h�����h����� �����! "���# $%��&'������()�+*,��� -".0/����+*0�+1#&2*43

56(��7*,(�8�.0 [�ÌyÜ@ÈÊÉØËØÕØÈÊÔUãEËGÌ:9<;=�[/ÌnÜUÈÊÉÖËGÕÖÈÊÔUã@ËØÌ%>?;@�

� � A � �D�CB � ��� EG�� � A � �D�QB � �J� E�V� �D� A � �IL � � MOW� � A � ���IH � �D� ER�� � A � �J�QB � �D� E�W�I�D� A �O� L �Q� M �� � A � ��B � �D� E��� � A � �D�QB � � EGN� ��� A � �IL � � M�F�O� A �"�J� B ��� E F� � A � ���CB � � EGS� �J� A � ��L � �D� M�N� �J� A � �J�CH � �D�� � A � �D�IL � �D� M ��I��� A �I��� H ��� P �� �J� A � ���IL � �J� � �J�� ��� A � ���IH � �D� P��� �D� A � �D��L � � MO��C�D� A �C�D� H ��� ���D�� � A � ����L � � MOV� ��� A � ����H � � � ���

[�ÌnÜUÈÊÉØËØÕØÈÊÔ@ã@ËGÌ%�];_^` ���!����� ^ba?cd^Ue fhg

²����Bfxqo\_� ( � eukoeU�¨wZeuq�eu����\'��¢¤��|Zsg�gq�egzj[¥e�kdqo��hj[ fx��kd��wx����{Ee�ko��sB|�eg���Bsgqo��koWx[�¦

����� �$����� �!�

7­|O{!\ {'sg|Ckd\'hCk�clegqd\ zx\ 'Z|x\@z%¯�koeBcy�ic��ªqosg[ koWx\�v � B WZeU®B\�kdsj¬O\�egcocy���g|Z\Ez�egc­\'®B\'|x��m&eBc(w�sBcocy��¬x��\g¦¢§\*WZeU®g\_zi\'®B\'��sgw�\Ezjkn�7s�[���hi\Ez¨�CwZegqoeg����\'����[.wZ��\E[.\E|Ckoe�ko��sB|Zc7�ªsg����s�����|x��kn�7s.zi�./�\Eqd\E|Bk(egwxwxqosBeg{�WZ\EcE¦VXWZ\ 'Zq�cykP��cP��|Zcdwx��qd\@z�sg�/kdWZ\­¬ZeBcy��{­qd\Ewxqd\@cy\E|Ckoe�ko��sB|�su�0�Ckdq�egcocy\E|jeu���gsBqd��kdWZ[ wZqd\@cy\E|Bko\Ezj��|�²����Bfxqd\�]g¯eBc>��kr�ªsg����s��(c>kdWx\P���-� $���$lsg��koWx��ceu���gsBqd��kdWZ[&¦�VXWx\(®B\'q�cy��sg|.�g��®g\E|.��|�kdWZ��cwZeuw�\'qr��ceg|���[�wxqos�®g\E[.\E|Cksg�koWx\�sg|Z\.wZqd\@cy\E|Bko\Ez£��| ;�]E��=¿¦jVXWx\¥cd\E{'sg|Zz£��[�wx��\'[�\E|Bk�e�ko��sB|§WOegc*¬�\'\'| zi\@cy���g|Z\Ez¡egc*��k��PeBc_kdWx\qo\Ecdfx��k�su�(e0�#�%$&�co{�Wx\Ezxfx����|x�Ieg���Bsgqo��koWx[�¦3¢§\3zi\@cd{'qd��¬�\�kdWx\@cy\¥kn�Ps°��[.wZ��\E[.\E|Ckoe�ko��sB|Zc���|¡kdWZ\j|x\'hCkkn�Ps¥cy\@{6ko��sB|Zc'¦

ÞkjK[�Þml

Page 11: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

������������ �������������������� �!�"���#�%$&� ?

Context 1Context 2 Context 2Context 1

^ [D[�D�

^ � [���

^ �\[�D�

^ �D[���9 �D[���

9 �\[�D�

9 � [���

9 [D[�D�

9 ����D�

9 �J��J�

9 �D��D�

9 [���J� 9 [D��J�

9 � ��D�

9 �\��J�

9 �D��D�

> [D[�D�

> � [���

> �\[�D�

> �D[���

> [���J�

> �D��D�

> �J��J�

> ����D�

> [!��D�

> ������

> �D��D�

> �!����

> [D��J�

> � ��D�

> �\��J�

> �D��D�

9 [!��D�

9 ������

9 �D��D�

9 �!����

^ [!��D�

^ ������

^ �D��D�

^ �!����

^ [���J�

^ �D��D�

^ �J��J�

^ ����D�

^ [D��J�

^ � ��D�

^ �\��J�

^ �D��D�

²����gfxqo\ I (>vIegwxwx��|x��su�0[�eukdqo��{'\Ec�X ¯�Y eu|Zz�T sg|�e(- #�C�wxqos¨{'\EcocysBqX�gqo��z%¦��� � � � ��� �x� ´�� �0�x� ´ ¹¤� ,�� ¸ ���Ö� � � |IkdWx��cl��[�wx��\'[�\'|Ckoeukd��sg|I�P\�WZeU®B\ cykdfZzi��\EzI��Wx��{�W£zxe�k�e�euqo\���|C®Bsg��®g\@z3��|I\@eg{�W�koeBcy�3kds&zi\!ko\'qo[���|x\WZs��}kds�cyWOeuqo\.{'sg[�wxfik�e�kd��sg|Oc­sBfikE¦ � |£wZWZegcd\3]B¯�eu���>sgw�\'q�e�ko��sB|Zc­{'sg|Z{'\'qo|RsB|x��m>X sBq6Yj¦�V+s3�B\'\Ew��si{'eg����knmg¯¨koWx\Ecd\*koeBcy�ic�euqo\*[¥euwxw�\Ez°{'��sCcy\_kds�koWx\�zxeukoe�kdWx\Em3fOcy\B¦ � |°wxWZegcd\��i¯x\@eg{�W&[ fx��kd��wx����{Ee�ko��sB|��|¨®gsB��®B\Ec e£[¥e�koqd��h±{!sB[.��|x�R�ªqdsB[ {!sg|Cko\!h¨k&]�eu|Oz±eI[¥eukdqo��h±{'sg[���|Z�R�ªqosg[ {!sB|Ckd\!h¨k��i¦©¢¤WZ\'| e{'sg[�wxfik�e�ko��sB|.|Z\'\EzZcrzxe�k�e_kdWZeuk7euqo\X|xsukzx��cykdqo��¬Zfikd\@z�sg|.kdWx\lqd���gWCk{!sB|Ckd\!h¨k@¯B{!sBwx��\Ec0su�%[���cdcd��|Z�*zZe�koe�ªqosg[ sB|x\¥{!sB|Bko\!h¨k*kds&kdWZ\.sgkdWx\Eq�WZeU®g\�kds�¬�\�wO\Eqy�ªsBqd[�\Ez�¦ � |£kdWZeuk�wxWZeBcy\B¯�kdWx\¥wx��eg{'\'[�\'|Ck�su�7kdWx\k�egcd�ic*�PeBc_zxqd��®g\E|£¬¨m°koWx\¥qd\@zifZ{!kd��sg|¡su�X{!sg[�[�fx|x��{'e�ko��sB|§{'sBcykE¦¥prqosizifZ{!koc*��|C®Bsg��®¨��|x�&fx|x[�sizi� 'Z\EzwOeuqdkoc­su� X 3 qo\Ecdw�¦ Y 5 egqd\�kdW¨fZc*{!sB[�wxfikd\@zIsB|£{!sB|Ckd\!h¨k�] 3 qd\@cyw�¦�{'sg|Ckd\'h¨k�� 5 ¦�²���|Oeu����mIeBzxzi��kd��sg|Zcsg�_wZWZegcd\I� euqo\°\!hi\E{'fikd\@z sg| kdWx\£{!sg|Cko\!h¨k���Wx\'qo\°[¥e�koqd��h T WZeBc�kds ¬�\Rzi��cnkoqd��¬xfiko\Ez%¯���|�sgq�zi\'qkos3qo\EzxfZ{!\ kdWx\¥{!sB[�[ fx|x��{'eukd��sg|Reg[�sgfx|CkE¦�VXWZ��c*{�WxsB��{'\�{!sB[�\Ecl�ªqdsB[:kdWx\��Öeg{!k­koWZe�k�egzZzi��ko��sB|R{!sCcnk��c���s��7\Eq�kdWZeg| {'sg[�[ fx|Z��{Ee�kd��sg|�{!sCcnk��ªsBqje§�g��®g\'| [¥e�koqd��h cd�.SE\g¦-VXWx\@cy\°{�WxsB��{'\Ec���\Eegz fZc�kos¡kdWx\wZWZegcd\6�¨¬Zegcd\Ezj[.��hi\EzC�¨wZeuq�eu����\'����[�wx��\'[�\E|Bk�e�ko��sB|3sg�>�CkoqoeBcdcd\'|&eu���gsBqd��kdWx[�¯xcdWxs���|j¬¨m&²+���gfZqd\ Ex¦

��� � � � �Ö�E� �� �x� ´ ¹�� ,�� ¸ � �ª� � � |RkdWx��c�cd\E{6ko��sB|R�P\¥zi\!k�eu���+koWx\�wxqo��|Z{'��wZ��\.su�Psgfxq*����cykn�¨¬Zegcd\EzIeg���Bsgqo��koWx[�¦ 9 [¥e�tnsgq*egcocdfx[�wikd��sg|£��c[¥eBzi\j��| kdWZ��c���[�wx��\'[�\E|Bk�e�ko��sB|�¦ � |Ozi\'\@z �7\&egcocyfZ[.\�kdWZeuk.�P\��¨|xs�� Wxs�� wxqosizifZ{6k�c�egqd\3cyWZegqd\@zsBfik ¬�\!�ªsBqd\�koWx\j¬�\'�B��|x|Z��|x�Rsu�PkdWx\3����cnk.cd{�Wx\@zifx����|Z�Ieu���gsgqo��kdWx[�¦�V/sRzx\!kd\Eqd[���|x\¥koWZe�k�cdWZegqd��|x�Z¯+�7\{'sg|Zcd��zi\'q7kdWx\*cd\'®g\E|jwxqosizifZ{!kocXegcPe�{�WOeu��|�¯¨��Wx��{�W3��cP¬xfZ����k(egc�ªsg����s�� (>VXWx\Eqd\­��cXe�qd\E��eukd��sg|�¬�\!kn�P\'\E|kn�PsIwxqosizifZ{!koc����(koWx\'��q.qd\@cyfx��koc.euqo\�¬�sukoW±sgw�\'q�eu|Ozxc�sg�leI�g��®g\E|±egzZzi��ko��sB|©��|©wxWZeBcy\&�x¦¡²+���gfZqd\��cdWxs��(c7kdWZ\�{�WZeu��|3sg��wZqdsizifZ{!koc�eg|Zz�koWx\�wxqosgw�sBcd\Ez&{!fikd«G����|x\�zi\!ko\'qo[.��|x��|x��kdWZ\�wx��eB{!\E[.\E|CkE¦

9­cl�ªsBq_�CkoqoeBcdcd\'|/¯O�P\�zi\@{!��zi\�kos3{'sg[�wxfiko\�eg{E{!fx[�fx��e�kd��sg|Oc(su�rwZWZegcd\j] {'��sCcy\ kdsjkdWx\.zxe�k�e¥kdWZ\'m��|¨®gsB��®B\g¦ #7sB|Z{!\Eqd|x��|x�§kdWZ\Iegzxzi��kd��sg|Oc¥su��wxWZeBcy\I�Z¯7[�sCcnkjsg�­koWx\'[ 3 + P ¯1+ 7 ¯ + � ¯ZT P 7 ¯(eu|Oz T 7 P 5{Eeu|¡¬�\j{!sB[.wZfikd\@z¡����koWxsgfik.eu|¨m£zxeukoe°[�s�®B\'[�\'|Ck egc��7\�{Eeu| cy\E\���|©²+���gfZqd\/�i¦&`�\E[¥eu��|x��|Z�Regz¨«zx��ko��sB|Zc­egqd\�wO\Eqy�ªsBqd[�\@z°sg|RkdWx\�{!sB|Bko\!h¨k­��WZ\'qo\�[¥e�koqd��h T WZeBc(kos3¬�\.zx��cykdqo��¬Zfikd\@z°kds�eU®gsB��z�\'h¨kdq�e{'sg[�[ fZ|x��{Ee�ko��sB|Zc'¦

[\[ ß^]_&_�`&a

Page 12: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

]@� ����� ��$ � ����������� � *�����������C� ÏoËØÕØÈ Í l ��������� C� ÏoËØÕÖÈ Í ÆiÌyá�ÙuÑ6ÕØÏ�Õ i(Ò�Ï�ÕØÈÓÏ�ÔUÐÊÌnÉ6� � h�� � h�� � h�� F h�� W h�� N h�� V ÆiÌná�ÙuÑ6ÕGÏ�ÕkilÒ6Ï�ÕÖÈÓÏ�ÔUÐÊÌnÉ X#� h XG� h X�� h X�F h XGW h X�N h X�V� �!���! l�!�# $%��&'�����!()�+*,��� -".0/����+*0�+1#&2*43 � �����! ���� $%��&'������()�+*,��� -l.,/����7* �71#&2*,3

� (��7*,(�8�.0 �� � (��7*0(�8�., ��

� � A 9 �D�IH 9 �D� ER� X#� A > �D�IH > �D� E�N� � A 9 ���IH 9 �D� E�� X�� A > �J�QB > �D� EGV� � A 9 �D�IH 9 �J� EG� X�� A > ���CB > �D� E�S� F A 9 ���CB 9 �D� EGF X�F A > �D�IH > �J� E�K�QW A 9:�J� B 9Q�D� E W X W A >Q��� H >Q�D� E �\[

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � N�A=XG�[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � � ; XG�[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ ��� A=X �[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � � ; X��[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � �6A=X�F[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � � ; X�F[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � ��A=XGW

� V A 9 �D��L � N M�� X�W A X���LCX�� M �� N A 9 �D��L � � M�F X�N A X��ZL > �D� M:�� � A � FCL � � M�N X�V A X�FZL > �D� M:W�Q� A �QW L �Z� M V

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � ��A=XGW[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � F�A=X�N[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � W6A=XGV

^ �D� A � �ZH � N^ �D� A ^ �D�CB � W^ �D� A ^ �D�IH � � ^ �D�^��J� A �QV H ��W ^��J�^ ��� A � FZH � N ^ ���^ �D� A � �ZH � V^ �D� A ^ �D�QB � F^��D� A ^��D� H ��� ^��D�

` ���!����� ^ba?cd^Ue f g

²����gfxqo\6E-(�prWZegcd\Ez¨�C¬ZeBcy\@z�[���hi\@z���[.wZ��\E[.\E|Ckd��sg|&su���CkoqoeBcdcd\'|3eu���gsgqo��kdWx[ �ªsBq #7sB|Bko\!h¨k*] 3 ��\'�àk 5 eu|Oz� 3 qd���gWCk 5 ¦

Q6 Q2 Q4 Q1 Q3 Q5 Q7

U2U1U4 U3 C12C21

CONTEXT 2

CONTEXT 1

²����gfxqo\ ��( #7WZeg��|�su��wxqos¨zxfZ{6k�c�eu|Zz�{!fxky«¿����|x\_�ªsgq­�¨kdq�egcocy\E|3eu���gsBqd��kdWZ[&¦

ÞkjK[�Þml

Page 13: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

������������ �������������������� �!�"���#�%$&� ]B]

V+sjzi\'kd\'qo[���|Z\*kdWx\ co{�Wx\@zifx����|x�¥sg�>sgfxqleg���Bsgqo��koWx[�¯x�P\�fZcy\�e�����cnk­su�0qd\@egzimjkoegcd�ic'¦PVXWx\Ecd\�k�egcd�¨cegqd\ (§eBzxzi��kd��sg|ZcE¯(wxqosizifZ{!kocE¯(cy\E|Zzxc3sgqjqo\E{'\'��®g\@c'¦ 9�zxzx��ko��sB|�eu|Zz�wZqdsizifZ{!kjk�egcd�¨c3euqo\R{!sB|Zcd��zi\Eqd\@zqo\EeBzim�egcPcys¨sg|3egc7¬OsgkdW&sgw�\'q�eu|ZzxcWZeU®g\­¬O\E\'|3wxqdsizifO{!\Ez�¦0²ZsgqX{!sB[.[�fx|x��{'eukd��sg|�sBwO\Eqoeukd��sg|ZcE¯Be koeBcy���cXqo\Eegzxm�sg|Z{'\�e.zZe�koe�kos�koqoeg|Zcy�ª\'q���cXwxqdsizifO{!\Ez�¦VXWx\­{�Wxsg��{!\Xsu��koWx\�qd\@egzim koeBcy��kos�\!hi\@{!fiko\���crzxqd��®g\E|.¬¨m.cy\E®g\'q�eu�x{'qd��kd\Eqd��ex¦�VXWx\�[�sBcykr��[�w�sgqdkoeg|Bk

��ckdWx\*eu��[ kds ¬Oeu��eu|Z{'\­{!sB[�wxfikoeukd��sg|Zc¬O\'kn�7\E\'|j{'sg|Ckd\'h¨kocPegc7��sB|x� egc7w�sBcocy��¬x��\g¦>VXWx\'|3{!sg[�[�fx|x��{'e�«ko��sB|°sgw�\'q�e�ko��sB|Zc�WZeU®B\�wxqo��sBqd��knmjs�®B\'q(egzZzi��ko��sB|ZcE¦(²+��|Zeg����m°egzxzi��kd��sg|Oc(��Wx��{�W°wxqos¨zxfZ{!\ e�qd\@egzim3koeBcy�WOeU®g\�wxqo��sBqd��knmjs�®B\'q�koWx\ sgkdWx\EqlsB|x\EcE¦l²xsB����s�����|x��kdWx\@cy\�qdfZ��\@c(�P\�sg¬ik�eu��|�kdWx\.eu���gsBqd��kdWx[ wxqo\Ecd\'|Ckd\@z��|°²����Bfxqd\��Z¦

����� � ��� ��� �����V+s���[�wx��\'[�\E|Bk kdWx\¥wxWOegcd\!«leu|Oz£����cnky�C¬ZeBcy\@zR[���h¨\@zC�CwOeuq�eu����\E�>®g\'q�cd��sB|Zc_su�PkdWx\¥¢¤��|xsg�BqoeBzR®�egqd��eu|CkE¯�P\��g\Ewik>kdWZ\lcdeg[�\�|xsuko��sB|¥su�%{!sg|Cko\!h¨kocreg|Zz�kdWx\­cdeg[.\(zZe�koe�zx��cykdqo��¬Zfikd��sg|¥wxqo\Ecd\'|Cko\Ez.��|�cd\E{!kd��sg|��Z¦�]g¦OPfikXcy\E®g\Eqoeg�xsgwiko��[��"SEeukd��sg|Zc7WZeU®g\l¬O\E\'|�[¥egzx\l��|�¬OsgkdW3��[�wx��\'[�\'|Ck�e�kd��sg|Oczifx\­kds kdWx\­wxqosgw�\'qdkd��\Ecrsg�koWx\�¢¤��|xsg�BqoeBzjv � B.¦x¢§\�zi\!k�eu���%kdWx\@cy\*sBwikd��[��.S@e�kd��sg|OcX��|&kdWZ��c(cd\E{!kd��sg|�¦

��� ) � � ��� �x� ´�� �0�x� ´ ¹¤� ,�� ¸ � �Ö� � VXWZ��c���[�wx��\'[�\'|Ckoeukd��sg|¡su�X¢¤��|xsg�BqoeBzI®�euqo��eg|Ck*WOeU®g\�¬O\E\'|¡��[.wZ��\E[.\E|Ckd\Ez eg��[���|x�°e�k�qo\EzifO{!��|x��kdWx\{'sg[�[ fZ|x��{Ee�ko��sB|3eu[�sBfx|CkE¦>V/s¥eB{�Wx��\E®g\­kdWx��cX�gsCeu���P\_fOcy\@z�koWx\_�ªsg����s�����|x�.sgwiko��[��"SEe�ko��sB|ZcE¦0²���qocyk��7\fOcy\@z�qo\Ezifx|Ozxeu|Ck�{!sg[�wxfxkoe�ko��sB|�¦ � |Zzi\E\Ez¥koWx\�{!sB[.wZfikoeukd��sg|�sg��� � {Eeu|�¬�\*wO\Eqy�ªsBqd[�\@z�sg|&{!sB|Bko\!h¨k­]eBcP¬OsgkdW&sgw�\'q�eu|Zzxc 3 Y 7R7 eu|Zz ��� 5 euqo\­eu��qo\Eegzxm¥qd\@zi��cykdqo��¬xfikd\@z¥�ªsBq7koWx\�{!sB[.wZfikoeukd��sg|jsg� ! �*eg|Zz ! *qo\Ecdw�\E{6ko��®B\'��mg¦ � k7eu����s��(c0fOc0kds�\'hi{�WOeu|x�B\PE*[¥eukdqo��h��BfOeuqdkd\'q�c���|Ocnko\Eegz.sg� �­¬�\!kn�P\'\E|�wxWZegcd\EcX](eg|Zz¥�i¦VXWZ\'|�wxWZeBcy\�����cl{'sg[�wxfiko\Ez&sg|x��m&sg|�sg|x\�{'sg|Ckd\'h¨klegcX�ªsBq­�Ckdq�egcocd\'|�¦VXWZ��c­{�Wxsg��{!\*fx|¨¬Zeg��eg|Z{!\@cXkdWx\eg[�sgfx|Ck­sg�7{!sB[�wxfikoeukd��sg|I¬�\!kn�P\'\E|£{'sg|Ckd\'hCk�cl¬Zfik_qo\EzxfZ{!\@clkoWx\�eu[�sgfx|Ck_su�7zxeukoej\!hx{�WZeu|Z�g\Ez�����kdWqo\'�Ceuq�z�kds&koWx\���[�wx��\'[�\'|Ck�e�kd��sg|§wxqo\Ecd\'|Ckd\@zI��| ;�]@��=G¦ OPsukoW§eg���Bsgqo��koWx[¥c(�ªsBq�{!sg|Cko\!h¨koc.]¥eu|Zz§�&euqo\wZqd\@cy\E|Bko\Ez3��|°²����gfxqo\�?Z¦

��� ) � � �Ö�E� � �0�x� ´ ¹¤� ,�� ¸ � �ª� � 9­c>�ªsBqr����cykn�C¬Oegcd\Ez���[�wx��\'[�\'|Ck�e�kd��sg|¥sg�/�Ckdq�egcocy\E|.eg���Bsgqo��koWx[�¯g�P\Xkdqo��\@z.kds 'Z|Oz¥e�{�WZeu��|¥su��wxqos¨zxfZ{6k�c'¦VXWZ\jcnkoqdfO{6kdfZqd\¥sB¬ikoeg��|x\@z¡��c |xsuk.cysIcd��[�wx��\g¯>eBc�cyWZs���|§��|©²����Bfxqo\I]E�Z¯/¬xfik.��k���c�cnko�����7w�sBcocd��¬x��\�kdszx\!kd\Eqd[���|x\ e�{'fiky«¿����|x\���|OzifZ{!��|x�je��gs¨siz&koegcd�&[�egwxwx��|x�Z¦(VXWZ\ eu���gsBqd��kdWx[ wxqo\Ecd\'|Ckd\@z&��|R²����gfxqo\�]B]�ªsB����s��(c�koWx��cl[¥euwZwx��|Z�Z¦_¢£\.{'eu|£cy\E\�kdWOe�klkoWx\�kn�7sjeBzxzi��kd��sg|Zc­su�rwZWZegcd\���koWZe�k�euqo\ {'sg[�wxfiko\Ez°sB|{'sg|Ckd\'h¨k(�.egqd\*s�®g\Eqd��euwZwO\@z¥¬¨m�kdWZ\�{!sg[�wxfxkoe�ko��sB|3sg� ! � sB|&{'sg|Ckd\'h¨k*]B¦

� � ;&�������P��!�7� ���­� �!53��+�!��#

� |IkdWx��c*cy\@{6kd��sg|/¯O�P\ \E®Ueg��fOe�kd\ kdWx\�kdWZ\'sgqo\!ko��{Eeu��{!sBcykocE¯���|Rkd\'qo[¥clsg�r[�\'[�sgqom3fOcdeg�g\ eg|Zz°\!hi\@{!fiko��sB|ko��[�\B¯�su��zi��/�\'qo\'|Ckj®B\'q�cy��sg|Oc�sg� �Ckdq�egcocy\E| eg|Zz ¢¤��|Zsg�gq�egz�eg���Bsgqo��koWx[¥c(Izxeukoe��CwZegqoeg����\'�(�CkoqoeBcdcd\'|�¯wZWZegcd\Ez¨«Xeu|ZzI����cnky�C¬ZeBcy\@z�[���hi\Ez¨�CwZegqoeg����\'�0�Ckdq�egcocy\E|�¯�zxe�k�eU�CwOeuq�eu����\E�%¢¤��|xsB�gq�egz%¯�wxWZeBcy\@z¨«Peg|Zz°����cykn�¬Oegcd\Ez3[.��hi\EzC�¨wZeuq�eu����\'��¢¤��|Zsg�gq�egz�¦

[\[ ß^]_&_�`&a

Page 14: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

]U� ����� ��$ � ����������� � *�����������C� ÏoËØÕØÈ Í l ��������� �� ÏoËGÕÖÈ Í" ÆiÌná�ÙgÑ6ÕØÏ�ÕkilÒ�Ï�ÕØÈÓÏ�Ô@ÐÊÌyÉ6� � h�� � h�� � h�� F ÆiÌná�ÙuÑ6ÕØÏ�Õ ilÒ6Ï�ÕÖÈÓÏ�ÔUÐÊÌnÉ X#� h X�� h X�� h X�F h XGW� �!���! l�!�# $%��&'������()�7*0�G� -l.,/����7* �71#&2*,3 � �����! ���# $:�G& �����!()�+*,��� -".0/����+*0�+1#&2*43

� (��7*0(�8�., �� � (��7*,(�8�.0 ��

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ 9 �D� ; X��[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � ��A > �D�

��� A 9Q��� H 9Q�D� E � X � A >O�J� B >Q�D� E V� � A � ��L � � MO� X�� A X���LCXG� M��� � A 9 �D�UH 9 �J� E�� X#� A > ���CB > �D� E�S

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ ��� A=X �[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � � ; X��

� � A 9 �D��L � � M�F X�� A X���L > �D� MOW^ ��� A � �UH � � ^ ��� X#� A XG��H X�� ^ �J�

[�ÌyÜ@ÈÊÉØËØÕØÈÊÔUãEËGÌ%^ �J��A=X��

� � A 9 ���ZB 9 �D� E�F X#� A > ���IH > �D� E��\[�ZF A 9O�J� B 9Q�D� E W X F A >O�D� H >O�J� E K

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � F ; XGW[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � F�A=X�F

^ �D� A � ��L � F M�N X�F A XGW�LCX�� MOV^ �D� A ^ �D�QB � � BQP�F X#� A X�FQB'XG� BQP��

[�ÌyÜ@ÈÊÉØËØÕØÈÊÔUãEËGÌ%^ �D��A=X��

� � A 9 �D�UH 9 �D� E�� X�F A > �D�IH > �D� E�N

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � ��A=X��

^ �D� A ^ �D�CH � �

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � ��A=X�F

�ZF A ��� L �Z� M �^ �D� A ^ �D�IH � F^ �D� A ^ �D�IH � � ^ �D�^ �D� A ^ �D�CH � F ^ �D�

` ���+����� ^ a?c ^Ue f g

²����gfxqo\ ��(�����cnky�C¬ZeBcy\@z�[.��hi\Ez°��[�wx��\'[�\'|Ckoeukd��sg|Isg�7�Ckdq�egcocd\'|Ieu���gsBqd��kdWZ[��ªsBq #7sg|Cko\!h¨k�] 3 ��\!�àk 5 eu|Oz� 3 qd���gWCk 5 ¦

ÞkjK[�Þml

Page 15: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

������������ �������������������� �!�"���#�%$&� ]@�

��������� �� ÏoËGÕÖÈ Í l ��������� C� ÏoËØÕØÈ Í" ÆiÌná�ÙuÑ6ÕØÏ�Õ ilÒ6Ï�ÕÖÈÓÏ�ÔUÐÊÌnÉ6� � h�� � h�� � h�� F h�� W h�� N h�� V ÆiÌná�ÙgÑ6ÕØÏ�ÕkilÒ�Ï�ÕØÈÓÏ�Ô@ÐÊÌyÉ X�� h XG� h X�� h X�F h X�W h X�N� �����! ���# $:�G& �����!()�+*,��� -l.,/����7* �71#&2*,3 � �!���! l�!�# $%��&'������()�7*0�G� -l.,/����7* �71#&2*,3

� (��7*0(�8�., �� � (��7*0(�8�., ��

� � A 9 ����H 9 �D� E�� X�� A > �J�6B > �D� EGW� � A � ��B 9 �D� EG� XG� A > �D�6B'X#� E�N� � A 9 �D��B 9 ��� E�� X�� A > �D�6B > �J� EGV� F A 9 �J��B � � E�F X�F A > ���UH X�� E�S

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � � ; XGW[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � W6A=X��[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ 9 �D� ; X�N[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ �ZN A >Q�D�[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ 9 �D� ; X��[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � V�A > ���

� � A � �IL � W M�� XG� A X�NCL > �D� M �� � A � FZL � N M�N X�N A X�WZLCX�� M:W� F A 9 �J�CL � V MO� XGW A X#�ULCX�F M:V� N A � NQB � W E�N� W A � �CL � N M�F

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � N�A=XG�

^ �D� A � NCH � F ^ �D���F A ��N H �QW E �

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � �6A=X�N[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � W6A=XGW

��N A ��F H �Q� E �^ �J� A � FCH � �^ �J� A ^ �J�CH � � ^ �J�^ ��� A � NCH � W ^ ���^��D� A ��N H ��� ^��D�

` ���!����� ^ a cd^Ue f g

²����gfxqo\ ?-(7prWZeBcy\@zC�C¬ZeBcy\@z&[.��hi\Ez���[�wx��\E[�\'|Ckoeukd��sg|°su�>¢¤��|xsg�BqoeBz�eu���gsgqo��kdWx[��ªsgq�#7sB|Bko\!h¨k ] 3 ��\'�àk 5eg|Zz�� 3 qd���gWCk 5 ¦

[\[ ß^]_&_�`&a

Page 16: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

]E� ����� ��$ � ����������� � *��

Q3

Q2

C11 Q5

Q7Q6

Q1 Q4

C12 C22 C21

Q3

CONTEXT 2

CONTEXT 1

U1

U2U3

²+���gfZqd\�]E��( #7WZeu��|3sg��wZqdsizifZ{!koc�eg|Zz3{'fiky«¿����|x\*�ªsBq�¢¤��|Zsg�gq�egz��reuqo��eu|CkE¦

� ��� � �!�� � ��� ����� � ������ ���! � � ��� ����������� � � � � �!��� ����� � � �

VXWZ\�qd\@cyfx��koc(su�0kdWx\j]E�¥egzZzi��ko��sB|Zc(eg|Zz3kdWx\ ��[ fx��kd��wx����{Ee�kd��sg|Oc�su�0kdWZ\��Ckdq�egcocd\'|�eu���gsgqo��kdWx[ WZeU®g\_kds¬�\�cnkosgqo\Ez ��|.kd\E[�wOsBqoegqdm*®�euqo��eg¬x��\EcE¦ � �O�P\�{!sB|Zcy��zi\Eq+koWx\�eu���gsBqd��kdWx[}wZqd\@cy\E|Bko\Ez ��|¥²����Bfxqo\­] 3 ��\!�àk 5 ¯Z]@�ko\'[�w�sgq�euqom­®�euqo��eu¬x��\Ec+euqo\|x\E\Ezi\@z���| wxWZeBcy\­]g¯A�(��|�wZWZegcd\X�x¯�eu|Zz �l��| wxWZeBcy\P�x¦K9lc��7\P{!sB[.wZfikd\7kdWx\wZqdsizifZ{!k�su�0[�eukdqo��{'\EcXsg��cd�.SE\U� zi��cnkoqd��¬xfiko\Ez&sg|�c��BfOeuqo\­�Bqd��zxcXsu�0cy�"S'\ "N# "/¯xeg����kdWx\@cy\_kd\E[�wOsBqoegqdm®�egqd��eu¬x��\Ec*egqd\�su�Xzi��[.\E|Zcd��sB|M�32A-%"/¦�¢¤��kdW e�|Zeg��®B\�w�sg����{'mI�ªsBq*kd\'[�w�sgq�euqom°®�euqo��eg¬x��\@c*eu����si{'eukd��sg|�¯koWx��cleu���gsBqd��kdWZ[ |x\'\@zxc -�<5')�32A-%" ,87_kd\E[�wOsBqoegqdm�\E��\E[�\'|Ckoc�sB|°\EeB{�W&wZqdsi{!\@cdcdsgq@¦ � |�kdWZ��c­cy\@{6kd��sg|/¯x�7\cykdfOzim.eg|Zz�{!sB[.wOeuqo\7koWx\lzi�./�\'qo\'|CkwZegqoeg����\'�i��[�wx��\E[�\'|Ckoeukd��sg|Zc����koW¥qd\E�Beuq�z kds*kdWZ\'��qsgwxkd��[.�"SEeukd��sg|Zc��|&kd\Eqd[¥cXsu��[�\'[�sgqom�fZcoeu�B\g¦� |£kdWx\¥zxeukoe��CwZegqoeg����\'����[�wx��\'[�\E|Bk�e�ko��sB|Zc_wxqd\@cy\E|Ckd\EzR¬¨mR²����Bfxqd\��&eu|Zz£�Z¯�koWx\�eu�����B|x[�\'|Ck*su�HX

eg|Zz Y ��[.wZ����\Ec7kdWZ\_eg����si{'e�ko��sB|�su�/kn�7s kd\'[�w�sgq�euqom�®Uegqd��eu¬Z��\@csg�+cy�"S'\0�32 "/#F�324-%"+¦�²ZfxqdkdWx\Eqd[�sgqo\g¯euk(kdWZ\�\E|Zz°su�0koWx\�{'sg[�wxfik�e�kd��sg|/¯ZkoWx\�eg�����g|x[�\E|Bk_su�DT ��|ZzifZ{'\Ec­eu|IsgkdWx\Eq(kd\E[�wOsBqoegqdm3®Uegqd��eu¬Z��\�sg�koWx\3coeu[�\jcd�"S'\g¦ � |±egzxzx��ko��sB| su��kdWZ\Ecd\j®�euqo��eg¬x��\Ec*koWZe�k�zisI|xs£|xsuk�zx\'w�\'|Zz±su�XkdWx\&{!sB[.wZfikoeukd��sg|�¯�¨kdq�egcocy\E|&eg���Bsgqo��koWx[ |x\E\EzxcPkoWxqd\E\*kd\E[.w�sgq�euqom�®�euqo��eu¬x��\EcP\EeB{�W�¬O\E��|x�¥sg�0cd�"S'\ �324-&"N#.�32DC "/¦rVXWZ��ceg���Bsgqo��koWx[ koWCfOc¥|x\'\@zxc�euk���\@egcyk <IKA�+7I2D>&" 73kd\'[�w�sgq�euqom©\'��\'[�\E|Bk�c¥w�\'q�wxqosi{!\@cdcdsgq@¦ #7sg|Z{'\'qo|x��|Z�¢¤��|xsB�gq�egz3®Uegqd��eu|Ck@¯¨kdWx\�eg���Bsgqo��koWx[ |x\E\Ezxc�sB|x��m�kn�Ps�ko\'[�w�sgq�euqom�®�euqo��eg¬x��\@cXsu�>cy�"S'\ �32A-%"N#N�32�C "��\EeBzi��|x�.kds¥e�kdsuk�eu�/eg[�sgfx|Ck�su� L4�+7?2DC "@7­ko\'[�wOsBqoegqdm¥\E��\E[�\'|Ckoc�w�\'q�wZqdsi{!\@cdcdsgq@¦� |>;�]E� =O�P\(wxqosgw�sBcd\Ez.��[�wx��\'[�\E|Bk�e�ko��sB|Zcrsg���¨kdq�egcocy\E|�eg|Zz�¢¤��|xsg�BqoeBz.eg���Bsgqo��koWx[¥c>fZcd��|Z��[.��hi\EzC�

wOeuq�eu����\E����cy[ ��Wx��{�W |x\'\@z�-1')�32 " ,873kd\E[�wOsBqoegqdm \E��\E[.\E|Ckoc�sg| \EeB{�W±wxqosi{!\Ecocdsgq.su�_{'sg|Ckd\'h¨k�]�eu|Oz� ')�32%" ,873ko\'[�wOsBqoegqdm \E��\E[�\'|Ckoc�sB| \@eg{�W±wZqdsi{!\@cdcdsgq�sg�_{!sB|Ckd\!h¨kj�i¦�VXWx\°��[�wx��\E[�\'|Ckoeukd��sg|Zc�wxqo\!«cd\'|Cko\Ez���|�koWx��c7wOeuw�\'q�{!��\Eegqd��m�qd\@zifZ{!\­kdWZ��cXeu[�sBfx|CkE¦>VXWx��c7qo\EzxfZ{6ko��sB|j{!sB[.\@c7�ªqdsB[ e {�WZeg|x�g\­��|3kdWx\{'sg[�[ fZ|x��{Ee�ko��sB|§w�sg����{!m§��Wx��{�W WOegc eg| ��|-AZfx\'|O{!\jfZwOsB|¡kdWx\3|CfZ[ ¬�\'q sg��|x\'\@zi\Ez§ko\'[�wOsBqoegqdm£\'��\!«[�\E|Bk�c'¦ � |�;�]E��=��7\lzx��z�[�\Ecocoeu�g\X�BqdsBfxwx��|x��zifxqo��|Z��{!sg[�[�fx|x��{'e�ko��sB|�wxWZegcd\Ec>kds�qo\EzifO{!\XkdWZ\(|¨fx[ ¬�\'qsg����e�ko\'|Z{'��\@c'¦rV+s¥zis�kdWOe�kE¯Z�7\*fZcd\Ez&¬x����kd\E[.w�sgq�euqom¥®�euqo��eg¬x��\EcPsu�>cy�"S'\ �32%"/¦��l\'qo\­koWx\*kd\E[�wOsBqoegqdm®�egqd��eu¬x��\Ec_euqo\�cy[¥eu����\'q 3 �324-&"§eBc_�7\.�g��®g\.��[�w�sgqdkoeg|Z{!\�kds3kd\E[.w�sgq�euqom�®�euqo��eg¬x��\.qo\'fZcd\���|Zcykd\@egzIsg�[�\@cdcoeu�B\­�BqdsBfxwx��|x�Z¦VXWx\¤wZWZegcd\6�¨¬Zegcd\Ez ��[�wx��\'[�\'|Ck�e�kd��sg|Oc su�°�Ckdq�egcocy\E| eg|Zz ¢¤��|xsB�gq�egz eu���gsBqd��kdWx[¥c£kdW¨fZc©|x\'\@z

L�')�32A-%" , 7 ko\'[�wOsBqoegqdm¡\E��\E[.\E|Ckoc�w�\'q�wxqos¨{'\EcocysBqE¦09­c koWx\�fZcd\3sg�l����cyk¥co{�Wx\@zifx����|x�¡eu����s��(c�e£¬O\'ky«

ÞkjK[�Þml

Page 17: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

������������ �������������������� �!�"���#�%$&� ]�I

���������C� ÏoËØÕØÈ Í l ��������� �� ÏoËGÕÖÈ Í" ÆiÌná�ÙgÑ6ÕØÏ�ÕkilÒ�Ï�ÕØÈÓÏ�Ô@ÐÊÌyÉ6� � h�� � h�� � ÆiÌná�ÙuÑ6ÕØÏ�Õ ilÒ6Ï�ÕÖÈÓÏ�ÔUÐÊÌnÉ X#� h X�� h X�� h X�F h XGW� �!���! l�!�# $%��&'������()�7*0�G� -l.,/����7* �71#&2*,3 � �����! ���# $:�G& �����!()�+*,��� -".0/����+*0�+1#&2*43

� (��7*0(�8�., �� � (��7*,(�8�.0 ��

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ 9 �D� ; X��[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � ��A > ���

^ �D� A 9 �J��L � � MO� X�� A X���L > �D� M �� � A 9 �D�UH 9 �D� E�� X#� A > �J�QB > �D� EGW� � A 9 �D�ZB 9 ��� E�� X�� A > �D�QB > �J� EGV

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ �Z� A=X �[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � � ; X�F

^ �D� A � ��L � � M�� X�W A X�FCLCX�� MOW��� A ��� B 9O�D� E � X � A >Q�D� B X � E N� � A 9 �J�6B � � E�F X#� A > ���IHYX�� E�S

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � � ; X�F[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � �6A=X��

^ ��� A 9 �D��L � � MOV X#� A X�FCLCX�� M�F

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � ��A > �D�

^ �J� A � ��L � � M�N X#� A X���H XG� PU�X�W A XGW�H X�� PR�

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ ��� A=X �

^ �D� A ^ �D�IH � � ^ �D�

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � �ZA=X��

^ �J� A ^ �J�CH � �^ �J� A ^ �J�ZH ^ �D� ^ �J�

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ ��� A=X W

^ ��� A ^ ���IH � � ^ ���^ �D� A ^ �D�CH � � ^ �D�

` ���+����� ^ a?c ^Ue f g

²����gfxqo\R]B]�(&�/��cykn�¨¬Zegcd\Ez¡[���h¨\@z©��[�wx��\E[�\'|Ckoeukd��sg|©sg�l¢¤��|Zsg�gq�egz eg���Bsgqo��koWx[ �ªsgq&#7sg|Ckd\'hCk�] 3 ��\'�àk 5eg|Zz�� 3 qd���gWCk 5 ¦

[\[ ß^]_&_�`&a

Page 18: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

]@E ����� ��$ � ����������� � *��ko\'q�qo\'fZcd\*su��kd\E[.w�sgq�euqom¥®�euqo��eg¬x��\EcE¯BkoWx\�����cykn�¨¬Zegcd\Ez3��[�wx��\'[�\E|Bk�e�ko��sB|&sg�0�CkoqoeBcdcd\'|&sg|Z��m�|Z\'\EzZcP�ªsgfxqko\'[�w�sgq�euqom¥®Uegqd��eu¬Z��\@cr�ªsBq�\Eeg{�W3wxqos¨{'\EcocysBq7s���|Z��|x� X}eu|Oz T.¯x�G¦ \g¦�¯¨kdWxsCcy\*su�+kdWx\�'Zq�cyk�{'sg|Ckd\'h¨kE¯xeu|Oz'O®g\ sB|�kdWxsCcy\�s���|x��|x� Y�¯�\@eg{�WIsg�>kdWx\@cy\�®Uegqd��eu¬Z��\@c(¬�\'��|x�jsg�cy�"S'\(�324-%"+¦�#7sg|O{!\'qo|x��|x��¢¤��|xsg�BqoeBz®�egqd��eu|CklkoWx\�����cykn�C¬Oegcd\Ez°��[�wx��\E[�\'|Ckoeukd��sg|Rsg�¢¤��|xsg�BqoeBz°|x\'\@zxc*��\Ecoc(ko\'[�wOsBqoegqdm�®�euqo��eu¬x��\Ec 3 sB|x��mI�sg��cd�.SE\ �324-&" 5 sg|�wxqos¨{'\EcocysBqoc7su�+kdWx\ 'Oqocyk({!sg|Cko\!h¨k�kdWZeg|3sB|&kdWxsCcy\*su�+kdWx\�cd\E{'sg|Zz3sg|Z\ 3 cnko�����\I 5 ¦

9­c��7\&�Peg|Ck kds¡\E®Ueg��fOe�kd\3kdWZ\3[¥e�hi��[ fx[ eu[�sgfx|Ck.su�­[�\E[.sBqdm¡|x\E\Ezi\@z±¬¨m sBfxq.eg���Bsgqo��koWx[¥c'¯��|Z{'��fZzx��|x� ��|xwxfxk3eu|Zz�sBfikdwZfikjzxe�k�ex¯X�P\°�������lsB|x��m {!sg|Ocy��zi\'q�kdWx\°[�\'[�sBqdm fZcdeg�g\�su�*wxqosi{!\Ecocdsgq�ceg����si{'eukd��|x�RkoWx��c¥[¥e�hi��[¥eg�(eu[�sgfx|Ckjeg|Zz±|Zsukjeu|�eU®g\Eqoeg�g\3®Ueg��fZ\g¦-²����Bfxqo\§]U�§cyWxs��(c.koWx\°|x\E\Ezi\@z[�\E[.sBqdm3cywZeB{!\*su��kdWZ\�zi��/�\'qo\'|Ck(®B\'q�cy��sg|Zc7�ªsBq "JV -.��WZ\'| �2®�euqo��\EcE¦ � |xwxfik­eg|Zz3sgfxwxfxk­zxe�k�e�euqo\k�eu�B\'|&��|Ckds¥eB{'{!sBfx|Ck���|&kdWZ��cX\@cnko��[¥e�ko��sB|�¦

0

64

128

192

256

320

384

0 1024 2048 3072 4096 5120 6144 7168

Mem

ory

spac

e (in

MB

)

Matrix sizes

Data-Parallel StrassenData-Parallel WinogradPhase-based Mixed-Parallel StrassenPhase-based Mixed-Parallel WinogradList-based Mixed-Parallel StrassenList-based Mixed-Parallel Winograd

²����gfxqo\.]@� (>v°\E[�sgqomjcywOeg{!\*|x\E\Ezi\@z��ªsBq�kdWZ\*zx��/�\'qo\'|Ck(��[.wZ��\E[.\E|Ckoe�ko��sB|ZcX��Wx\E| "JV -i¦¢§\�{'eg|§cd\'\�sg|£kdWZ��c 'O�gfxqo\.kdWZeuk_koWx\�zxeukoeU�¨wZeuq�eu����\'����[�wx��\'[�\'|Ck�e�kd��sg|§su�(�CkoqoeBcdcd\'|§eu���gsgqo��kdWx[

��c­kdWx\��Psgq�cnk_��|£kd\Eqd[¥c*sg�[�\'[�sgqom°fZcoeu�g\B¦ � eukoe��CwZegqoeg����\'��¢¤��|Zsg�gq�egzReu|ZzRwxWZegcd\6�¨¬Zegcd\EzI[.��hi\EzC�wOeuq�eu����\E����[�wx��\'[�\E|Bk�e�ko��sB|Zcl|Z\'\Ez°kdWx\.cdeg[.\�cywZeB{!\B¦(VXWx\'|I�P\)'O|Zz�kdWZ\ ����cykn�¨¬Zegcd\Ez�[���h¨\@zC�CwOeuq�eu����\E���[�wx��\'[�\'|Ckoeukd��sg|ZcE¦/¢§\�{'eg|�{!sB|Z{!��fZzi\7kdWOe�k0����koW.e­����[���ko\Ez.eU®�eu����eg¬x��\7[�\'[�sgqom�cywZeB{!\PwO\Eq0wxqdsi{'\EcocysBq3 �ªsgq���|Zcykoeu|O{!\���I�EBv>O�sg|�²����Bfxqo\�]U� 5 ¯i[���hi\EzC�¨wZeuq�eu����\E����cd[ fOcy��|x������cyky«¿¬Zegcd\Ez3eu���gsgqo��kdWx[¥cPeu����s��(c7kdsWOeu|Zzi��\*¬x���g�g\Eq�cy�"S'\@cXsu�0[�eukdqo��{'\EcPkoWZeu|�zxe�k�eU�¨wZeuq�eu����\E��®g\'q�cd��sB|Zc'¦

ÞkjK[�Þml

Page 19: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

������������ �������������������� �!�"���#�%$&� ]D�� ��� � ������� �� � � � � ���� ��� � � � � �i��� ,�, ´ , + �x� � � �+´ � � ¹ �x� �Ö� / �7´ ���i� � ¸ � + ¸ ¹ ´ ,Ø�7­fZq�{!sB[�[ fx|x��{'eukd��sg| [�sizi\'�X��c�koWx\3{'��eBcdcd��{'eu��� :���� ��Wx\'qo\� ��c kdWx\&|x\'kn�7sBqd�£��eukd\'|O{!mg¯� kdWx\[�\@cdcoeu�B\*cd�"S'\g¯�eu|Oz��3koWx\ ��|¨®g\Eqocd\�su�0kdWZ\�|x\!kn�Psgqo�j¬Oeu|Zzi����z¨koW�¦�9lclsgfxq(|Z\!kn�Psgqo�j��clcd����k�{�Wx\Ez%¯O�7\zxs.|Zsuk(WZeU®B\lkos�k�eu�B\*{'sg|Ckd\E|Ckd��sg|&��|Ckds¥eB{'{!sBfx|CkE¦� |M;�]�I@=G¯ie�{'��eBcdcd� 'O{'eukd��sg|¥su��sBwO\Eqoeukd��sg|Zc��|¨®Bsg��®g\Ez���|3�CkoqoeBcdcd\'|�eg���Bsgqo��koWx[ ��cPzisg|Z\l¬�\E{'egfZcd\l\EeB{�W

sg�>kdWxsCcy\ [�eUm�WZeU®B\�zi�./�\Eqd\E|Bk_\!hi\E{'fikd��sg|£cyw�\'\@z%¦l¢§\.{Eeu|Izi��cykd��|x�gfx��cdW°kn�7sj[�eg��|£{!��egcocy\@c(�vIeukdqo��h[�fx��kd��wx����{'eukd��sg|¡eu|ZzR[¥e�kdqo��h£egzxzx��ko��sB|�¦ koWx\.�ªsgqo[�\'q*��c�{'sg[�wxfiko\Ez£¬¨m�kdWx\¥qosgfiko��|Z\ �������I� � su�7kdWx\p O �\9­������¬xq�euqom­��Wx����\rkdWZ\7��e�kdkd\'q+��c�WZeg|Zzi��\Ez�¬¨m�e([�sizi� 'Z\Ez�®g\'q�cd��sB|�su�Ze(p O �\9­����|Ckd\Eqd|Oeu�gqosgfxkd��|x\g¦�i��|Z{'\­sBfxqX¬x��s¨{���cd�"S'\_��cX{'egqd\'�ªfx����m�{�WxsCcy\E|�¯¨�7\*zis.|xsukXWZeU®B\­{'eB{�Wx\_\�/�\E{6k�c'¦ �ls���¯i�7\_�g��®g\_zi\!k�eu����\@z[�sizi\E��cP�ªsBq�\Eeg{�W&su��kdWZ\_¬Oegcd��{_sgw�\'q�e�ko��sB|ZcX��|C®Bsg��®g\@z���|&sBfxq�zi�./�\Eqd\E|Bkleg���Bsgqo��koWx[¥c'¦

+ �i�'� � � �,ª� � � ,��Ø�g�i� � ¸ � � | ; ��=Oe_[�s¨zx\'�x��c>�g��®g\E| �ªsBq>kdWZ\l�i{'e"� 9(p 9 #�� qdsBfikd��|x\ �������I� � ¯U��Wx��{�W��clfZcd\Ez&��|°sBfxqleg���Bsgqo��koWx[ kos�{!sg[�wxfxkd\�koWx\ cd\'®g\E|3��|x|x\Eql[¥e�koqd��h3[�fx��kd��wx����{'eukd��sg|ZcE¦ ��\!k(¬�\6�kdWx\�|¨fx[�¬O\Eq_sg�qos��(c'¯�eu|Zz���koWx\�|CfZ[ ¬�\'q*su�7{'sg��fx[�|Zc_su�rkoWx\.wxqdsi{'\EcocysBql�gqo��zReu|OzRegcocdfx[�\kdWZeuk(��|¨®Bsg��®g\Ezj[¥e�kdqo��{!\Ec�egqd\�c��CfZeuqo\_eg|Zz3su�>cy�"S'\ � ¦0¢£\�WOeU®g\_kdWx\*�ªsB����s�����|x�.{'sBcyk�[.sizi\E���������� V&���� ��'���� ����� � � � ""! �#� � $ ! �@� , : ' �&%' :(� '% ,G� 7 :*)+�,%' :"� '%.- � � k/! J

��Wx\'qo\ �!�U�§��c&e©qosgfiko��|Z\£eu����s�����|x�±fZc�kos eB{��Cfx��qd\IkdWx\§\!hi\E{'fikd��sg|-kd��[.\£su��e cy\ �Cfx\'|Ckd��eu�qdsBfikd��|x\@c¥�ªsgq&e �g��®g\E|-cy\'kjsu� {Eeu������|x� wZegqoeg[�\!kd\EqocE¯ ��� ��� � ��c¥koWx\£cd\��Cfx\E|Ckd��eu�l{'sgfx|Ckd\EqdwOeuqdksu� ����� ��� � ¯:k ��c�kdWZ\°¬x��si{��±cd�"S'\g¯Peu|Zz�� '% eg|Zz�� '% 3 qd\@cyw�¦0� %' eg|Zz�� %' 5 egqd\3�ªfx|Z{!kd��sg|Zc¥sg��gqo��z°kdsgw�sg��sg�Bm&eg|ZzI{'sg[�[ fx|Z��{Ee�kd��sg|°wOe�kyko\'qo|°�ªsgq_¬Zeg|Zzi����z¨kdWReg|Zz°��e�ko\'|Z{'m&��|£e�qds�� 3 qo\Ecdw�¦{!sB��fx[�| 5 ¬ZqdsCegzx{EegcykE¦

+ �i�'� � � �x¹�¹��Ö� � ¸ � OP\E{'egfZcd\�su�%sgfxq7zxe�k�e�zi��cnkoqd��¬xfiko��sB|�¯Ceg����egzxzx��ko��sB|Zcreuqo\�\!hi\@{!fiko\Ez���s¨{Eeu����m.����kdWi«sgfik&{!sg[�[�fx|x��{'e�ko��sB|Zc¥egc�\!hiwx��eu��|x\Ez ��|¤�i\E{6ko��sB|¤�Z¦�]g¦�VXWZ\R{!sCcnk�sg�*eg|�eBzxzi��kd��sg|���c��g��®g\E|¬¨m

�21�343ZV(���� � ' �5�.� �#� � ""! �#� � $ ! ,�J��� � � � 6 �i�@� � � �i��� ,�, ´ ,�� ,�� ¸ � �ª� � ©�VXWZ\[¥eu��|�zi�./�\'qo\'|Z{'\¬�\!kn�P\'\'|��Ckdq�egcocd\'| eg|Zz�¢¤��|xsg�BqoeBz�eu���gsgqo��kdWx[¥c���c+kdWx\E��q�|¨fx[ ¬�\'q0sg�Zegzxzi��kd��sg|Oc'¦OPfik*¬�sukdW§eu���gsBqd��kdWx[¥cl|x\'\@z%��wZqdsizifZ{!koc*eu|OzI�jqo\Ezx��cykdqo��¬Zfikd��sg|Zclkds&eu�����B|Rcdsgfxq�{!\.zxe�k�ejeu|OzI¬xqo��|Z�¬Oeg{���kdWZ\�qd\@cyfx��k(sg|�{!sB|Ckd\!h¨k�]g¦V+s ¬�\_cykd������eg¬x��\�kds.wO\Eqy�ªsBqd[ egzZzi��ko��sB|Zc7����koWxsgfik�{!sB[�[ fx|x��{'eukd��sg|Zcsg|Z{'\l[¥e�koqd��{!\@cegqd\_eu�����B|x\Ez%¯

koWx\¥¬x��si{��Rcy�"S'\�WZeBc_kdsI{�WZeg|x�g\�zifxqo��|x�&kdWx\¥qo\Ezi��cykdqo��¬xfxkd��sg|�¦�²Zsgq���|Zcnk�eu|Z{'\g¯/²����gfxqo\�]E��cdWxs��(c_kdWx\|Z\'� zxe�k�eIzi��cnkoqd��¬xfiko��sB|§sg��[�eukdqo��h+X ��Wx\E| " V -x¦ 9­c���|©²����gfxqo\ Ii¯0cdfx¬Zco{!qo��wik��g��®g\EcE¯��ªsBq�\EeB{�W¬Z��si{���¯i��k�c�[¥e�koqd��h��CfZeuqdkd\Eq�eu|Zz&cyfZwO\Eqoco{!qo��wxk�{'sgqoqd\@cyw�sg|Ozxc7kds.kdWx\�¬Z��si{��C«¿{'mi{!����{_zi��cnkoqd��¬xfiko��sB|�¦

[\[ ß^]_&_�`&a

Page 20: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

]@� ����� ��$ � ����������� � *��Context 2Context 1

9 [D[�D�

9 � [�D�

9 FD[���

9 ND[���

9 �\[�D�

9 �D[�D�

9 W [���

9 V [���

9 [DF�J�

9 � F�J�

9 FDF�D�

9 NDF�D�

9 �\F�J�

9 �DF�J�

9 W F�D�

9 V F�D�

9 [!��D�

9 ����D�

9 F!����

9 N!����

9 �D��D�

9 �!��D�

9 W�����

9 V�����

9 [�W�J�

9 �DW�J�

9 F�W�D�

9 N�W�D�

9 �JW�J�

9 ��W�J�

9 WDW�D�

9 VDW�D�

9 [���D�

9 �D��D�

9 F�����

9 N�����

9 �J��D�

9 ����D�

9 WD����

9 VD����

9 [DN�J�

9 � N�J�

9 FDN�D�

9 NDN�D�

9 �\N�J�

9 �DN�J�

9 W N�D�

9 V N�D�

9 [D��D�

9 � ��D�

9 FD����

9 ND����

9 �\��D�

9 �D��D�

9 W ����

9 V ����

9 [�V�J�

9 �DV�J�

9 F�V�D�

9 N�V�D�

9 �JV�J�

9 ��V�J�

9 WDV�D�

9 VDV�D�

²����Bfxqd\.]E��(>vIeuwxwZ��|x�¥sg�+[¥e�koqd��hJX sB|&eH- #�C¥wxqdsi{'\EcocysBq7�Bqd��z3eu�àkd\EqXkdWx\�eg�����g|x[�\E|Bk(cykd\Ew�¦V+s*wO\Eqy�ªsBqd[ cdfZ{�W.e­qo\Ezx��cykdqo��¬Zfikd��sg|�¯g�7\PfZcd\Ez kdWx\�qo\Ezi��cykdqo��¬xfxkd��sg| qosgfxkd��|x\Xsu���x{'e-�\9(p 9 #�� ;�]��@=G¯

¬Oegcd\Ez§sg|§kdWZ\j{'eukd\'qowx������euq�eg���Bsgqo��koWx[ ��Wx��{�W ��c�\�Q�{!��\'|Ck��ªsBq�[�sBcyk�{!sB[.[�fx|x��{'eukd��sg|§wZe�kdkd\'qo|�¦ � kocwZqd��|Z{!��wx��\���cXkdWx\ �ªsg����s�����|x�O¦X�¨sBfxq�{!\�eu|Oz&zx\Ecykd��|Ze�ko��sB|°wxqdsi{'\EcocysBqoc�egqd\�{'sg|Zcd��zx\'qo\Ez&egcle¥cd��|x�g��\�����cykE¦%7eg{�W wxqdsi{'\EcocysBq�{'sg[�wxfiko\Ec.koWx\R{!sB[�[ fx|x��{'eukd��sg| wZe�kdkd\Eqd|�¯P�G¦ \B¦�¯7kdWx\Ieg[�sgfx|Ck�su�*zxe�k�e¡��k�WOegc.kds\'hx{�WZeu|x�B\�����kdW£eg|¨m3sgkdWx\Eq­��|¨®gsg��®g\@z�wxqos¨{'\EcocysBqE¯O��|O{!��fZzi��|x�j��kocd\'���n¦ 7­|Z{!\ kdWx��c_wZe�kdkd\'qo|R{'sg[�wxfiko\Ez%¯\@eg{�W�wZqdsi{!\@cdcdsgq+WZegc ����� � �R� '�� �D��$7����kdW.sukoWx\'q�wxqosi{!\Ecocdsgq�c%kds_\!hx{�WZeg|x�g\7zxe�k�ex¦+²����gfxqo\_]'�­cdWxs��(c+Wxs��e*����cnksg���*wxqdsi{'\EcocysBqoc�qosg����c>�����g\(e�{Ee�kd\EqdwZ������egq0kds�\E|Zcdfxqd\XkoWZe�kPeu���Z{!sB[.[�fx|x��{'eukd��sg|Zcregqd\Xw�\'qd�ªsgqo[�\Ez%¦� fxqo��|Z�.\E®g\E|&cykd\EwZc'¯�� wZqdsi{!\@cdcdsgq�c7\!hi\E{'fikd\�cd\'���à«¿{'sg[�[ fx|Z��{Ee�kd��sg|Oc'¯i�G¦ \B¦�¯i[�\E[.sBqdm3{!sgwZ��\@c'¦V+s°\Ecykd��[¥e�kd\.koWx\�{!sCcnk�su�PkdWZ\¥qd\@zi��cykdqo��¬xfikd��sg|§��|C®Bsg��®g\@zI��|£koWx\�zxeukoeU�¨wZeuq�eu����\'�>eu���gsBqd��kdWx[¥cE¯%�7\

'Oqocyk(WZeU®g\*kosjzi\'kd\'qo[���|Z\�kdWx\�{!sg[�[�fx|x��{'e�ko��sB|�wZe�kdkd\'qo|ZcE¦ � |IkdWx��cl{Eegcd\�\Eeg{�W�wxqosi{!\@cdcdsgq�sg�0{'sg|Ckd\'hCk] 3 �G¦ \B¦�¯ " 7jcd\'|Zzi��|x�£wxqdsi{'\EcocysBqoc 5 cy\E|Zzxc��I¬x��s¨{��ic�su�(cd�.SE\��32�C "M#M�32DC " kosI�Rwxqosi{!\@cdcdsgq�c'¦°VXWx\\E|Ckd��qd\�[�eukdqo��h���clkoW¨fZc­{'sg[�[ fZ|x��{Ee�ko\Ez%¦_V/s&cy��[�wx�����ªm°sgfZq_eg|Zeu��micy��c(�P\�eBcdcdfx[�\�kdWZ\'qo\ ��clsB|x\g¯%eu|OzsB|x��mjsg|Z\g¯O{'sg[�[ fZ|x��{Ee�ko��sB|3w�\'q_cnko\'w°su��kdWZ\ {'eukd\'qowx������euq(eg���Bsgqo��koWx[�¦ �/\Eegzx��|x�¥fOc�kds�koWx\��ªsB����s�����|Z�[�sizi\E���ªsgq�koWx\�eu�����B|x[�\'|Ck�su�>e�[¥eukdqo��h��ªqosg[ e "N# "�wxqosi{!\EcocdsgqX�gqo��z�kds�e "N#N-%"�wxqos¨{'\EcocysBqX�gqo��z

� 1 � V���� V � 7 � : -%" 7 �7­|Z{'\[¥e�koqd��{!\@c+egqd\Peu�����B|x\Ez�¬OsgkdW�eg���Bsgqo��koWx[¥c�w�\'qd�ªsgqo[ �XwxqosizifZ{6k�c�\'®g\E|����xkdWx\P\!hi\@{!fiko��sB|�sgq�zi\'q

��c_zi��/�\'qo\'|Ck@¦­VXWx\@cy\�wxqdsizifO{6kocl��|¨®Bsg��®g\ �CfZegqyko\'q�c�su�r[¥e�koqd��{!\@c(sg�cy�"S'\(� zx��cykdqo��¬Zfikd\@zIsg|£e " # -&"wZqdsi{!\@cdcdsgq��gqo��z%¦K9­crcoeu��z ¬�\!�ªsBqd\XkoWx\�¬x��s¨{��.cy�"S'\XWZeBc>¬�\'\'|¥{�WZeg|x�g\@z�zxfxqd��|x�_kdWZ\(qd\@zi��cnkoqd��¬xfiko��sB|.cykd\Eweg|Zz3��c�|Zs���\��CfZeg��kdsYk� V �32�C "+¦�VXWZ\�{!sBcyk�[�s¨zx\'�/{'sgqoqd\@cyw�sg|Zzx��|x� kds¥wxqosizifZ{!kocX��cPkdWx\E|

ÞkjK[�Þml

Page 21: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

������������ �������������������� �!�"���#�%$&� ]@?

Step 5 Step 6 Step 7 Step 8

Step 1 Step 2 Step 3 Step 4

7 6 5 4

3210 0 1 2

7

6 5 4

3

345

7 0 1

6 2

2107

6 5 4 3

5 6 7 0

1234 3 12

4

5 6 7

0

6 7 0 1

5 4 3 2 4 3 2

5

6 7 0

1

²����Bfxqo\.]E��( #7sg[�[�fx|x��{'e�ko��sB|3wOe�kyko\'qo|3sg�+koWx\�{'eukd\Eqdwx������egqXeg���Bsgqo��koWx[�¦

� ������� V L����� � � � � � � � ��� � 7"�� ��� � 7-%"�� � � -����

: L � O � - T 7 ) � '7 ' :"� 7 '' - : � 7�� ' )�� '7 ' :(� 7 '' - � Jv°sizi\E��c­euqo\�zi��cykd��|x�Bfx��cdWx\@z&�ªsgqleBzxzi��kd��sg|Zc(eBc�]E�¥egqd\�{'sg[�wxfiko\Ez���|I�Ckdq�egcocy\E|°eu|Zz¡]�I ��|°¢¤��|xsu«

�BqoeBz%¦89lc7�ªsgqXwxqosizifZ{6k�c'¯CkoWx\Ecd\_eBzxzi��kd��sg|ZcP��|¨®gsg��®g\��CfZeuqdkd\Eqocsu�/[¥e�koqd��{!\@cPsu�+cd�.SE\0� zi��cykdqo��¬xfikd\@z�sB|e "N#N-%"�wxqosi{!\EcocdsgqP�gqo��z%¯i��\EeBzi��|x�.kds

� ���� ������ �1�343 V <I> � ���� � � �5�.� ��� � 7"�� ��� � 7-&"������ ¯xeg|Zz��� V ��� � �� 31�343 V <�K � ���� � � �5�.� � � � 7"�� � � � 7-&"������ J

� �0�7\�cdfx[ koWx\Ecd\�{!sCcnk@¯i�7\*sB¬ikoeg��|

�� �� ���� ���� ������ � V LO ���� � O ��� ����� � � � -&" ! � � � C " ! � � - T T : <I> O ���� � O �5�.� � � � -&" ! � � � C " ! T T: L4�+7C ) � '7 ' :"� 7 '' - : � � 7 � :3<9C " ) � '7 ' :"� 7 '' - : �&" 7 � �

�ªsBql�Ckdq�egcocy\E|&eg|Zz

[\[ ß^]_&_�`&a

Page 22: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

�g� ����� ��$ � ����������� � *��

� � �� ����� V ��� � �� 3 V LO �� � � O ��� ����� � � � -%" ! �#� � C " ! � � - T T :3<IK O ���� � O �5� � � � � -%" ! �#� � C " ! T T: LA�+7C ) � '7 ' :(� 7 '' - : � � 7 � :3<9C " ) � '7 ' :(� 7 '' - : � " 7 � �

�ªsBqXkdWx\�¢¤��|xsg�BqoeBzj®�egqd��eu|CkE¦

��� � � ) + � � ´ ¹ ��� �i��� , , ´ ,����'���x�@� ´ � � ,�� ¸ � �ª� � ©�¢¤WZ\'|3e�zxe�k�e���c[���cdcd��|x� sB|je {'sg|Ckd\'hCk7kds�\!hi\E{'fikd\*e {'sg[�wxfik�e�kd��sg|/¯B�P\lWZeU®B\(kos w�\'qd�ªsgqo[ e ��|Cko\'qy�{'sg|Ckd\'h¨k0{'sgw¨mg¦ � |.¬�sukoW���[�wx��\'[�\'|Ckoeukd��sg|ZcE¯u�P\7fOcy\X��s¨{Eeu����m�¬Z��si{��¨��|Z�_cd\'|Zz.eu|Oz qo\E{'\'��®g\7�ªfx|Z{6ko��sB|Zc�kdsw�\'qd�ªsgqo[ koWx\Ecd\ {!sB[�[ fx|x��{'eukd��sg|ZcE¦ OP\E{EeufZcd\_kdWx\�kn�Ps¥cyfZ¬i«G�Bqd��zxcXWZeU®B\�cdeg[.\�cd�"S'\�eu|Oz&coeu[�\�cdWZeuw�\g¯zZe�koe�\!hx{�WZeu|Z�g\�{'eg|¥¬O\_zisg|x\l��|�wOeuq�eu����\E�x¬O\'kn�7\E\'|¥wxqosi{!\@cdcdsgq>wZeg��q�cE¦+prqosi{!\@cdcdsgq�� ���#k����R���-T����� �ªqosg[ {!sg|Cko\!h¨k�]_\'hx{�WZeu|x�B\EcXzxeukoe.����koW�wxqdsi{'\EcocysBq�� ���#k����R���-T�����: "� /��|&{'sg|Ckd\'hCk­�x¦� |§¬OsgkdW¡®g\Eqocd��sg|ZclkdWx\Eqd\�euqo\&]E���¨\'|Zz���`�\@{!\'��®g\.sgw�\'q�e�ko��sB|Zc_��Wx\'qo\�e3[¥eukdqo��hIsg�Pcd�.SE\(�32A-%"§��c

{'sg[�[ fZ|x��{Ee�ko\Ez��ªqosg[ ej{'sg|Ckd\'h¨klkdsjkdWx\.sukoWx\'q@¦ � k*�g��®g\@c(fZc*e3{!sg[�[�fx|x��{'e�ko��sB|I{!sCcnk_[.sizi\E�+\ �CfZeu�kos

��� � 3 V ���^V K-%" 7 � 7 � :3<�� � J��wxqosizifZ{6k�cXeuqo\_{'sg[�wxfiko\Ez3sg|3kdWx\�'Zq�cnk�{!sB|Ckd\!h¨k(eu|Ozj�.sg|jkdWx\�cd\E{'sg|Zzjsg|x\B¦89­cX\EeB{�WjwxqosizifZ{6k

zx\Eeu��cX����kdW�[¥e�kdqo��{!\EcPsg�0cd�.SE\ �324-i¯i�P\*WZeU®g\��������� V C O ���� � O � � � � � � � � -%" ! � � � -&" ! �@�324- T : -1')�324-5, 7 � '' : - �324-�324-&" � '' T

V C ' �� � � '+� � ��� � ��k ��k �@�32A-A,G, : -A� 7 � '' : >&"&� '' J#7sB|Z{!\Eqd|x��|x��egzxzx��ko��sB|ZcP�7\_WZeU®g\­kds�zi��cnko��|Z�gfx��cyW&wxWZeBcy\'«0eg|Zz3����cnky�C¬ZeBcy\@z�eu���gsBqd��kdWZ[&¦ � |Zzx\'\Ez&egc

wZWZegcd\P�­��c0|xsuk>WZeg|Zzi��\Ez kdWx\�coeu[�\7�XeUm*��|.¬�sukdW���[�wx��\E[�\'|Ckoeukd��sg|ZcE¯UkdWx\�{'qd��kd��{'eg�iwZe�koW ®�euqo��\@c'¦ � krWZegc]U��egzxzx��ko��sB|ZcX�ªsgqlkdWx\ wxWZegcd\Ez¨�C¬ZeBcy\@z3eg���Bsgqo��koWx[ eg|Zz�sg|x��mR]@�.�ªsgq�koWx\ ����cnky�C¬ZeBcy\@z3sB|x\g¦P9lcocdfx[���|x�koWZe�kleg���%sBwO\Eqoeukd��sg|ZcP�Psgqo�¥sg|&[¥e�kdqo��{!\EcXsg�0cd�"S'\ �324-x¯i�7\*�g\'k

����� �� �1�343 V <I- O ���� � O �5� � � � � -%" ! � � � -%" ! T T V <I-S' ���� � ' �5�.� �Rk ��kU,G, J��� V ���1�343 V <�� O ���� � O �5� � � � � -%" ! �#� � -%" ! T T V <�� ' ���� � ' �5�.� �Rk ��kU,G, J

� �/�P\*cyfx[ koWx\Ecd\­{'sBcykocE¯¨�7\­�g\!k7kdWx\­�ªsg����s�����|x�.{!sCcnkP[.sizi\E���ªsgqP\Eeg{�Wj®g\'q�cd��sB|¥su�/[���h¨\@zC�CwOeuq�eu����\E��¨kdq�egcocy\E|3eu���gsBqd��kdWZ[

ÞkjK[�Þml

Page 23: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

������������ �������������������� �!�"���#�%$&� �x]

� � � ���� ���� ������ � V CW' ���� � '���� ����� �#k ��k �@�32A-A,E, : <�-S' ���� � ' � � � ��k ��k0,E,:0-5� 7 � '' : K-%" 7 � 7 � : >&" � '' :3<�� � J� � V ��� ���� ������ � V CW' ���� � '���� ����� �#k ��k �@�32A-A,E, : < � ' ���� � ' � � � ��k ��k0,E,:0-5� 7 � '' : K-%" 7 � 7 � : >&" � '' :3<�� � J��� � � � + � � ´ ¹ ��� �i��� , , ´ , � � ��¸ �Z���x¹����i� �Ö� � �@�

� | koWx\I¢¤��|xsg�BqoeBz±®�euqo��eu|Ck¥su� �Ckdq�egcocd\'|�eu���gsgqo��kdWx[�¯7kdWx\I¬Oegcd��{&sgw�\'q�e�ko��sB|Zc¥euqo\�kdWZ\Icdeg[.\IeBc¥��|�¨kdq�egcocy\E|j®g\Eqocd��sg|�¦ � �+�P\�{!sg|Ocy��zi\'qPkoWx\�{!qo��ko��{Eeu��wZe�koWZcXsu��wxWZeBcy\'«>eg|Zz3����cnky�C¬ZeBcy\@z���[�wx��\E[�\'|Ckoeukd��sg|Zc�P\(WZeU®B\�(0��cd\'|Zzxc �Uqo\E{'\'��®g\@c'¯O]@��eBzxzi��kd��sg|Zc7eu|Zz¥��wxqos¨zxfZ{6k�cr�ªsBqrkdWZ\­wxWOegcd\6�C¬Oegcd\Ez.��[�wx��\'[�\E|Bk�e�ko��sB|eg|Zz£]E��cy\E|Zzxc �Uqo\E{'\'��®g\EcE¯�?.eBzxzi��kd��sg|ZcE¯i�¥wxqosizifZ{6k�cP�ªsgqXkoWx\�����cykn�C¬Oegcd\Ez���[�wx��\'[�\'|Ckoeukd��sg|�¦VXWx\.{!sCcnk_[.sizi\E�/sg�>kdWx\��jwxqosizifZ{6k�c'¯OkdWZ\�{!sB[�[.sB|°wZeuqdk­sg�r¬OsgkdWR{'qd��kd��{'eg�/wZeukdWZcE¯�WZegc­eu��qd\@egzim

¬�\'\E|j�B��®B\'|¥��|3cd\E{6ko��sB|JIi¦Ê�i¦ �x¦ � �/�P\­zi\'koeg����kdWZ\­qo\'[¥eu��|x��|x� {'sg[�wOsB|x\'|CkXsu�%koWx\Ecd\_{'qd��kd��{'eg�ZwZeukdWZcE¯¨�7\WOeU®g\

� ��� ����� � 3 V ��� V � �+7C " 7 � : � �r¯Zeu|Oz� � V ���� � 3 V ��� V K5�+7-%" 7 � : < ��� �

�ªsBqXkdWx\�{'sg[�[ fx|Z��{Ee�kd��sg|Oc�eu|Zz

� ��� ����1�343 V <I- O ���� � O �5� � � � � -%" ! � � � -%" ! T T VB<I-S' ���� � ' �5�.� ��k �RkU,G,�¯xeg|Zz� � V ���1�343 V �

O ���� � O �5�.� � � � -&" ! � � � -%" ! T T V � ' ���� � ' �5� � ��k �RkU,E,���ªsBq�egzZzi��ko��sB|ZcE¦ � �(�P\�cyfZ[ eg���kdWx\@cy\j{'sBcykoc 3 {!sB[.[�fx|x��{'eukd��sg|ZcE¯�eBzxzi��kd��sg|Zc�eu|Zz§wxqosizifZ{!koc 5 �7\

sB¬ikoeg��|3kdWZ\*�ªsg����s�����|x�.[.sizi\E��c��ªsgq�sgfZq�[.��hi\EzC�¨wZeuq�eu����\'����[�wx��\'[�\E|Bk�e�ko��sB|ZcXsu��¢¤��|xsB�gq�egz3®�euqo��eg|Bk

� ��� ����� V�� � � �� 3 V C ' ���� ��'���� ����� ��k ��k �1�324-A,E, :3<I-S' ���� � ' �5�.� ��k �RkU,G,:0-A� 7 � '' : �C " 7 � 7 � : � "&� '' :"> � J

[\[ ß^]_&_�`&a

Page 24: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

�B� ����� ��$ � ����������� � *����� V ���� V�� � � �� 3 V C ' ���� ��'���� ����� ��k ��k �1�324-A,E, : � ' ���� � ' � � � �Rk ��k0,E,:0-A� 7 � '' : K-&" 7 � 7 � :"> "&� '' : < ��� J��� � � � � ¸ � �i� �Ö� ¸ � �.� � ¹¤� � ��, µ � �Ö�

V�eu¬Z��\@cl]_eu|Zz&��qo\EcdwO\@{6kd��®g\E��m��B��®B\­e cdfx[�[¥euqom�su��koWx\_zx��/�\'qo\'|Ck�{!sg[�w�sg|x\E|Ckoc7su��koWx\_{'sg[�wxfik�e�ko��sB|eg|Zz&{!sg[�[�fx|x��{'e�ko��sB|3[�sizi\E��cP�ªsBqXkdWx\�zi�./�\'qo\'|Ck�®g\'q�cd��sB|ZcPsu�0�¨kdq�egcocy\E|&eg|Zzj¢¤��|xsB�gq�egzjeg���Bsgqo��koWx[¥c(zZe�koe��CwZegqoeg����\'�Z�¨kdq�egcocy\E|�¯gwxWOegcd\Ez¨«0eu|Zz¥����cnky�C¬ZeBcy\@z�[.��hi\EzC�¨wZeuq�eu����\'�O�CkoqoeBcdcd\'|/¯BzxeukoeU�¨wZeuq�eu����\'�i¢¤��|xsu«�BqoeBz%¯¨wxWZeBcy\@z¨«eu|Zz3����cykn�¨¬Zegcd\Ez3[���hi\Ez¨�CwZegqoeg����\'��¢¤��|xsg�BqoeBz%¦

��������� ������������������������ ���! #"�$�%'& ()"�"�*+&!*, .-

-0/ 1 2 3 -#/ 1 24 &5�6()75758 -:9;(�&6( < =�?> =F@> = � A�B =�?> =F@>4 &5�6()75758 -DCFE�()758 G =�?> =�?> = � AIH =�?> =�?>4 &5�6()75758 -DJK*,7L& G =�?> =�?> = � A�M =�?> =�?>N *,-� .O)�6()"D9;(I&6( < =�?> =F@> = � AIP =�?> =F@>N *,-� .O)�6()"QCRE�()758 G =�?> =�?> = � AIH =�?> =�?>N *,-� .O)�6()"QJK*,7L& G =�?> =�?> = � S =�?> =�?>

V+eg¬x��\¥] (r�¨fx[�[¥euqom�su�>{!sB[.wZfikoeukd��sg|�{!sBcykocE¦

�����T�U���V�XW)������Y��:����������I��� Z � [ �� \��VW.]4 &5�6(�75758 -Q9;(I&6( ^)_�`ba.cId�ef_ >�?> `g_ �?>>ih j dlk�enm >�?> `gm �?>>�h `polk � m4 &5�6(�75758 -qCRE�()758 r�_Yc)s6k � `bs�_ >> t k�m >> ` j�u m4 &5�6(�75758 -qJK*,7L& r�_Yc)s6k � `bs�_ >> t k�m >> ` j�u mN *,-� )O)�6(."Q9v(�&6( ^)_�`ba.cId e _ >�?> `g_ �?>> h j dlk e m >�?> `gm �?>> h `polk � mN *,-� )O)�6(."qCRE�(�758 w)_YcId'k � `bs�_ >> t k�m >> `bxlw�mN *,-� )O)�6(."qJ�*,7L& r�_Yc)s6k � `bs�_ >> t k�m >> ` j�u m

V�eu¬x��\�� (�¨fZ[.[¥egqdm�sg��{'sg[�[ fx|Z��{Ee�kd��sg|�{!sCcnk�c'¦

ÞkjK[�Þml

Page 25: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

������������ �������������������� �!�"���#�%$&� �g�

� |&{!sg[�[�fx|x��{'e�ko��sB|j{!sCcnk�c'¯C®�eu��fx\Ec7su� � '' ¯5� '7 ' ¯5� 7 '' ¯5� '' ¯5� '7 ' ¯ieu|Oz� 7 '' {'eu|j¬O\_qd\Ewx��eg{!\@z¥¬¨m.kdWx\E��qeB{6kofZeu��®�eg��fx\@c 3 egcocyfx[���|x��kdWOe�kl�P\�WZeU®g\�e¥kdqo\'\�¬ZeBcy\@z3¬ZqdsCegzx{Eegcyk 5 ¯Zqd\@cyw�\E{!kd��®g\'��m�������� 7 '(" ,� # �@2%"/¯'����� � 7 '(" ,�� :=<I,�#.�@2%"/¯�'����� � 7 '(" ,�� :=<I,�#.�@2%"/¯������ � 7 ' " ,�� �r¯�'�������� 7 '(" ,��:H<�,�#.�r¯ueu|Oz='�������� 7 '(" ,��:H<�,�#.�r¦V�eu¬Z��\��.�g��®g\@cPkdWx\*|x\E� {'sg[�[ fx|Z��{Ee�kd��sg|�{!sCcnk�c'¦

�����T�U���V�XW)������Y��:����������I��� Z � _ m4 &5�6()75758'-Q9;(�&6( ^ `ia��?s������� � �+k���� ` j �5cId'k j dlk��?s������� � �+k����F` j �K`golk �4 &5�6()75758'-qCFE�()758 r)c.s!k � `bs������� � �+k����Kc6k t k������� � �+k����F` j�u4 &5�6()75758'-qJK*,7L& r)c.s!k � `bs������� � �+k����Kc6k t k������� � �+k����F` j�uN *,-� .O��6(."D9v(�&6( ^ `ia��?s������� � �+k���� ` j �5cId'k j dlk��?s������� � �+k����F` j �K`golk �N *,-� .O��6(."QCRE�(�758 w.c�dlk � `bs������� � �+k����Kc6k t k ������ � �+k!��� `gwN *,-� .O��6(."QJ�*,7L& r)c.s!k � `bs������� � �+k����Kc6k t k������� � �+k����F` j�uV�eu¬Z��\��-(�ifx[�[�egqdmjsu��{'sg[�[ fZ|x��{Ee�ko��sB|&{'sBcykoc�egcP�ªfZ|Z{6ko��sB|Zc�su�,� eu|Zz �%¦

� e�k�eU�CwOeuq�eu����\E�(��[�wx��\'[�\E|Bk�e�ko��sB|Zc�sg���¨kdq�egcocy\E|-eu|Zz�¢¤��|xsg�BqoeBz WZeU®g\Ieg��[�sBcyk�koWx\£coeu[�\I{!sCcnk[�sizi\E�ئ � |Zzi\E\Ez�koWx\lsB|x��m.zi�./�\'qo\'|Z{'\l��crkoWx\l|¨fx[�¬O\Eqsu�/eBzxzi��kd��sg|Zc 3 ]@��®ic'¦P]@I 5 ¦^9lcrkdWx\*{!sB[�wx��\'hi��knmsg��eBzxzi��kd��sg|.��c���|H&('X�+7I,!¯�kdWZ��c>zi�./�\'qo\'|Z{'\P��c0|x\E�g�����B��¬x��\P����koW.qo\'�Ceuq�z*kds_koWx\Xqd\@cnk>su�OkdWx\�{'sBcyk0[�sizi\E�ئ²Zsgq�koWx\°cdeg[.\�qo\EeBcysB|�¯�P\°{!��eu��[ kdWZeuk¥kdWx\�[���h¨\@zC�CwOeuq�eu����\E�P��[�wx��\'[�\'|Ckoeukd��sg|Zc�sg���¨kdq�egcocy\E| eu|OzkoWx\.����cnky�C¬ZeBcy\@z°��[.wZ��\E[.\E|Ckoe�ko��sB|Isu�7¢¤��|xsB�gq�egz�WZeU®B\�{'sg[�wZeuq�eu¬Z��\ {'sBcyk­[�sizi\'��cE¦)#7sB|Z{!\Eqd|Z��|x�jkdWx\wZWZegcd\6�¨¬Zegcd\Ezj��[�wx��\'[�\'|Ck�e�kd��sg|�sg��¢¤��|xsg�BqoeBz%¯¨kdWZ\�qd\@zifx|Zzxeg|Ck�{'sg[�wxfik�e�ko��sB|&��|ZzifO{!\Ecle�qo\EzxfZ{6ko��sB|sg�/koWx\�{!sB[.[�fx|x��{'eukd��sg|�{!sBcykE¦²����gfxqo\�]@I*cdWxs��(c>��|Zcykoeu|Cko��eukd��sg|Zc>sg��cdsg[�\l{'sBcykr[�s¨zx\'��c>����koW�\!hiw�\'qo��[�\E|Bk�eu����m�[�\Eegcdfxqo\Ez�®�eu��fx\Ec

�ªsBq �%¯ �§eg|Zz3cd\��Cfx\E|Ckd��eu��\!hi\E{'fikd��sg|3kd��[�\EcPsu��wxqdsizifO{6kocXeg|ZzjeBzxzi��kd��sg|ZcE¦>VXWx\Ecd\*®Ueg��fZ\EcX{!sBqdqo\EcdwOsB|Zzkos�e�{!��fZcykd\'qsu��p�\E|Ckd��fx[ �d�y� {!sB|x|x\@{6kd\@z.¬Cm.e�²ZeBcnk�%>koWx\'qo|x\'k|x\!kn�Psgqo��¦/¢§\l{'eg|¥cy\E\XkdWZeuke*[.��hi\EzC�wOeuq�eu����\E�(��[�wx��\'[�\E|Bk�e�ko��sB|�eg{�Wx��\'®B\Ecj¬O\'kykd\Eq3w�\'qd�ªsgqo[¥eu|Z{'\°kdWOeu|�zxeukoe��CwZegqoeg����\'����[�wx��\E[�\'|Ckoeukd��sg|Zc��WZ\'|jsB|x\_WZegckds.[ fx��kd��wx��m�[¥e�koqd��{!\@c7zi��cykdqo��¬xfxkd\Ez3sg|&zi��cªtnsB��|Ckd\@z��Bqd��zxcE¦�VXWx��cP�Beg��|&{'eu|j¬O\*�gqo\Eeukd\EqkoWZeu|�C�K�L sg|°e�{!��fZcykd\Eq(cyfZ{�W�egcPkoWx\�sg|Z\*cd��[ fx��e�ko\Ez3��|°²����Bfxqo\.]�Ii¦" # �P��X����4 ��#3����5&"%$ �X�0"/�'��#

²Z\'� �7sBqd�¥WZeBcX¬O\E\'|&wxqosgw�sBcd\Ezj�ªsgqPkdWx\���[�wx��\'[�\'|Ckoeukd��sg|&su�0|¨fx[�\'qo��{Eeu���g\Eqd|x\E��cPsB|3Wx\'kd\EqdsB�g\'|Z\'sgfOcwZ��euky�ªsgqo[¥c'¦ � |J; �x¯���=G¯ukdWx\�eufxkdWxsBqoc�wxqos�®g\7kdWx\ �lp>«¿{'sg[�wx��\!ko\'|x\@cdc0sg�ZkdWZ\XzZe�koeu«¿zx��cykdqo��¬Zfikd��sg|.wxqosg¬x��\'[�ªsBqlkoWx\�{'��eBcdcd��{'eu�+[¥e�kdqo��hC«¿[ fZ��ko��wx����{'e�ko��sB|Iwxqosg¬x��\'[:����kdWRzx��/�\'qo\'|Ck*wxqdsi{'\EcocysBqoc(cdw�\'\EzReu|ZzRwxqd\@cy\E|Cke&w�sg��m¨|xsg[���eu�r{!sB��fZ[.|x«G¬ZeBcy\@zRWZ\'fxqo��cykd��{u¦3VXWx\�eu���gsBqd��kdWx[¥c_wxqo\Ecd\'|Ckd\@z£egqd\�®B\'qomI\�Q�{!��\'|Ck�¬xfxk�kdWx\zx��cykdqo��¬Zfikd��sg|3fZcy\@zj��cXWx���gWx��m¥��qdqo\'�Bfx��egqPeg|Zz���\EeBzxc7kds�Wx���gW3qd\@zi��cykdqo��¬xfikd��sg|&{!sCcnk�c7��WZ\'|&fZcy��|x��sukoWx\'q�B\'qo|x\'��cP¬O\'�ªsgqo\ 3 eu|Oz&eu�àkd\Eq 5 koWx\*[�eukdqo��h¨«¿[¥e�kdqo��h�wxqdsizifO{6kE¦

[\[ ß^]_&_�`&a

Page 26: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

�u� ����� ��$ � ����������� � *��

0

20

40

60

80

100

120

140

160

180

0 512 1024 1536 2048 2560 3072 3584 4096 4608 5120

Tim

e (in

Sec

.)

Matrix sizes

Data Parallel StrassenList-based Mixed Parallel Winograd

²����gfxqo\ ]�I (�VXWx\'sBqd\'kd��{'eu��w�\'qd�ªsgqo[�eg|Z{!\lsu�/zxeukoeU�¨wZeuq�eu����\'�O�CkoqoeBcdcd\'|/¯BwxWOegcd\!«>eu|Zz¥����cnky�C¬ZeBcy\@z�[.��hi\EzC�wOeuq�eu����\E��¢¤��|xsB�gq�egz�sB|°e.{'��fOcnko\'q�su�0p0\'|Ckd��fx[ �y�y� {!sB|x|x\@{6kd\@zj¬¨mje¥²Oegcyk %>kdWx\Eqd|Z\!k(|x\'kn�7sBqd��¦

�¨\E���Pegzxegwikd��|x�j����¬xqoegqd��\Ec�kds&®Uegqd��eu¬Z��\���sBeBzxc(WOeU®g\ eg��cds�¬�\'\'|§cnkofZzi��\Ez ; �g��=G¦_p>euq�eu[�\!ko\'q�c 3 ¬Z��si{��cd�"S'\eg|Zz*�gqo��z*cdWZeuw�\ 5 euqo\r{!sB[.wZfikd\@z�e�k/qofx|i«Gkd��[�\g¦ �ls��7\E®g\EqE¯'kdWx\Pzxe�k�e(zi��cnkoqd��¬xfiko��sB|���c+cnko�����Bqo\'�Bfx��euq@¦VXWZ��cXsBwikd��[��.S@e�kd��sg|&��[.wZqds�®B\Ec7kdWx\�w�\'qd�ªsgqo[¥eu|Z{'\­¬Zfik(��k(��cX|xsgkXqo\Eeg����m¥kosg��\'q�eu|Ck7kos�Wx���BW&zx��/�\'qo\'|Z{'\Ec¬�\!kn�P\'\E|jkoWx\�wxqosi{!\Ecocdsgq�c7cdwO\E\Ez%¦V+slzi\Eqd��®g\>koWx\WZ\!kd\EqdsB�g\E|x\'sBfZc%eu���gsBqd��kdWx[ �ªqosg[ sgfxq�WZsg[�sg�B\'|x\EsgfZc%��[�wx��\'[�\'|Ckoeukd��sg|�sg�Z�Ckdq�egcocy\E|

eg���Bsgqo��koWx[�¯/�P\�WZeU®B\�kdsI[�s�®g\�koWx\j{'fiky«¿����|x\jzx\'w�\'|Zzi��|x�£sg|¡kdWZ\�qd\E��eukd��®g\�cdwO\E\Ez¡sg�XwZqdsi{!\@cdcdsgq�c*��|\@eg{�W&{!sg|Cko\!h¨kE¦>²����Bfxqo\�]@E�cdWxs��(ckdWx\�{�WZeg��|3su�/wZqdsizifZ{!koc�eg|Zz¥koWx\�wxqosgw�sBcd\Ez�{'fiky«¿����|x\�zi\'kd\'qo[���|Z��|x�koWx\�wx��eg{'\'[�\'|CkE¦H9lcXkoWx\�[¥e�koqd��h3[�fx��kd��wx����{'eukd��sg|���cXkdWx\ [.sCcnk�ko��[�\ {'sg|Zcdfx[���|x�.koegcd�jsg�>�¨kdq�egcocy\E|� ceg���Bsgqo��koWx[�¯/eu|§\�Q�{'��\E|Bk eg���Bsgqo��koWx[ WZeBc_kds°\'®B\'|x��mI¬Zeg��eg|Z{!\�kdWZ\¥[�eukdqo��h£[ fx��kd��wx����{Ee�kd��sg|¡��sBeBzR¬�\!«kn�P\'\E|¡kdWx\jkn�7sI{'sg|Ckd\'h¨kocE¦I²+���gfZqd\°]D��cyWZs��(c*kdWx\&cyWOeuqo��|x�°sg�Xkoegcd�ic�¬�\!kn�P\'\E|©{!sg|Cko\!h¨koc �g��®g\'|¡¬¨mkoWx\�����cyk eu���gsBqd��kdWZ[&¦3¢£\3{'eu| cd\'\�koWZe�k \@eg{�W {!sB[�wxfikoeukd��sg|§koeBcy�£\!hi\E{'fikd\@z¡sg| {!sB|Ckd\!h¨k�� 3 qd���gWCk 5{'sgqoqd\@cyw�sg|Ozxc'¯ueBc>[ fO{�W�egc>w�sBcocy��¬x��\g¯gkds*kn�Ps*koegcd�ic>su��kdWZ\lcdeg[�\��C��|Zz�\'h¨\@{!fiko\Ez�sB|�{'sg|Ckd\'h¨k(] 3 ��\!�àk 5 ¦

Q6 Q2 Q4 Q1 Q3 Q5 Q7

U2U1U4 U3 C12C21

CONTEXT 1

CONTEXT 2

²����Bfxqd\�]@E-(r�Ckdq�egcocd\'|�{�WZeu��|3sg�+wxqosizifZ{!koc(eg|Zz3{'fiky«¿����|x\*�ªsBqXkdWx\�WZ\!kd\EqdsB�g\E|x\'sBfZc7eg���Bsgqo��koWx[�¦

ÞkjK[�Þml

Page 27: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

������������ �������������������� �!�"���#�%$&� � I

��������� �� ÏoËGÕÖÈ Í l ��������� C� ÏoËØÕØÈ Í" ÆiÌná�ÙuÑ6ÕØÏ�Õ ilÒ6Ï�ÕÖÈÓÏ�ÔUÐÊÌnÉ6� � h�� � h�� � h�� F h�� W h�� N h�� V ÆiÌná�ÙgÑ6ÕØÏ�ÕkilÒ�Ï�ÕØÈÓÏ�Ô@ÐÊÌyÉ X�� h XG� h X�� h X�F� �����! ���# $:�G& �����!()�+*,��� -l.,/����7* �71#&2*,3 � �!���! l�!�# $%��&'������()�7*0�G� -l.,/����7* �71#&2*,3

� (��7*0(�8�., �� � (��7*0(�8�., ��

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � ��A > �D�

� � A 9 �D��H 9 �D� E�� X�� A > �D�UH > �D� E�N��� A 96��� H 96�D� E �

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � F�A=X��

� W A 9 �D��H 9 �J� E�� X�� A > �J�6B > �D� EGV��N A 96��� B 9:�D� E F

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � W ; XG�[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � W6A=X��

� V A � �CL � F M � X�� A X���L > �D� M:W� � A 9 �D�IL � W M��^ �D� A � V�H � � PR� XG� A > ���ZB > �D� E�S� F A 9 �J��B 9 �D� EGW

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � F ; X��[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � F�A=XG�

X � A >���� H >��D� E �\[

� W A � �ZL � � MO� X�F A X���LCX�� M:V� � A 9 �D�CL � F M�F^���� A �QW H �O� ^���� X � A >��D� H >��J� E K^ �D� A � V�H � � P��

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � �ZA=XG�

�QV A ��N L �O� M N X � A X F B X � BQP �

[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � �ZA=X��[�ÌnÜUÈÊÉØËØÕÖÈÊÔUã@ËØÌ � ��A=XG�

^ �D� A ^ �D�QB � W^ �D� A ^ �D�CH � V ^ �D�^ �J� A � ��H � � ^ �J�^ �D� A ^ �D�IH � � ^ �D�

` ���!����� ^ a cd^Ue f g

²����gfxqo\¥]�� ( �(\'kd\EqdsB�g\'|Z\'sgfOcX[.��hi\Ez���[�wx��\E[�\'|Ckoeukd��sg|�su��CkoqoeBcdcd\'|°eu���gsBqd��kdWZ[ �ªsgq�#7sB|Ckd\!h¨k�] 3 ��\'�àk 5eg|Zz�� 3 qd���gWCk 5 ¦

[\[ ß^]_&_�`&a

Page 28: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

��E ����� ��$ � ����������� � *��� ��¥�£�X�0�'�)��#�+����� ��"/5&��+"

V+s§\!hiwO\Eqd��[�\'|Ckoeg����m ®g\Eqd���ªm koWx\���[�wZeB{6k¥su�_[.��hi\EzC�¨wZeuq�eu����\'����cd[ sg| wZeuq�eu����\E�7��[�wx��\'[�\'|Ckoeukd��sg|Zc�sg��¨kdq�egcocy\E|¡eg|Zz ¢¤��|xsg�BqoeBz¡eu���gsgqo��kdWx[¥cE¯+�P\�q�eu|¡kd\@cnk�c sB| Wxsg[�sB�g\'|Z\'sgfOc�eu|Zz¡WZ\!kd\EqdsB�g\E|xsgfZc�wx��euky«�ªsBqd[¥cE¦�VXWZ\��ªsgqo[�\'q��c�� �� ��$& ���_��Wx��{�W���c7e�{!��fZcykd\Eqsu� �lp©\6�¨®g\E{!kdq�e_|xsizi\Ec 3 p0\'|Cko��fx[ �d�y� ���B��v ��S����kdW-� I�E§v>O sg�­[�\'[�sgqom©w�\'q¥|xsizi\ 5 {!sB|x|x\E{!kd\@z©koWxqdsBfx�gW�e¡²ZeBcnk&%>koWx\'qo|x\'k�|x\!kn�Psgqo� ®¨��e �(ppqdsi{!fZqd®B\��B�g�B�¥cy����ko{�WZ\EcE¦(VXWx\���e�kyko\'q­��c­¬xfx����k_fxw�sg|°koWx\�{'sg|x|x\@{6ko��sB|°su�0kn�7sjWxsg[�sg�B\'|x\EsgfZc�{'��fOcn«ko\'q�c�¬¨m±e¡²Zegcyk&%>koWx\'qo|x\!k�����|x��¦-VXWx\ 'Zqocyk�{!��fZcykd\'q¥��c�{'sg[�wOsCcy\@z±sg�­�¡p0\'|Ckd��fx[ prqds±�g�g�§v ��S{'sg|x|Z\E{6ko\Ez�koWxqosgfx�BWReu| %rkdWx\Eqd|x\'k_|Z\!kn�Psgqo��¯���Wx����\ kdWx\�cd\E{!sB|Zz°��|Z{!��fZzi\@c_�jp0\'|Ckd��fx[ �d� � Ig��v ��S{'sg|x|Z\E{6ko\Ez£kdWZqdsBfx�gW¡e�²Zegcyk)%>koWx\'qo|x\!k�|x\!kn�Psgqo��¦�VXWZ��c���[�wx����\@c�eI�i¦Ê��I&cdwO\E\Ez§qoeukd��s�¬O\'kn�7\E\'|§kdWx\kn�Ps¥{!sB|Bko\!h¨kocE¦²Zsgq�eu����\!hiwO\Eqd��[�\'|CkocE¯¨�7\_fZcd\Ez��ªsgq�{!sB[.[�fx|x��{'eukd��sg|ZcPe ®B\'q�cy��sg|jsu��koWx\ O �\9 #X�¥����¬Zqoegqdm�sg|jkdsBw

sg�7v°p � 3 �\9lv sg| � ��� ��$& ���3eu|Zz§v°p � # � sg|RkdWx\�Wx\'kd\EqdsB�g\'|Z\'sgfOclwx��e�kd�ªsgqo[ 5 ¦�VXWZ\.p O �\9­�R®�]g¦ �����¬xqoegqdm ��c¥fZcd\Ez �ªsBq���|Z|x\'q¥[¥e�koqd��h [�fx��kd��wx����{'eukd��sg|ZcE¦�VXWx\Icd\��Cfx\E|Ckd��eu��O � 9_�±�B\'qo|x\'����c�koWx\°sg|x\�B\'|x\Eqoeukd\@zI¬¨m 9>V � 9_��¦ �l\'qo\.�P\¥{!sB[.wOeuqo\�w�\'qd�ªsgqo[¥eu|Z{'\�sg�Peu���>su�PkdWZ\�eg���Bsgqo��koWx[¥c_wxqd\@cy\E|Ckd\Ez£��|�i\E{6ko��sB| Ix¦

� ��� � ����� � ��� � ��� � �� ���!���!� ������������

¢§\ q�eu|°\@eg{�W°��[�wx��\E[�\'|Ckoeukd��sg|Isg�7�Ckdq�egcocy\E|Ieu|ZzI¢¤��|xsB�gq�egz°eg���Bsgqo��koWx[¥c�sg|R�3eu|Zz°�C��wxqosi{!\Ecocdsgq�csg� � �� ��$&*��6¦r²����Bfxqo\Ec*]E��cdWxs��(c7kdWx\�qo\Ecdfx��kd��|x��\!hi\E{'fikd��sg|OcPkd��[.\@c�sg|���eu|Zz��C��wxqosi{!\@cdcdsgq�c'¦

7­|j��wxqdsi{'\EcocysBqocE¯UkdWx\­[�\Eegcdfxqo\Ez�\'h¨\@{!fiko��sB|�kd��[�\Ec7[�euko{�W�koWx\(koWx\'sBqd\'kd��{'eu�Zkd��[�\Ecrsg�/²����Bfxqd\ ]@Ii¦VXWZ��c�qo\Ecdfx��kl{!sBqdqosg¬�sgq�e�ko\Ec7sgfxq(eg|Zeu��micy��c'¦>pWZegcd\6�¨¬Zegcd\Ez3��[�wx��\'[�\E|Bk�e�ko��sB|Zc�eg{�Wx��\'®B\�cy�����BWCkd��mj¬�\!kyko\'qw�\'qd�ªsgqo[¥eu|Z{'\�kdWOeu|¥����cykn�¨¬Zegcd\Ez���[�wx��\E[�\'|Ckoeukd��sg|ZcE¦>VXWx��c7[¥eUm�¬�\­zxfx\(kos�e�¬�\!kdkd\EqPcdm¨|Z{�Wxqosg|x�"SEeukd��sg|¬�\!kn�P\'\E|&{'sg|Ckd\'h¨kocX��|3koWx\�wxWZeBcy\!�C¬ZeBcy\@z�®B\'q�cy��sg|�¦¢§\*sg¬ik�eu��|¥kdWx\�coeu[�\l�¨��|Zz�sg�/qo\Ecdfx��k�cPsg|3�C��wxqosi{!\@cdcdsgq�c'¦+¢£\*{'eg|3eg��cds�cd\'\lkdWZeukPfZcd��|x����kd��[�\Ec

[�sBqd\�wxqosi{!\@cdcdsgq�c'¯U�P\l{'eg|¥zi\Eeg�x����koW¥�*kd��[�\Ecr¬Z���B�g\'q>[¥eukdqo��{'\EcE¦�VXWx\­cd{Eeu��eu¬x������knm�sg��sBfxqreu���gsBqd��kdWZ[�c��| ko\'qo[ sg�([�\'[�sgqom§{'sg|Zcdfx[�wiko��sB| ��c kdW¨fZc�®g\Eqd� 'Z\Ez�¦R²xsBq.{'sg[�wZeuqo��cysB|§wZfxqdw�sBcd\��P\jq�eu|©e°wZfxqd\�x{'e-�\9(p 9 #�� {'s¨zx\ koWZe�k�eu�����g|Zc­su�PX eg|Zz Yj¯/{!sB[�wxfikd\@c*e �������I� � sg|§eu����wZqdsi{!\@cdcdsgq�cleg|Zz°koWx\'|qo\Ezx��cykdqo��¬Zfikd\@cXkdWx\�qd\@cyfZ��k T.¦�¢§\�{Eeu|°cy\E\�kdWOe�k(koWx\ \'h¨\@{!fiko��sB|&kd��[.\�su�0kdWZ��c­{!sizi\���c(®B\'qom3{!��sBcd\*kdskoWxsBcd\�sg�(sgfxq�[.��hi\EzC�¨wZeuq�eu����\'�r��[�wx��\E[�\'|Ckoeukd��sg|�¦ OPfik@¯>egc��P\j�������Pcd\'\���| koWx\j|x\'h¨k.cd\E{6ko��sB|�¯0sgfxqeg���Bsgqo��koWx[¥cX{'eu|&\EeBcy����m�¬O\�eBzxeuwxkd\Ezjkds�qofx|�sg|&Wx\!ko\'qosg�B\'|x\EsgfZcPwx��e�kd�ªsgqo[�cE¦

� ��� � � ��� ����� �!� �!���� � �� ���!���!� � ������� ��

V+s�®�eu����zZe�kd\sgfxq�WZ\!kd\EqdsB�g\E|x\'sBfZc�®g\Eqocd��sB|­sg�¨kdWx\7����cnky�C¬ZeBcy\@z_[.��hi\EzC�¨wZeuq�eu����\'�g�CkoqoeBcdcd\'|�eu���gsBqd��kdWZ[&¯E�7\{'sg[�wZegqd\@z���kXsB|jkn�Ps.Wx\!ko\'qosg�g\E|x\'sBfZcPwx��e�ky�ªsBqd[¥cE¦>²+��q�cnk@¯i�7\*cy��[ fZ��eukd\Ez3Wx\'kd\'qosg�B\'|x\E��knm¥sB|M� �� ��$& ���6¦V+s3eB{�Wx��\'®g\ kdWx��c_wxqos¨{'\EcocysBqocl¬O\E��sB|x�g��|x�jkds�{!sg|Cko\!h¨k���\!hi\E{'fikd\.\EeB{�WR{!sB[�wxfikoeukd��sg|Zeg�/k�egcd�Icy\E®g\Eqoeg�ko��[�\B¦(VXWx\ |¨fx[�¬O\Eqlsu�r{'sg|Zcd\E{'fikd��®g\�\'h¨\@{!fiko��sB|Zclzx\ 'Z|x\@c�koWx\ �dcyw�\'\@z�qoeukd��sD��¬O\'kn�7\E\'|I{'sg|Ckd\'hCk�c'¦ � kWOegc­kds�¬�\.|Zsukd��{!\@zIkoWZe�k�koWx\�|x\!kn�Psgqo�°��c_cykd�����rWxsg[�sB�g\'|Z\'sgfOcl��|RkoWx��c*wx��euky�ªsBqd[�¦�²+���gfZqd\&])? 3 kosgw 5wZqd\@cy\E|Bk�c�kdWx\lqd\@cyfx��kocrsu��kdWx��cr\'h¨w�\'qo��[.\E|CkE¦�¢£\l{Eeu|�cy\E\XkdWZeukrkdWZ\(�Beg��|.��|O{!qo\Eegcd\Ec>eBc0kdWZ\lcyw�\'\@z.qoeukd��s¬�\!kn�P\'\E|¥kdWZ\(kn�Ps�{!sB|Ckd\!h¨k�c'¦ 7l�/{!sBfxqocd\g¯B�7\­��s¨sCcy\�ko��[�\_��Wx\'|�kdWx\_\'|Ckd��qo\l{!��fZcykd\'qP��cWxsB[�sg�g\E|x\'sBfZcE¦

ÞkjK[�Þml

Page 29: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

������������ �������������������� �!�"���#�%$&� ���

0

50

100

150

200

250

300

350

400

0 1024 2048 3072 4096 5120 6144 7168 8192 9216 10240

Tim

e (in

Sec

.)

Matrix sizes

Data Parallel StrassenData Parallel WinogradPhase-based Mixed Parallel StrassenPhase-based Mixed Parallel WinogradList-based Mixed Parallel StrassenList-based Mixed Parallel WinogradPure ScaLAPACK version

0

20

40

60

80

100

120

140

160

180

0 512 1024 1536 2048 2560 3072 3584 4096 4608 5120

Tim

e (in

Sec

.)

Matrix sizes

Data Parallel StrassenData Parallel WinogradPhase-based Mixed Parallel StrassenPhase-based Mixed Parallel WinogradList-based Mixed Parallel StrassenList-based Mixed Parallel Winograd

²����gfxqo\�]@�-(�#7sg[�wZeuqo��cysB|.sg�%kdWx\­\!hi\E{'fikd��sg|�kd��[.\­su�%koWx\­zi�./�\'qo\'|Ck7WxsB[�sg�g\E|x\'sBfZc>��[�wx��\E[�\'|Ckoeukd��sg|Zcsg��kdWZ\_[¥eukdqo��h�wxqosizifZ{!kX�B\'qo|x\'��wxqo\Ecd\'|Ckd\@z���|jkoWx��cXwZeuw�\'qXsg|�� 3 kdsBw 5 eu|Zz3�B� 3 ¬�sukdkdsg[ 5 wxqosi{!\Ecocdsgq�csg� � �� ��$&*��6¦VXWx\E|¥�7\Xqofx|�sgfxq>[���hi\EzC�¨wZeuq�eu����\E�i��[�wx��\'[�\E|Bk�e�ko��sB|Zc 3 WxsB[�sg�g\E|x\'sBfZc>eu|Oz�WZ\!kd\EqdsB�g\E|x\'sBfZc 5 eu|Oz

e°�i{Ee-�\9(p 9 #�� {'sizi\�kdWZeuk�wO\Eqy�ªsBqd[¥c_koWx\�eu�����B|x[�\'|Ck�su��X eu|Zz+Yj¯/koWx\¥[ fx��kd��wx����{Ee�ko��sB|¡eg|Zz£kdWx\qo\Ezx��cykdqo��¬Zfikd��sg|¡su�PkdWZ\�qd\@cyfZ��k�sB|§sBfxq {'��fZcykd\Eq�su��{!��fZcykd\EqocE¦3²����gfxqo\°])? 3 ¬�sukdkdsB[ 5 cyWZs��(c_kdWZeuk�kdWx\

[\[ ß^]_&_�`&a

Page 30: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

�g� ����� ��$ � ����������� � *��

-40

-30

-20

-10

0

10

20

30

40

0 512 1024 1536 2048 2560 3072 3584 4096

Gai

n (in

%)

Matrix sizes

Speed Ratio = 1Speed Ratio = 2Speed Ratio = 3

-40

-30

-20

-10

0

10

20

30

40

0 256 512 768 1024 1280 1536 1792 2048

Gai

n (in

%)

Matrix sizes

Gain over homogenous versionGain over ScaLAPACK version

²����gfxqo\ ])?�(<B*eg��|Zcjsu�*kdWZ\RWx\'kd\'qosg�B\'|xsBfZc���[.wZ��\E[.\E|Ckoe�ko��sB|¤s�®g\Eq¥kdWx\RWxsg[�sg�B\'|x\EsgfZc�®g\'q�cd��sB| sB|e�cd��[�fx��e�kd\@z§Wx\!ko\'qosg�g\E|x\'sBfZc_wx��e�ky�ªsBqd[ ����koW zi��/�\'qo\'|Ck cdwO\E\Ez¡q�e�ko��sCc 3 kdsBw 5 ¦ B*eu��|§sB¬ikoeg��|Z\Ez§s�®g\Eq�x{'e-�\9(p 9 #�� eg|Zz3Wxsg[�sg�B\'|x\EsgfZcP®B\'q�cy��sg|3¬Cm�kdWx\�Wx\'kd\EqdsB�g\'|ZsgfZcP®g\Eqocd��sg| 3 ¬Osgkykosg[ 5 ¦

�Ceu��| ��c¥{!��\Eegq¥¬O\'kn�7\E\'| kdWx\°WZsg[�sg�B\'|x\EsgfZc.®g\Eqocd��sg|±sg�lkoWx\°[���hi\Ez ��[�wx��\E[�\'|Ckoeukd��sg| su���Ckdq�egcocy\E|eg���Bsgqo��koWx[ egcX�P\'����egcPkoWx\ �i{Ee-� 9lp 9 #�� {'s¨zx\*eg|ZzjkdWx��c�\'®B\'|3�ªsgq(cd[¥eu���%[�eukdqo��{'\EcE¦

ÞkjK[�Þml

Page 31: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

������������ �������������������� �!�"���#�%$&� ��?

� 8 1 ����� ���£��5j � ���P53��"+�!��#

VXWZ\�&(')�+79b cEd ^ ,�eBcym¨[�wikdsgkd��{j{'sg[�wx��\!hi��knm§su�­�¨kdq�egcocy\E|±eu|Zz©¢¤��|xsB�gq�egz¡eu���gsBqd��kdWZ[�c�{'eg|©sg|x��m¡¬O\eB{�Wx��\'®g\@z¥���+koWx\�zi\E{'sg[�wOsCcy��kd��sg|3��c�egwxwx����\@zjqo\E{'fxq�cy��®g\E��mB¦/¢§\*{Eeu|&\EeBcy����m�zx\'qo��®B\­qo\E{'fxqocd��®g\l®B\'q�cy��sg|Ocsg�>sgfxq­[.��hi\EzC�¨wZeuq�eu����\'�%��[.wZ��\E[.\E|Ckoe�ko��sB|ZcE¦ � |Zzi\E\Ez�\EeB{�W°su�0kdWZ\�cd\'®g\E|°��|Z|x\'q­wxqdsizifO{6kocl��|¨®Bsg��®g\Ec�e[¥eukdqo��h£zx��cykdqo��¬Zfikd\@z£sB| {!sg|Cko\!h¨k�]¥eu|Zz¡e&[¥e�kdqo��h§zi��cnkoqd��¬xfiko\Ez§sg| {'sg|Ckd\'hCk �x¦jVXWx\�egcocdfx[�wikd��sg|Zc�P\�[¥egzx\�kds�¬Zfx����z°sBfxq­eg���Bsgqo��koWx[¥c(euqo\*kdW¨fZc­qd\@cyw�\E{!kd\@z%¦�¢£\(tnfZcnk_WZeU®g\�kds���qo��ko\�®g\Eqocd��sg|Zc���WZ\'qo\koWx\&qd\@cyfx��k���c.zi��cnkoqd��¬xfiko\Ez sg|±{'sg|Ckd\'hCk��i¦ ²Zsgq.��|Zcykoeg|Z{!\�cdfZ{�W±e£����cnky�C¬ZeBcy\@z [���hi\@zC�CwZegqoeg����\'�7��[.«wZ��\E[.\E|Ckoe�ko��sB|Isu�X�Ckdq�egcocy\E|Reu���gsBqd��kdWZ[ {Eeu|I¬�\¥zi\'qo��®B\Ez�sg�rkdWx\�cdWZeuqo��|Z�jsu�rwZqdsizifZ{!koc_wxqo\Ecd\'|Ckd\@z°��|²����gfxqo\(�u�Z¦+v°sBqd\Es�®g\'q+kdWZ\�¬x��si{�� cd�"S'\���c0kdWx\�sB|x��m�wZeuq�eu[�\'kd\'q�kos�egzxegwikE¯geBc>e�zi��cnkoqd��¬xfiko��sB|.eg����s�����|Z�koWx\�{'sg[�wxfik�e�ko��sB|Isu�eBzxzi��kd��sg|Zcl����koWxsgfik*{!sB[.[�fx|x��{'eukd��sg|ZclWZegclkds3¬O\��g\'wxk­fx|Ckd���+kdWx\�[.sCcnk_��|Z|x\'q��\'®B\'�G¦�k V �32A- V "+¯u��Wx\'qo\'�%��c�koWx\X|CfZ[ ¬�\'q0su��qd\@{!fxq�cy��sg|�cnko\'wZc���c>e¤cy�"S'\Xcdeukd��cn�ªm¨��|x�­kdWx��c>{!sB|Zzi��kd��sg|�¦

Q6 Q2 Q4 Q1 Q3 Q5 Q7

U2U1U4 U3 C12C21

CONTEXT 1

CONTEXT 2

²����gfxqo\(�u��(��CkoqoeBcdcd\'|�{�WZeu��|.su��wxqdsizifO{6koceu|Zz�{!fxky«¿����|x\X�ªsgq>®B\'q�cy��sg|.��Wx\Eqd\PkdWZ\�qd\@cyfx��k¡sg|¥{!sB|Bko\!h¨k�i¦

OPfik���%e*qd\@{!fxq�cy��®g\P[���hi\@zC�CwZegqoeg����\'�¨��[�wx��\E[�\'|Ckoeukd��sg|�su�/�CkoqoeBcdcd\'|�eu���gsBqd��kdWx[}qo\EzxfZ{!\@c0kdWx\�|¨fx[.«¬�\'q*sg�7wxqosizifZ{6k�c'¯%��k���|CkoqdsizifZ{'\Ec*e3|xsg|i«¿|x\E�g�����B��¬x��\�eg[�sgfx|Ck*su�7{'sg[�[ fx|Z��{Ee�kd��sg|Oc'¦ 7­|§ejwZ��euky�ªsgqo[����kdW&�ÖeBcnk(wxqosi{!\@cdcdsgq�c7eg|Zz�e�cy��s���|x\'kn�7sBqd��¯xcdfZ{�W�egc6� �� ��$&*��6¯xkoWx\��Beg��|&��|&ko\'qo[�c(su��{'sg[�wxfik�e�ko��sB|{Eeu|¡|xsgk�zi��®C��zi\�sBfik�koWx\3{'sBcyk�su�XkdWx��c {'sg[�[ fZ|x��{Ee�ko��sB|¡eg[�sgfx|Ck ��|Z{!qo\EeBcy\B¦3p0\Eqy�ªsBqd[¥eu|O{!\���c�kdW¨fZc�Psgq�cyk>kdWZeg|�kdWx��csg��e�|xsg|i�Cqo\E{!fZqocd��®B\X��[�wx��\'[�\'|Ck�e�kd��sg|/¦�¢§\(wx��eu|�kds cykdfZzxm���Wx\'kdWx\Eq7qo\E{'fxq�cy��sg|�{'eg|��[�wxqos�®g\_wO\Eqy�ªsBqd[¥eu|O{!\_sg|°e.wx��euky�ªsBqd[ ����kdW°e.Wx���gW°{!sB[�[ fx|x��{'eukd��sg|��u{!sB[.wZfikoeukd��sg|�q�e�ko��sO¦

� � ��#&�7�!5&"/�'��# � #3���35��5j�0� 1 �����¥"

� |�kdWx��cPwZeuw�\'q@¯B�P\_WZeU®g\(wZqd\@cy\E|Bko\Ez�[���h¨\@zC�CwOeuq�eu����\E�Z��[.wZ��\E[.\E|Ckoe�ko��sB|Zc7su���CkoqoeBcdcd\'|�eg|Zz¥¢¤��|xsg�BqoeBzeg���Bsgqo��koWx[¥c0��|�koWx\lco{!sBwO\(sg��{!����\E|Bky�¨cd\'qo®g\Eq0egwxwx����{Ee�ko��sB|Zc 3 �G¦ \g¦�¯gkoWx\lzi�./�\Eqd\E|Ck7zxe�k�e*��|¨®Bsg��®g\Ez.��|�{!sB[�«wZfikoeukd��sg|Zcregqd\�eg��qo\EeBzim�zi��cnkoqd��¬xfiko\Ez�sg|¥zi��cÖtnsg��|Bkr�Bqd��zxc 5 ¦�¢£\({�WZsBcd\7koWx\Ecd\(eu���gsBqd��kdWZ[�c0¬�\E{EeufZcd\XkdWZ\'megqd\�{'sg[�wOsCcy\@z.su�%cy\E®g\'q�eu�izxeukoe��C��|Zzi\'w�\'|Ozxeu|Ck>koeBcy�ic0��Wx��{�W¥euqo\�\EeBcy��\'q0kos�cd{�Wx\@zifx��\(eu|Zz.wx��eg{'\({!sB[�«wOeuqo\Ez&kds&ej{'��eBcdcd��{Eeu�+[�eukdqo��h°[�fx��ko��wZ����{'eukd��sg|�¦ 7­fxq­eg���Bsgqo��koWx[¥clegqd\.cy��[�wx��\.eu|ZzReu����s�� kdWZ\�fZcd\ sg�WZ���BW�wO\Eqy�ªsBqd[¥eg|Z{!\�|¨fx[�\'qo��{'eu�O�g\Eqd|x\E��c 3 \B¦ �O¦�¯i�i{'e"� 9lp 9 #�� 5 ¦ 9(�àkd\Eq7�B��®¨��|x� zi\'koeu����c7eg¬OsBfikkoWx\lwxqosu«w�sBcd\Ez�[���hi\@zC�CwZegqoeg����\'�¨eu���gsBqd��kdWZ[�c0eg|Zz�koWx\'��q>qo\'��e�kd\@z ��cocyfx\@c'¯��P\X�BeU®B\rkdWZ\'sgqo\!ko��{Eeu�i[�sizi\'��c0su�OkdWx\@cy\eg���Bsgqo��koWx[¥c'¦3¢£\3eu��cysIcdWxs��P\Ez§Wxs�� eu| ��[.wZ��\E[.\E|Ckoe�ko��sB| su�XkdWZ\�����cykn�C¬Oegcd\Ez§®g\Eqocd��sB|£sg�l�Ckdq�egcocy\E|k�euqo�g\'kd��|x�(Wx\'kd\'qosg�B\'|xsBfZc+wx��euky�ªsBqd[¥c�{'eu| ¬O\�zi\Eqd��®g\@z*�ªqdsB[ kdWx\PWxsB[.sB�g\E|x\'sBfZc���[�wx��\E[�\'|Ckoeukd��sg|�¦�¢£\{'��eg��[ kdWOe�k_�CkoqoeBcdcd\'|Ieg���Bsgqo��koWx[���c([�sBqd\���|Ckd\Eqd\@cnko��|x��koWZeu|IkdWx\ {'��eBcdcd��{'eu�/[�eukdqo��h¨«¿[¥e�kdqo��h3wxqosizifZ{6k�ªsBqreu|.��[�wx��\'[�\E|Bk�e�ko��sB|.sB|¥e­WZ\!kd\EqdsB�g\E|x\'sBfZc�wx��euky�ªsBqd[�¦+V+s�®�eu����zZe�kd\XsBfxqreuwxwZqdsCeg{�W�¯��P\�{!sg[�wZegqd\@zkoWx\'sBqd\'kd��{'eg�x[�s¨zx\'��ceu|Oz�\!hiwO\Eqd��[�\'|Ckoeg�xqo\Ecdfx��k�crsg��sBfxqr�Psgqo� ����kdW�zxeukoe��CwZegqoeg����\'�i��[�wx��\E[�\'|Ckoeukd��sg|Zcsg�0�CkoqoeBcdcd\'|�eu|Ozj¢¤��|xsg�BqoeBz3eg���Bsgqo��koWx[¥c'¦%>hiw�\'qo��[�\'|Ck�c({!sBqdqosg¬�sgq�e�ko\Ez�sgfZqXkdWx\Esgqo\!kd��{'eg��eu|Zeg��micd��cE¦

[\[ ß^]_&_�`&a

Page 32: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

�B� ����� ��$ � ����������� � *��¢§\Ieu��[ kds©zx\'®g\E��sBw±qo\E{'fxqocd��®g\&®g\'q�cd��sB|Zc sg�­sBfxq¥[���hi\Ez¨�CwZegqoeg����\'�X��[.wZ��\E[.\E|Ckoe�ko��sB|ZcE¦ � |Ozi\'\@z

koWx\�{'sg[�wx��\!hi��knm&su� &('X�+7;b cEd ^ ,({Eeu|°sB|x��m&¬�\ qo\Eeg{�WZ\Ez&���>koWx\.�¨kdq�egcocy\E|�zx\E{!sB[�wOsCcy��kd��sg|°��clegwxwx����\@zqo\E{'fxq�cy��®g\E��mB¦I¢£\&eu��cysRwx��eg|©kos£cykdfZzim eg|©sukoWx\'q.�PeUm£kds£wO\Eqy�ªsBqd[ [¥e�koqd��h [ fx��kd��wx����{Ee�kd��sg|±sg|±eg|WZ\!kd\EqdsB�g\E|x\'sBfZc�wx��euky�ªsBqd[ fOcy��|x��[���hi\@zC�CwZegqoeg����\'����cy[�¦/VXWx\���zi\@e*��c�kos��g\'\Ew.kdWx\(eg���Bsgqo��koWx[ zi\Ecd���g|x\@z�ªsBq�kdWZ\�Wxsg[�sB�g\'|Z\'sgfOc_{Eegcd\¥�g��®g\E|§��| cy\@{6ko��sB|§�O¦ �x¦ ��eg|Zz§\!hi\E{'fikd\¥��k sB|¡eIwx��euky�ªsBqd[ cyfZ{�W eBc*kdWx\sB|x\¥fZcd\Ez§��|¡cd\E{!kd��sg| �¨¦ OPfik ��|£koWx��c {'eBcy\.koWx\jeu�B�gqo\'�Ce�kd\�wOs��P\'q�c*su�PkdWZ\j{!��fZcykd\'q�c��������¬O\ �Cfx��kd\\ �Cfx��®�eg��\E|Bk@¯%egc­[.sBqd\.cy��s��}wxqosi{!\Ecocdsgq�c��������>¬�\���|¨®gsg��®g\@z%¦ 7­fxql�ªfxkdfxqo\��Psgqo��'Z|Zeu����mI{!sg|O{!\'qo|ZclkdWx\egfikdsB[¥e�kd�"SEeukd��sg|�su�xkoWx\(cd{�WZ\Ezifx����|x�­wZqdsi{!\@cdc0eg|Zz�koWx\X��[�wx��\'[�\E|Bk�e�ko��sB|�sg�OsukoWx\'q0|¨fx[�\'qo��{'eu�¨�B\'qo|x\'��c'¦

� �� ��X�0��#3�P��"

;�]= � ¦ OXeg����\'mB¯A��¦ ��\E\g¯Beg|Zz ��¦B�i��[�sg|/¦��lcd��|x� �Ckdq�egcocd\'|� c89(���gsgqo��kdWx[ kds 9�{'{'\'��\'q�e�kd\PkoWx\l�¨sg��fiko��sB|sg���/��|Z\Eeuql�im¨cykd\E[¥c'¦�� �D� ��� � �\� @�@�������2����� � � ���u¯i� 3 � 5 ( � I5�@�¨���i]g¯C�Beg|£]@?�?x]B¦

; � = � ¦ OPeg�+eg|ZzIv ¦ �leg��|Z\EcE¦ 9(wZwxqdsCeg{�Wx\@cX�ªsgq � |Ckd\E�gq�e�kd��|x��V�eBcy�°eu|Oz � eukoe¥p7euq�eu����\E����cy[�¦6������� � � �9� �&��� � � i�¯ E 3 � 5 ( �U�u�¨�u�O¯C�Bfx��«Î�i\'w¡])? ?g�x¦

; ��= 7�¦HOP\EeufZ[.sB|CkE¯ � ¦HO7sBfZzi\'kE¯�9�¦ ��\'�Bqoeg|Zz%¯X²P¦7`(eBcnko\'����sO¯Xeg|Zz��¦P`�sg¬�\'qdkE¦ �(\'kd\EqdsB�g\'|Z\'sgfOc[¥eukdqo��h¨«¿[�eukdqo��h*[ fZ��ko��wx����{'e�ko��sB|�¯�sgq+wZeuqdkd��kd��sg|Z��|x�­elc��BfOeuqo\r��|Ckoslqo\E{6k�eu|x�B��\@c(��lp>«¿{'sg[�wx��\!ko\'|x\@cdceg|Zz±egwxwxqosUhi��[¥e�ko��sB|±eu���gsBqd��kdWZ[�cE¦ � |���� ������� ������ ����,�$ �-�2� � � ��� �����!�.����� � �N� �%$& �&� )9� ��� ����� � � ��������� ������� ������� ���������U¯�wZeg�g\@cX��?g���¨�g� Ii¦ � %�%�% #7sg[�wxfxkd\'q_�¨si{!��\!knm�prqo\EcocE¯Z�u�g�Z]g¦

; ��= 7�¦ OP\EeufZ[.sB|CkE¯ � ¦�O7sBfZzi\'kE¯B²P¦¨`(egcykd\E����sZ¯Ceu|Zz��¦B`(sg¬�\'qdkE¦�vIeukdqo��h.[ fx��kd��wx����{Ee�kd��sg|jsg|¥Wx\'kd\'qosu«�B\'|x\EsgfZcXwZ��euky�ªsgqo[¥c'¦ ��������� �� O¯�]U� 3 ]E� 5 (�]@�g�B�U��]@� Ix]g¯g�u�B�x]B¦

; I =*p0¦ O+tnsBqocykoeBz%¯�²P¦�vIeu|x|Z\g¯�V*¦/�¨sBqd\E®C����¯�eu|Zz£v ¦ �re�tÎko\'q�cy��{u¦ %^Q�{!��\'|Ck vIe�koqd��hRv°fx��kd��wx����{Ee�ko��sB|sB|I� � v � #7sB[.wZfikd\EqocE¦X]E� 3 ] 5 ( �B��EU�¨�B�Z]g¯g�Ceu|¨fZegqdm°]@?�?B�x¦

; E�= �¦ �%¦ OP��eB{��C�ªsgq�z%¯x�Z¦�#7Wxsg�G¯�9 ¦ #7��\Eeuqomg¯�%(¦ � 9ZS'\E®g\EzxsZ¯Z�O¦ � \'[�[�\'�G¯ � ¦ � Wx������sB|�¯O�O¦ � sg|x�CeuqoqoeZ¯�%¦ �leg[�[�egqd����|x�Z¯ B.¦ �(\E|xqdmB¯ 9 ¦up0\!kd��kd\'kE¯ ��¦B�Ck�eu|x��\'mB¯ � ¦g¢¡eu���g\EqE¯geg|Zz�`�¦ #­¦g¢¤WZeg��\Emg¦ �2�! #" '�$" �%'& $��2$(�*)�� � � �!¦� � 9lv ¯+])?�? �¨¦

; ��= %l¦(#PegqdsB|.eg|Zz�²7¦¨�¨fiko\'q@¦Op>euq�eu����\'� %>h¨ko\'|Zcd��sB|�su��e � m¨|Zeg[���{Xp0\'qd�ªsgqo[¥eu|Z{'\(²xsBqd\@{'egcykd��|x�*V/s¨sB�ئ� | � ���)�I�8�� 6 �-��� �� � Ii ������$&�m� � � � ������� �!�.��\� � � � �%$& �&� )9� *�2� ����� � � ���u¯ � eBcy�G¯i�gfx��mj�u�B�B�x¦

; ��=��%¦�#7WZeukykd\EqÖtn\E\g¯K9�¦ �/\'¬�\E{���¯/p0¦/p>eukd|Zeg��eZ¯/eu|Zz¡v ¦+VXWxsukdkd\!koWxsizi�G¦3`�\@{!fxq�cy��®g\ 9(qoq�eUm �/eUmBsgfik�ceg|Zz¥²ZeBcnk7p0egqoeg����\'�xvRe�kdqo��h�v°fZ��ko��wx����{'e�ko��sB|�¦ � |J�H���@���2�2� � ��� $6�� +�P�.� � ���� �," � �1������" � �i � '����$&�m� � � � ��� �����!�.��$" � � � �&�! ���1$ � � �," ��� ���! �2��X� ����$6¯i�xeu��|Bkd«¿vReu��sZ¯x²xq�eu|O{!\g¯i�Bfx|x\¥]@?�?�?Z¦

; ?�= #­¦ #7Wxsgf�¯(�¦ � \'|x�O¯Zeu|Oz-.¦x¢¡eu|x�O¦ 9}vIegcocy��®g\E��mjp0egqoeg����\'�%v°\'kdWxsiz&�ªsBq(vIe�koqd��h3v°fx��kd��wx����{Ee�«ko��sB|<OXeBcy\@z°sg|¡�Ckdq�egcocy\E|� c­v°\'kdWxsiz%¦�V/\@{�Wx|x��{'eu�0`�\EwOsBqyk��.� �/*� O7« 9lvR�¨«*?B��«�]��¨¯ #7\E|Bko\'q­�ªsgq�x{!��\'|Ckd� 'O{ #7sg[�wxfxkd��|x�Z¯ZVXWZ\0�l|x��®g\'q�cd��knm�sg�0�Ckosg|¨mJOPqos¨sg��¯(�(s�®g\E[ ¬�\'q*])? ?g�Z¦

;�]E��=*²P¦ � \Ecdwxqo\S�eu|Zz ²7¦r�¨fxkd\'q@¦±v°��hi\Ez±p0egqoeg����\'� � [�wx��\'[�\'|Ckoeukd��sg|Zc.sg�(kdWx\°V+sgw ��\'®B\'���Ckd\Ew sg��¨kdq�egcocy\E| eu|Oz±¢¤��|xsg�BqoeBz±vIeukdqo��h±vIfx��ko��wZ����{'eukd��sg| 9(���gsgqo��kdWx[¥cE¦ � | � ���@�?�6� ���1��2� �0� �� ���� �����!�.��� � � � � �%$& �&� );� �2� �H���@�2�&$2$&� ��� Ii ������$&�m� �3�!����6��+� ���4��¯x�ieu|�²xq�eu|Z{'��co{!sO¯�9(wZqd���O�u�B�x]B¦

ÞkjK[�Þml

Page 33: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

������������ �������������������� �!�"���#�%$&� �Z]

;�]g]=6O_¦ � fx[���kdqo\Eco{!f�¯0�Z¦ �r¦+`�si{�W�¯+eg|Zz � ¦/V+qom¨cykdq�eu[�¦�²0egcyk vIeukdqo��h£vIfx��ko��wZ����{'eukd��sg|R9(���gsBqd��kdWZ[�csB|°v � v � 9lqo{�Wx��kd\@{6kofxqd\B¦D��� �����!�.��$" � � � �&�! ���1$ � � � " � � �#� �2�� � � �-$6¯O� 3 � 5 ( Iu�U�1���Z¯O]@?�?u�O¦

;�]@� =_p0¦(²���cd{�Wx\Eq°eu|Zz-`�¦�prqdsB¬O\Eqyk@¦ %^Q�{!��\'|Ck�prqdsi{'\Ezifxqo\Ec3�ªsBq �­cy��|x��vIe�koqd��h 9(���gsgqo��kdWx[¥cE¦ � |"W� � �U� ���� � ���A��� � �&$6� � � �H��������� � � � ���g¯C®Bsg��fx[�\�]'��sg��� ��#X�%¯BwZeg�g\@c>�Z]@�U�C�¨�5�i¦U�¨wZqd��|x�g\Eqy«�>\Eqd��eu�O¯-OP\'qo����|�¯%])? �U�O¦

;�]E��=6O_¦�B_qoeUmicdsg|­eg|Zz*`�¦ �reg|�zi\DB_\'� tn|�¦�9��l���BW�pr\'qd�ªsgqo[¥eu|Z{'\0peuq�eu����\E�B�CkoqoeBcdcd\'| � [�wx��\E[�\'|Ckoeukd��sg|�¦��� �����!�.��� �H���@�2�&$2$&� ��� � *��2$6¯�E 3 ] 5 ( �U��]@�x¯O]@?�?�EZ¦

;�]'��= � ¦ �O¦ �l���BWZeu[�¦�%rh¨wZ��sB��ko��|x��²0eBcnkvIeukdqo��h.v°fx��kd��wx����{'eukd��sg|�¢¤��kdWx��|�kdWx\��/\'®B\'�Z�6O � 9_��¦OV+\E{�Wx|Z��«{Eeu��`�\EwOsBqyk�V�`(� ?�« ?g�u�O¯ � \'wxkXsg� #X�B��#7\'|Ckd\EqXsu�K9lwxwx����\@z3vRe�kdW/¦ ��#7sgqo|x\'���$�(|x��®�¦�¯-9lwxqd���>])?B��?Z¦

;�]@I =��%¦ �lfZcdcy« ��\Ezx\'qo[�eg|�¯ %(¦��Ceg{!sB¬Zcdsg|�¯x�O¦O�BsgWx|OcysB|�¯ 9 ¦OV�cdegsZ¯Oeg|Zz°V*¦OV+fxqo|¨¬xfx���ئ(�CkoqoeBcdcd\'|� c�9l��«�Bsgqo��koWx[ �ªsBqvIeukdqo��h�vIfx��ko��wZ����{'eukd��sg| (0v°sizi\'����|x�Z¯ 9(|Zeg��micd��cE¯Beg|Zz � [.wZ��\E[.\E|Ckoe�ko��sB|�¦�V/\@{�Wx|x��{'eu�`(\'w�sgqdk # #X�C«ÎV�`X« ?�E�«�]'� �¨¯"#7\E|Bko\'q�ªsgq #7sB[�wxfikd��|x���x{!��\'|Z{'\EcE¯ 9lqd�Bsg|x|x\ �le�ko��sB|Zeu� �/eg¬�¦�¯�])? ?�EZ¦

;�])E�=6O_¦�� � �Bcykdq��g[�¯7p�¦ ����|x�Z¯�eg|Zz #­¦��reu| �/sBeg|�¦ B�%vIv°«*OXegcd\Ez �/\'®g\E�l�0O � 9_� ( �l���BW¤p�\Eqy«�ªsBqd[¥eg|Z{!\ vIs¨zx\'� � [.wZ��\E[.\E|Ckoe�ko��sB|Zc_eu|Zz°p0\Eqy�ªsBqd[¥eu|O{!\ %r®�eu��fZeukd��sg| OP\'|O{�Wx[¥euqo��¦­V/\@{�Wx|x��{'eu�`(\'w�sgqdk �lv � �­²/« ? Ix¦�]@�x¯.�([�\ � �l|x��®g\'q�cd��knmB¯ 7_{6kdsB¬O\Eq�])? ? Ii¦

;�]���= �¦7pqdm¨�����*eu|ZzCO­¦PV/sBfxq�eu|Z{�Wx\@euf�¦ ²Oegcyk3`�fx|Cko��[�\MOP��si{�� #7mi{!����{ � e�koe±`�\Ezi��cykdqo��¬xfxkd��sg|-sB|vIfx��ko��wZqdsi{!\@cdcdsgq�cE¦(���4� ��� ����� 8������� �!�"����� � �Q� �%$& �&� );� �2� ����� � � ���u¯���I 3 ] 5 ( Eg�����g�i¯9lfx��])? ?��i¦

;�]E��=��%¦r`(eu[¥egcd�Xeu|¨mg¦� �!� � �# � � ���4��$ �^��� �"� �! � � � � � �K��$�, � � �N� � �>������� �!�.��#�%$&� � ��� � �4� �.��� ��� ���� � � �0" � � �#� �2�� � � �-$6¦prW � kdWx\@cy��cE¯ �(|x��®g\Eqocd��knm¥su� � ������|Zsg��c�e�k �(qo¬Zeg|Ze�« #7WZeu[�wZeg���B|�¯�])?�? Ex¦;�])?�=*V*¦i`(egfx¬�\'qXeu|Zz B.¦i`��Z|x�g\EqE¦+�x{�Wx\EzifZ����|x� sg� � eukoe�p>euq�eu����\E�Ov°sizifx��\Ec�ªsgq��i{'��\E|Bko� '�{ #7sB[.wZfiky«

��|x�O¦ � |j� � 9­v ¯g\@zi��kdsgq@¯ �H���)�2�2�2� � ����$�� � � �(���"�� � �@ �������� �2� � �U������� �!�"�����H���@�2�&$2$&� ���H ���� ��� ���� � � � � �Z� � � ����� � ��� � ��¯��ieg| 9l|Ckdsg|Z��sO¯xV/\'hieBc'¯xvReuq�{�WR])? ?�?Z¦;Ê�u��=0��¦��O¦�`(s¨{�WZ\ eu|Oz �Z¦��O¦ � sg|x�CeuqoqoeZ¦ � \Ewx��s�mC��|x� p0egqoeg����\'� �(fx[�\Eqd��{'eg� �/��¬xq�euqom `(sgfi«

ko��|Z\Ecjkds #7��fZcykd\Eq #7sg[�wxfxkd��|x�±��|�e±�¨\E��� 9�zxeuwiko��|Z�±²ZeBcyWx��sg|/¯�9(wxqo�����u�B�B�x¦}�ifx¬x[���kdkd\@z¤kdsp>egqoeg����\'� #7sB[.wZfikd��|x�Z¯������ ������������������� ������� �! "��#�$�1� ���#�%� �$& �(' �D���� A� � ����)(*�+ '�, � ��*("�"#*��-�.�-�/�0�1 �5��2���34����&A� ���%�#4#5 �X� ��6 ¦

;Ê�i]=_¢}¦B`��g|Zco{�W�eg|Zz ��¦B�Ckoqoegf#7Z¦xVXWZ\ �/\'®B\'�x��O �\9­�.²�sgqo[¥c+sg�Opegqoeg����\'�i²0eg{!kdsBqd�"SEeukd��sg|�vI\!kdWZs¨zZc'¦� | � ¦ �Z¦�%r®�eu|ZcE¯ B.¦ `�¦+�Bsgfx¬�\'qdkE¯0eu|Zz ²P¦ �Z¦+p0\!kd\EqocE¯0\Ezi��kdsBqocE¯ ��� �����!�.�� � �Z� � � ���98(�u¯>wZeg�g\Ec� I@� ?B�i¦ %r��cd\'®¨��\Eql�i{!��\'|O{!\*prfx¬x����cyWx\EqZO_¦ ��¦�¯/])?g� ?x¦

;Ê�g� = ��¦u�Ckdq�egcocy\E|�¦ B*egfZcocy��eu|)%����[���|Zeukd��sg| � c��lsukD7­wikd��[¥eu�G¦� � � ��&�%$� ���H� � �-�� �� �.,u¯Z]E� 3 � 5 ( � Iu���� I�EZ¯%])?�E ?x¦

;Ê�u��=*v©¦gVXWxsukdkd\'kdWxsizi�G¯C��¦"#7WOe�kyko\'qØtn\'\B¯geu|Zz 9�¦"��\'¬�\E{���¦xV+fx|x��|x���Ckdq�egcocd\'|� c0vIe�koqd��h�v°fx��kd��wx����{Ee�ko��sB|�ªsBq(v°\'[�sBqdm&% Q¥{'��\E|Z{!m�¦ � | �H���@�2�2����� ����$ �� 0 �@�-����2� �Z� � � ��� � � 8g¯ 7­qd��eu|OzisZ¯(�(s�®R])?�?B�x¦

[\[ ß^]_&_�`&a

Page 34: Impact of Mixed–Parallelism on Parallel Implementations ...cecile/ENSEIGNEMENT/IPAR/Exposes/Strassen.pdf · ISSN 0249-6399 ISRN INRIA/RR--4482--FR+ENG apport de recherche INSTITUT

Unit e de recherche INRIA Lorraine, Technopole de Nancy-Brabois, Campus scientifique,615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES NANCY

Unit e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES CedexUnit e de recherche INRIA Rhone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN

Unit e de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY CedexUnit e de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex

EditeurINRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France)��������� ���� ���������� ��� ���

ISSN 0249-6399