efficient block cyclic data redistribution · vuu#vfe qd_bea9p]_xndcts vfe...

Post on 08-Sep-2020

2 Views

Category:

Documents

0 Downloads

Preview:

Click to see full reader

TRANSCRIPT

HAL Id: inria-00073925https://hal.inria.fr/inria-00073925

Submitted on 24 May 2006

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

Efficient Block Cyclic Data RedistributionLoïc Prylli, Bernard Tourancheau

To cite this version:Loïc Prylli, Bernard Tourancheau. Efficient Block Cyclic Data Redistribution. [Research Report]RR-2766, INRIA. 1996. �inria-00073925�

ISS

N 0

249-

6399

ap por t de r ech er ch e

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Efficient Block Cyclic Data Redistribution

Loïc Prylli and Bernard Tourancheau

N 2766Janvier 1996

PROGRAMME 1

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

/(021 354+6798%:;:=<�>2?(@BADCE7-?(>27F@�GH0JI�7F>2?(4FK�C2>LINM

O"PRQ2SJP�T2U+U#VXWZY�[ZP]\F^`_ba�VF\Fa9cdP]VFe,fdT2P�ThgigkjVFgiVFeml�ndToepVFe�qdV�qdQJrdrtsVFVFeuldP`sVFepVuT2c`vVFa�epwJe�aujVuU#VFeqd_be�a9P]_xndctsVFeO"PRQ2y]VFa{z{VF|}TLO

z~T2fdfdQJPRa�qdVXP]VF\F^`VuP]\F^`V�r,�`�2�2�2�.Y��JT2rL�J_bVuP,Wu�L�2��Y��2��fdThS2VFe

�����F�u�m�'�o�2����U+f`giVuU�Vur2a�_xr`S�gb_xr`VuT2P�TogbS2VundPpT��oVuP�r`VFgbe�QJr�qd_be�a9PR_�ndc`apVFq�U#VuU�QJP]w fdT2P�ThgigbVFg\FQJU+fdc`a�VuP]e�P�To_be�VFe�a-^`V¡fdP]Q`n`giVuU¢Qo£�q'Toa9T¤qd_be�a9P]_xndc`a�_bQJrBQo£,U�Toa9P]_b\FVFe�T2r`qB�2VF\Fa�QJPRe�TLU#QJr`Sa9^`V�fdP]Q`\FVFe�e�Q`P]eu¥§¦tgbQJ\F�o¨�\Fw`\Fgb_b\©qd_be�a9PR_�ndc`ap_iQ`r�e�VFVuU#eªa�QBe9c`_ba¬«VFgbg�£kQJPªU#Q2e�aªTogbS2QJP]_ba9^dU#em¥¦c`a�Q`r`V¡^dToe,apQ�\F^`QJQ2epV¬TªS2Q`QJq�\FQ`U�fdP]QJU#_be�V�£5QJP.a9^`V�e�_b­FV�Qh£{a-^`V¬n`gbQJ\F�2e�®=apQ�To\¯^`_bVF�2VªTS2Q`Q`q�\FQJU�fdc`a-Toa�_bQJr°T2r`q�\FQ`U�U#cdr`_i\uTha�_bQJr¤V-±¬\F_bVur`\Fw}TLr`q²T+S2QJQ`q�gbQJThq²ndTogxT2r`\F_xr`SJ³F¥,´�^`_be\F^`Q2_b\FV�^`VuTm�J_bgbw�qdVufdVur`qde~QJr�VuTh\¯^#Q`fdVuP�Toap_iQ`r�lJepQ�_ba_ieVFe�epVur2a�_xTog%a�QXnµVZTLn`giVDa�Q�S2Q,£�P]QJU¶QJr`Vqd_be�a9PR_�ndc`ap_iQ`r#a�QNT2r`QLa9^`VuP��2VuP]w,·dc`_b\¯�Lgbw2¥�¸�V"fdP]VFepVur2a�^`VuP]V"a-^`V¹TogbS2QJPR_ia-^dU#eº«V�_xU�f`gbVuU#VurLa�VFq_xr�a-^`VZ»d¼[,½�[,O%[N¼�¾ gb_xndP�T2PRw2¥�[¿\FQ`U�f`gbVFvJ_ba]w¬epa9c`qdw¬_beNU+ToqdVNa-^dToa�fdP]Qo�2VFea9^`VNV-±ª\F_iVur`\FwQo£(QJcdP�epQ2gxc`a�_bQJr�¥"´_xU#_xr`S�P]VFe9c`gba�etQJr#a9^`VN�Àr2apVFg�O�T2P�ThS2QJr�TLr`q�a9^`VN¼�PpTuw#´��oÁ�\FQJP�PRQJndQJPpToa�VFea9^`VDP]VFe9c`gba�eu¥�¸ÂV�e9^`Qo«�a9^`V�SJTh_�r+a9^dToa�\uTLr�ndVDQJn`a9To_xr`VFq¬c`ep_�r`S#a9^`V~S2QJQ`q�q'Toa-TDqd_iepa9P]_xndc`a�_bQJr«�_ba9^¤��r`cdU#VuP]_b\uTogº�oVuPpr`VFgie,T2r`q¤QJcdPP]VFqd_be�a9P]_xndc`a�_bQJr°P]QJc`ap_�r`VFem¥ÃÅÄJÆ(Ç]È#É��oÊ��2��fdTLP�TogbgbVFgd\FQJU�fdc`a�_xr`S'l2fdT2PpTogbgiVFgdgb_xr`VuT2PtTogbS2VundP�Tdl2fµVuP]e�QJrdTogb_b­FVFq¬Togbg˨�a�Qo¨ÌThgigd\FQ`U�¨U#cdr`_b\uToa�_bQJr�l%q'Toa9T�PRVFqd_iepa9P]_xndc`a�_bQJr�l�ÍNOHÎ�l'n`gbQJ\F�o¨�\Fw`\Fgb_b\�q'Toa9T�qd_be�a9P]_xndc`a�_bQJr

Ï5Ð#ÑÒ]Ó]ÔuÕ ÑÒ�Ö�×�ÓÌØ=ÙuÚ

Û`Ü�ÝmÞ�ß�àkÜ]á¹â(ã]ä,å-æ�Ü]çká�èmé�ÝuÞ�ßpàkÜÌá�ê]ß-ëë�èué¬ì�í�îºïZðZñ%íòï.ÛJó-ß�é#ð�ôxí�îºôöõ"÷Jìøܹá=Þ�ùpú�ù�ûýü(Ü]çká�þuéoù�éoê2ÿÜÝoù�Þ�ü ܺì�í�îºïtê]ß�éuá=Þ�ù-á���ô;ì ï���ü Ü����ºî��'î(ì��'îºïtù-ê]á=û ß�éNñ �ºñ�ì���ü ùì�ñ'ñ.ÝuÞ�ß���Þ�ù-ëëܺñ �ºî(ñ ��õ¬ê]ßpéuá=Þ�ù9áñ �ºî���������ïo÷

� Û`ù��2ßpÞ�ù9ákßpûýÞ�Ü�Û ô�������î(õ�ì�í�îºï����� �!"�Fñ%êÌßpü Ü�íºßpÞ�ëù�ü ܺï¯èuÝ(ÿÜ�Þkû ÜÀèmÞ�Ü$#uÜ(Û`ó9ßpé%�'&(�*)� ���)�+�Û",$��í¡ì�Ü�#mÜ.-/10 �32146587�9�:�;=<39�>"?�@�A876582�BC:171?6@�A"76582�D�9EA">39.7�B<�>

Unité de recherche INRIA Rhône-Alpes46 avenue Félix Viallet, 38031 GRENOBLE Cedex 1 (France)Téléphone : (33) 76 57 47 77 – Télécopie : (33) 76 57 47 54

� �t!X�p$'�%&(��)+*Z���9�.�������������!X�t$¿!#�.#���"�t$§$%�����ø����"�t$��X�{& )X�p����$�{��'&(�"�p������t$

���Ä �����Ä�� ½ �ý_xU�f`gxT2rLa9Toap_iQ`r#qdV¹r`QowJT2c`v,q��ËTogbSµjVundP]Vtgb_�rtsVuTo_xP]Ve9cdP giVFe"U+To\¯^`_xr`VFe"fdT2PpTogbg5jVFgbVFeNjTUBsVuU#Q2_xP]Vqd_be�a9P]_xndctsVFVNfdQ2epV¹gbVNfdPRQJn`g5jVuU#Vq'c�\¯^`QL_iv#qdVgxT,qd_be�a9PR_�ndc`ap_iQ`r�qdVFe�qdQJrdrtsVFVFe"fdQJcdP�gbVFeU�Tha9P]_b\FVFeVFa¹gbVFe�2VF\Fa�VucdPRe¹e-cdP�gbVFe�qd_��ZsVuP]VurLa�e"fdP]Q`\FVFe�e�VucdP]em¥��Dr`VDqd_iepa9P]_xndc`a�_bQJr�n`gbQ`\-¨�\FwJ\Fgb_b·dc`Ve�VuU#n`gbV�\FQ`r2�2Vur`_xPDfdQJcdP~gxT�f`gxcdfdT2P]aDqdVFeZThgiSLQJP]_ba9^dU#VFeul%U�To_beNcdr�\FQ`U�fdP]QJU#_be�VFepaNr"sVF\FVFe�e-To_xP]Vq'T2r`etgbV�\¯^`QL_iv#qdV�gxT.a9To_bgbgiVDqdVFe"n`gbQJ\Fe�®kfµQJcdP�Tm�2Q2_xP�jTDgxTD£kQ2_be�qdVFe"\uTogb\uc`gbeVFa�\FQJU�U#cdr`_b\uToa�_bQJr`eV-±¬\uTh\FVFe�VFa¬cdr`V�ndQ`rdr`V�PJsVufdT2PRa�_ba�_bQJr�qdV�\F^dT2P]SLVu³F¥º½�V¤\¯^`QL_ivBQJf`a�_xU�Thg¹VFepaXqd_��,sVuPRVur2a�fdQ`cdP\F^dTo·dc`V�TogbS2QJP]_ba9^dU#V2lHVFaX_bg�VFepaXqdQ`r`\�VFe�e�VurLa�_bVFg�qdV�fdQJc`�LQ2_xP�fdToe�e�VuP�q�� cdr`V�qd_be�a9P]_xndc`a�_bQJr jTg�� T2c`a-P]V�a9PFjVFeXP�T2f`_bqdVuU#VurLau¥ �~QJc`e¬fdP`sVFe�Vur2apQJr`e�_b\F_�gbVFe¬TogbS2Q`P]_ba9^dU#VFe�qdV�P]VFqd_be�a9P]_xndc`a�_bQJr · c`Vr`QJc`eDTu�LQJr`e_�U+f`g�TLr2adsVFe�q'T2r`e�gxT�n`_xn`gb_bQ2a9^�jVF· c`V�»d¼[,½�[,O'[D¼�¾#¥��Dr`V�sVFa-c`qdVNqdV�\FQJU�f`gbVFvJ_badsV�`_bVur2a#Vur`e9c`_ba�V�fdP]Q`c`�2VuP,g��ýV-±¬\uTh\F_iadsV�qdVFe#e�Q2gxc`a�_bQJr`e#\F^`Q2_be�_bVFeu¥�½�VFe�fdVuPÀ£kQJP�U+T2r`\FVFe#QJn`a�Vur`c`VFee9cdP+��rLa�VFg�O�T2P�ToS2Q`r°VFa�¼�P�Tmw©´��oÁ�\FQJP�PRQJndQJPRVur2a#r`Q2e¬P`sVFe9c`gba9Toa�eu¥���Q`c`e�U�QJr2a-P]QJr`e�gbV¤SJTo_xrQJn`apVurJc�Vur�c`a�_bgb_ie-T2r2a°cdr`V�ndQJrdr`V°qd_be�a9P]_xndc`a�_bQJr qdVFe�qdQJrdrtsVFVFe¡Tm�2VF\°�©r`QowJT2c`v�qdV°\uTogb\uc`gr`cdU#VuP]_b·dc`VXVFaDr`Q2e£kQJr`\Fa�_bQJr`e�qdV#P]VFqd_be�a9P]_xndc`a�_bQJr�¥� É%�u�FÇ������Ä¿� \uTogb\uc`g�fdT2P�Togbg5jVFgbV2lºThgiS'jVundP]V#gb_xr"sVuTo_xP]V�fdT2P�TogbgkjVFgbV2l \FQJU�U#cdr`_b\uToa�_bQJr°Vur sVF\F^dT2r`S2Va�QLa9TogdfdVuP]e�Q`rdTogb_iedsVFV2l2P]VFqd_be�a9PR_�ndc`ap_iQ`r�qdVqdQJrdrtsVFVFeulEÍDO�Î�l2P`sVufdT2P]ap_iap_iQ`r,qdVqdQJrdrtsVFVFe�VurLa9P]VFgxTo\�sVFVfdT2P~n`giQ`\Fe

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

. / ��'&(��!+*��������.

´�^`_be�fdT2fµVuP�qdVFep\uP]_xndVFe#a9^`V�e�QLg�c`ap_iQ`r�Qo£�a9^`V+q'Toa9T�P]VFqd_be�a9PR_�ndc`ap_iQ`r�fdP]QJn`gbVuU�T2P]_be�_xr`S�«N^`Vur_xU�f`gbVuU#VurLa�_xr`S¬gb_xr`VuT2P�ThgiSLVundP�T�_�r�T.qd_be�a9P]_xndc`a�VFq¤e�wJepa�VuU�¥2[Dgba9^`QJc`SJ^ªT#n`_ba¹e-fdVF\F_xTogb_b­FVFq(l�a9^`VfdP]Q`n`giVuU TLr`q�_iapeNe�QLg�c`ap_iQ`r�\FQJrLa9To_xr`e,fdQ2_xrLa�e�Qo£�S2Vur`VuP�Tog _�rLa�VuP]VFepaZP]VFS`T2P]qd_xr`S�q'Toa9T�\FQJU�U#cL¨r`_b\uToa�_bQJr°fdToapa�VuP�r`e_xr�q'Toa9T+fdT2P�TogbgbVFgºgxT2r`S`cdToS2VFeu¥¸�V�fdQ2_xrLaNQ`c`a¹a-^dToaa9^`V�fdT2fdVuP~_beNr`Q2aNThqdq'P]VFe�e�_xr`Sªa9^`V�fdP]QJn`gbVuU�Qo£"^`Qo«�a�Q�qdVFa�VuP�U�_�r`V

T�P]VFgbVF�oT2r2a,q'Tha9T�qd_be�a-P]_xndc`a�_bQJr�®5VF�2Vur�_Ë£¹VFvdfdVuPR_�U�Vur2a9Thg¹P]VFe9c`gba�e�TLP]V�S2_b�2VurÅ£kQJPD��rJcdU#VuPR_i\uThg�oVuPpr`VFgie-³Fl�ndc`aD^`QE« a�Q#_�U+f`giVuU�Vur2a�T#SL_i�LVur�P]VFqd_be�a9P]_xndc`a�_bQJr�¥�¥´�^`V¡fdPRQJn`gbVuU Qh£{q'Tha9T�P]VFqd_be�a-P]_xndc`a�_bQJrBQJ\F\ucdPRe�TheXepQJQJrÂToe�w2Q`c°qdVuTog«�_ba9^BT2P�P�Tmw`e,QJr

fdT2PpTogbgiVFg�qd_be�a9P]_xndc`a�VFq°U#VuU#QJPRw¬\FQJU+fdc`a�VuPFlL£ PRQJU��2VF\Fa�QJP]ea�QªUXc`gba�_˨�qd_xU#Vur`e�_bQJrdThg"T2P�P�Tmw`eu¥J� aT2fdf`gb_bVFe�ndQ2a-^�a�Q�q'Toa-Tm¨ÌfdT2P�TogbgbVFg�g�TLr`SJcdToS2VFe,e9c`\F^°Toe�ÍDO�Î�TLr`q²apQ¡»dOH|²Á�fdP]Q2SJPpT2U#eD«�_ba9^U#VFepe9ToS2V-¨ÌfdToepe�_xr`S'¥(��r°a-^`V10�PRe�aN\uThe�V#a9^`V�P]VFqd_be�a9P]_xndc`a�_bQJrÂ_be,_�U+f`gi_b\F_ba�_xr°T2P�P�Tmw�e�a-Toa�VuU#VurLa�egb_x�oV�24365§«N^`VuPRV72�VFa"5 T2P]V�a]«QZU+Toa9P]_b\FVFe"«~_ia-^#qd_��øVuPRVur2atqd_iepa9P]_xndc`a�_bQJr`eu¥d��r+a9^`Ve�VF\FQJr`q\uToepV�Tªgb_xndP�T2PRw�£�cdr`\Fa�_bQJr°^dToe,a�Q�nµV#\uTogbgiVFqBa�QªqdQ�a9^`V#e9T2U#V�QJfdVuPpToa�_bQJr°QJP~_ba,\uT2r°Togbe�Q�ndV^`_bqdqdVur°Toa�a9^`V�nµVFS2_xrdr`_xr`S²T2r`q�Vur`q¤Qo£HTLrªQJf`a�_xU#_b­FVFq²P]Q`c`a�_xr`V�\uTogbgk¥¸�VZfdPRVFe�Vur2aD^`VuP]V,a9^`V,TogbS2QJP]_ba9^dU TLr`q¬_xU�f`gbVuU#VurLa9Toa�_bQJrÅQo£ºa9^`V,P]VFqd_be�a9PR_�ndc`ap_iQ`r�P]QJc`ap_�r`V

a9^dTha¹_beNc`epVFq�_�r�»d¼{[Z½�[,O%[N¼�¾98;:%l �=<Ì¥?>,cdPepQ2gxc`a�_bQJrª_beNT�qdw rdT2U�_i\�TLfdfdP]QJTo\F^�_xrªQJPRqdVuPa�Q\FQJr`epa9P�c`\Fa"a-^`V�\FQJU�U#cdr`_b\uToa�_bQJr+e�VFa�eT2r`q�a9^`Vur#V-±ª\F_iVurLa�gbwª\FQJU�U#cdr`_b\uToa�V�a9^`VuUÅ¥@>,cdP"ThgiSLQo¨P]_ba9^dU c`e�VFe�e�VF�2VuPpTog�e�a9P�ToapVFS2_bVFe¹qdVufµVur`qd_xr`S�QJrªa9^`V�T2U#Q`cdr2aQo£ºq'Tha9T�a�Q�ndV,\FQJU�U#cdr`_b\uToa�VFqT2r`q°QJr¤a9^`V�a9T2P]SLVFaZT2PR\¯^`_ba�VF\Fa9cdPRV�\uT2fdT2n`_bgb_ba�_bVFe#_�r}QJP]qdVuP~a�Q¡nµV#�2VuP]w�£�Toe�am¥'�ka�P�cdr`e~£kQJPNTLr2wr`cdU#ndVuP~Qo£HfdPRQJ\FVFe�epQJP]eul'U+T2�2_xr`S¬Tm�oTo_bg�TLn`giV#a9^`V#fdQ2epe�_xn`_igb_ba]w�Qo£�gbQJToqd_xr`S�T2r`q�qdQo«NrL¨�gbQJToqd_xr`S£�P]QJUBAha�Q#QJr`V�fdP]Q`\FVFe�e�QJPapQ@Am£�P]QJU�U�T2rLw¬QLa9^`VuP]eu¥»`VF\Fa�_bQJr���S2_b�2VFe�TBndP]_bV-£XqdVFep\uP]_xf`a�_bQJr�Qo£XP]VFgxToapVFq «QJP��Leu¥"»`VF\Fa�_bQJr ��_xrLa9P]Q`q'c`\FVFe�a9^`V

»d¼[,½�[,O%[N¼�¾ q'Tha9T�qd_iepa9P]_xndc`a�_bQJr°U#Q`qdVFgieDT2r`q�r`Q2a9Toap_iQ`r`e{c`epVFq�_xrªa9^`_beZfdTLfdVuPF¥'»`VF\Fap_iQ`r!:fdP]VFepVur2a�e,a-^`VXTogbS2QJPR_ia-^dU#eNa-^dToaN«VuP]V#c`e�VFq¤£5Q`P�a9^`V#P]VFqd_be�a9P]_xndc`a�_bQJr}Qo£"q'Toa9TªT2r`q�epVF\Fa�_bQJr°�fdP]VFepVur2a�e,ap_�U�_�r`S°P]VFe9c`gba�e,QJn`a-To_xr`VFq�QJr¤qd_��øVuPRVur2a�U�Th\¯^`_xr`VFe¡® rdT2U#VFgbw�a9^`V�¼�PpTuw�´~�oÁ T2r`qa9^`V#��rLa�VFgºO�T2PpToS2QJrd³F¥

C � �"������"!ED§�~&��

Î%QJPT�gbQJr`S#a�_xU#V2ldP]VFqd_be�a-P]_xndc`a�_bQJr�«�Toe\FQJr`e�_bqdVuP]VFq��LVuP]w�qd_˱¬\uc`gbaN_xrªa9^`VDS2Vur`VuP�Thg�\uThe�V2ldT2r`qU#QLe�a�_xU�f`gbVuU#VurLa9Toa�_bQJr`e�T2P]V�P]VFepa9P]_b\Fa�_xr`S©a-^`V�fdQ2e�ep_�n`gbV°qd_be�a9P]_xndc`a�_bQJr`e�apQ}n`gbQ`\¯�BQ`P#\FwJ\Fgb_b\qd_be�a9PR_�ndc`ap_iQ`r`eF8 �dl��JltW2W2ltWElHG�<Ìl QJP,_xrÂe�QJU#V�_xU�f`gbVuU#VurLa9Toap_iQ`r`e�Thgig�n`gbQ`\¯�h¨�e�_b­FVFe¬^dToq�apQ²ndVU#c`gba�_xf`gbVXQh£òVuTo\F^ªQ2a9^`VuPRe�a�Q�VuToepV,e�QJU#V�U#VuU�QJP]w¬To\F\FVFepeNQJfµVuP�Toa�_bQJr`em¥»`QJU#V²PRVF\FVur2a�«QJP��Leªe9^`Qo« a9^dToaª_ba�\uT2r�ndV°qdQJr`V�Toaª\FQJU�f`_bgbV�a�_xU#V�_xr�a-^`V�S2Vur`VuP�Thg

\uToepVI8KJ`ltWu�=<¹QJP.qdVFe�\uP]_xndVªa9^`V�To\F\FVFe�e�Qh£~TLP�P�Tmw�VFgbVuU#VurLa�e#«�_ba9^°qd_��(VuP]VurLaXepa9P]_bqdVFeL8 �=<Ì¥�¦c`a

î(î�éNM'O 0 )�)

: � ���� ���N( ������� ���# ����($���( #�� ��*@( �����@�$��*

S2Vur`VuPpTogbgiw�a9^`VFepV�«QJP��Le¬Toqdq'P]VFepe�VFq�P]V-¨�qd_be�a-P]_xndc`a�_bQJr�Qo£XT2PpP�TmwJe#«�_ba9^�T 0'v`VFq§rJcdU#ndVuP+Qo£fdP]Q`\FVFe�e�Q`P]e¬Toa#\FQJU�f`_bgbV-¨�a�_xU#V2lQJP�a9^`V�\FQ`U�f`_bg�Tha�_bQJr�Qo£,q'Toa9T¤VFgbVuU#VurLa�e�To\F\FVFe�e#£kQJP#n`gbQJ\F�o¨\Fw`\Fgb_i\+qd_iepa9P]_xndc`a�_bQJr�¥�¸�V�fdP]QJfµQ2e�V#T�P�cdrL¨�a�_xU#V�T2fdfdP]Q`To\¯^ÅQo£�q'Toa9Tm¨ÌP]VFqd_be�a-P]_xndc`a�_bQJr°«�_ba9^¤r`Q\FQJU+f`_igbV-¨�a�_xU#V+_�rL£kQJP�U+Toa�_bQJr�T2r`q�T#�oT2P]_xT2n`gbV�rJcdU#nµVuPQo£"fdP]QJ\FVFepe�QJP]eu¥

������ø� ������������!#���� !#��$%�'&(��)+*,���-�. �Z.! &(�t!X�p$'�%&(��)+*Z���9�.

´�^`V�»d¼[,½�[,O%[N¼�¾�gb_xndP�T2P]w°c`e�VFe,a9^`V�n`gbQJ\F�h¨�\FwJ\Fgb_b\¬q'Tha9T�qd_be�a-P]_xndc`a�_bQJr°QJr°T+�J_xP]a9cdThg�SJP]_bqQo£NfdP]Q`\FVFe�e�Q`P]e�_xrÂQJP]qdVuPDa�Q°P]VuTo\F^°S2Q`QJq°gbQJToq`¨ÌndTogxT2r`\FV2lHS2QJQ`q°\FQJU�fdc`a9Tha�_bQJr°V-±¬\F_bVur`\FwBQJrT2PpP�TmwJe�T2r`q T2r�VF· cdTog#U#VuU#QJPRw�c`e9ToSLV}nµVFa]«¹VFVur fdP]QJ\FVFepe�QJP]eu¥~[ZPpP�TmwJe�T2P]V�«DP�T2fdfdVFq n2wn`gbQ`\¯�Le"_xrªTogbgdqd_�U�Vur`e�_bQJr`e\FQJP�P]VFe9fµQJr`qd_xr`S�a�Q,a-^`V{fdPRQJ\FVFe�epQJP�SJP]_bq(¥J´~^`V¹Î�_bSJcdPRV�W_bgigxc`e�a-P�Toa�VFea9^`V+QJP]SJT2r`_b­uToap_iQ`r°Qo£¹a-^`V¬n`gbQJ\F�o¨�\Fw`\Fgb_b\�qd_be�a9P]_xndc`a�_bQJrÂQo£NT��oÁ T2P�PpTuw`e,QJr°T��hÁ�S`P]_bq�Qo£��fdP]Q`\FVFe�e�Q`P]eu¥´�^`V�qd_be�a9PR_�ndc`ap_iQ`r°Qo£¹T�U�Tha9P]_bv�_beNqdV$0�r`VFq©nLwª£kQJcdPNU+To_xr�fdT2P�TLU#VFa�VuP]e� %T¬n`gbQ`\¯�ª«�_bqda9^

e�_b­FV2l�����T�n`gbQJ\F�°^`VF_bSJ^2a,ep_i­FV2l�����a9^`V�r`cdU#ndVuPDQo£�fdP]QJ\FVFepe�QJP~_�rÂT�P]Qo«�l����������'a-^`V¬rJcdU#nµVuPQo£�fdP]Q`\FVFe�e�QJP]e,_xr�Tª\FQLg�cdU+r�l������! "TLr`q�£kVF« Q2a9^`VuPRe,a�Q¬qdVFapVuP�U#_xr`V2l(«N^`Vur°Tªe9cdnL¨ÌU+Toa9P]_bv�_bec`e�VFq(l�«D^`_i\F^¤VFgbVuU#Vur2a,Qh£"a9^`V�S2gbQJndThgòU�Tha9P]_bv¬_be,a9^`V�e�a9T2PRa�_xr`S¡fµQ2_xr2a�T2r`q¤«N^`_b\¯^°fdP]Q`\FVFe�e�Q`P_ba,ndVFgbQJr`S2eDa�Q'¥

(0,1) (0,2)

(1,0)

(2,0)

(3,0)

(0,4) (0,5)(0,3)(0,0)

(2,3)

Blocks owned by the processors [0,0]

Grid of Processors [2,3] Block - Matrix

0 1 2

1

0 (2,0)

(1,0)

(3,0) (3,3)

(1,1)

(3,1)

(1,4)

(3,4)

(1,2) (1,5)

(3,2) (3,5)

(0,3)

(2,3)

(1,3)

(0,1) (0,4)

(2,1) (2,4)

(0,2) (0,5)

(2,2) (2,5)

(0,0)

Î�_bSJcdP]VDW� o´�^`V¹n`gbQ`\¯�D\Fw`\Fgb_i\~q'Toa9Tqd_be�a9PR_�ndc`ap_iQ`r�Qh£�T��oÁ T2P�PpTuwDQJr,TD��"��S`P]_bq�Qo£�fdP]Q`\FVFe�e�Q`P]eu¥

��r�»d¼[,½�[,O%[N¼�¾#l2a-^`V¹V-±ª\F_iVur`\Fw+Qo£øP]VFqd_be�a-P]_xndc`a�_bQJr�_bet\uP�c`\F_xTog�Toe�_xr�TLr2w#q'Toa9TDfdT2P�ThgigbVFgT2fdfdPRQJTo\F^�ndVF\uT2c`epVN_ba"e-^`QJc`gbq¡nµVZr`VFS2gb_bS2_xn`gbV�QJP�Tha"gbVuToe�ate9U�Togbg'\FQ`U�fdT2P]VFq�apQ,a9^`V�\FQJU�fdc`a-Tm¨a�_bQJr+_ba"«�Toe�qdQJr`V{£5QJP ¥2´�^`_be¹_betVFe9fdVF\F_xTogbgbw�qd_˱¬\uc`gba¹ep_�r`\FVDa9^`VNPRVFqd_iepa9P]_xndc`a�_bQJrªQJfµVuP�Toa�_bQJr�^dThe

ôöíºî�ôxõ

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

a�Q�ndV+qdQJr`V#qdw rdT2U�_i\uThgigbw2l «�_ba9^°r`Q�\FQJU�f`_bgbV-¨�a�_xU#V¬Q`P�e�a-Toa�_b\�_xrL£kQJP�U�Toap_iQ`r�¥(´�^`_be,qdw rdT2U�_i\T2fdfdPRQJTo\F^B_�U+f`gi_bVFe¤a9^dToa#«V�qdVuTog£�P]QJU a9^`V²nµVFS2_xrdr`_xr`S�«�_ba9^�a9^`V�U�Q2e�a#S2Vur`VuP�Tog~\uToe�V�Qo£P]VFqd_be�a-P]_xndc`a�_bQJr°TogbgiQo«VFq�nLw¬QJcdP{\FQJr`e�a9P�Th_�rLa�euldrdT2U�VFgiwª\Fw`\Fgi_b\#«�_ba9^�n`gbQ`\¯�LeNQh£ºe�_b­FV¡®���� �u³HQJrT�������� " �����! ��`_�PRa9cdTog'S`P]_bq�a�Q�\Fw`\Fgb_i\,«~_ia-^�n`giQ`\F�2eQo£ºep_i­FV�®������ ���x³�Q`r¬T ���� ��� " ������ �J_xP]a9cdThgSJPR_iqo¥@>,cdPP]QJc`ap_�r`VFe�T2PRVNTogbe�Q�c`e9T2n`gbV,«N^`Vur+qdVuTogb_xr`S�«�_ba9^+e9cdnL¨ÌU�Toa9PR_i\FVFe�®kndc`aqdQ#r`Q2aa9T2�oV_xrLa�Q¡Th\F\FQJcdr2ae�a-P]_bqdVFe9³F¥|²Q`P]VFQo�2VuPFlJr`QªgxToa�Vur`\Fw�^`_bqd_xr`S�a�VF\F^dr`_i·dc`VFeDQJPQo�2VuP]gxT2fdf`_xr`Sª\uT2r�ndV#c`e�VFq²nµVFaÌ«VFVurÅa9^`V

P]VFqd_be�a-P]_xndc`a�_bQJr�T2r`q�a9^`V�fdP]VF�`_bQJc`e\FQJU�fdc`a9Tha�_bQJr#ndVF\uT2c`e�VDa9^`VFe�VDP]QJc`a�_xr`VFe¹TLP]V�_xr`qdVufdVur`qdVurLa®kPRVuU�T2P��.a9^dToa�_ba�qdQJVFer`Q2a"fdP]VF�2VurLa"a-^`V{c`epV�Qo£�a-^`VFe�V�a�VF\F^dr`_b· c`VFet_�r`ep_iqdV,a-^`V{PRVFqd_iepa9P]_xndc`a�_bQJrP]Q`c`a�_xr`V�_ba�e�VFgË£�l�Toe~_ia~_ieDVFvdf`g�Th_�r`VFq°_xr�epVF\Fa�_bQJr�:'¥Ë�2³F¥

� � �"!#��$%�'&(��)+*,���-�,���� �N�~&(�]�����

´�^`V+«N^`Q2gbV¡fdPRQJn`gbVuU Qo£q'Toa9T�P]VFqd_be�a9PR_�ndc`ap_iQ`r�_be�£5Q`PDVuTo\¯^°fdP]Q`\FVFe�e�Q`P,a�Q!0�r`q�«D^`_i\F^Âq'Toa9Te�apQJP]VFq�gbQ`\uTogbgiwB^dToe�a�Q�nµV�e�VurLa,a�Q�a9^`V�QLa9^`VuP]e�T2r`q©PRVFe9fdVF\Fa�_b�2VFgbw©^`Qo«�UXc`\F^¤q'Toa9Tª_ba�«~_igbgP]VF\FVF_b�2V�£�P]QJU�a-^`V�Q2a9^`VuPReZT2r`q¤«N^`VuP]V�_iaD«�_bgbgºepa�QJP]V,_ba,gbQJ\uThgigbw2¥�¸§^`Vur�a-^`V�\FQJU�U#cdr`_b\uToa�_bQJrndc �(VuP]e¬TLP]V�ndc`_bgia�®kP]VFe-fdVF\Fa�_b�2VFgbw�PRVFe�VuP]�2VFq'³Fl�a9^`VFw�^dTm�2V�a�Q�ndV¤a9P�T2r`eR£5VuP�PRVFq nµVFaÌ«VFVur a9^`VfdP]Q`\FVFe�e�Q`P]eu¥

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

1�É 32 ���F�d�54xÉ06�É.7¬��8�ħÊ(�d�F� �FÄ`�u� �5£�«¹V�Toe�e-cdU#V�a9^dToa�a-^`V�q'Toa9T°T2P]V�e�apQJP]VFq�\FQJrLa�_˨SJc`Q`c`e�gbw _xr§TBn`giQ`\F� \Fw`\Fgb_b\°£ The9^`_bQJr�QJr�a9^`V�fdP]Q`\FVFe�e�Q`P]eulta9^`V�fdP]QJn`gbVuU _be�a9^`Vur�a�Q 0�r`q«N^`_b\F^�q'Tha9T�_ba�VuU#e�e�a�QJPRVFq�QJr�fdP]Q`\FVFe�e�Q`P ��9�«~_igbgndVªe�Vur`q©apQ�fdP]QJ\FVFepe�QJP �;:E¥�´�^`VFe�Vªq'Toa9T_ba�VuU#e�^dTm�2V#a�Q�nµV¡fdTo\F�oVFq�_xrÂQJr`V¬U#VFepe9ToS2V�ndV-£kQJP]V�ndVF_xr`S°e�VurLa�a�Q � : _xrÂQJP]qdVuP.a�Q�Tm�2Q2_bqe�a-T2P]a]¨ÌcdfªqdVFgxTmwJem¥

>,cdPºTogbS2Q`P]_ba9^dU e�\uT2r`e�Toa(a9^`Ve9TLU#V�a�_xU#Va9^`V¹U+Toa9P]_bv,_xr`qd_b\FVFe"Qo£'a9^`Vtq'Toa9TDn`gbQJ\F�2e�e�a�Q`P]VFqQJr � 9 T2r`q�a9^`QLe�V�a9^dToat«�_bgbgønµV,e�a�QJPRVFq�QJr � : ¥L|°QJP]VNfdP]VF\F_be�VFgbw2ld«VN�hVFVuf�a]«Q,\FQJcdrLa�VuP]eulLQJr`V\FQJPpP]VFe9fdQ`r`qd_�r`S�apQ �09 � eNq'Toa-T#gbQJ\uToap_iQ`r�_xr¤a9^`V�S2gbQJndTog�U�Toa9P]_bv�T2r`q¤a9^`V�QLa9^`VuP�apQ �:��ýe�Q`r`V2¥¸�V�_xr`\uP]VuU#VurLa�a9^`VuU�fdP]Q2SJP]VFepe�_b�2VFgbw¡nLw¡n`gbQ`\¯��The�_xr�T�U#VuPRS2VNepQJP]a�_xr�Q`P]qdVuPta�Q�qdVFa�VuP�U�_�r`Va9^`V+QE�LVuP]gxT2f�T2P]VuThe�Toe�e9^`Qo«Nr¤_xrÂÎ�_iS`cdP]V¬�©®5a9^`V#\FQJU�f`gbVFvJ_ba]w�_be�gb_xr`VuT2P�_xr�a-^`V¬rJcdU#nµVuP,Qo£n`gbQ`\¯�Le9³F¥´�^`Vur�«V�fdTo\F�Âa9^`V�q'Toa-T°_ba�VuU#eª\FQJP�P]VFe9fµQJr`qd_xr`S©apQ�a9^`V�Qo�2VuP]gxT2f�T2P]VuToe#_xr�QJr`VU#VFepe9ToS2V,a�QªndV�epVur2a�apQ �;:2¥

< í�ß9á=û ê]Üá>=Eù-á�á>=uû ç$�-ÜÀéEÜÀÞ�ùpüøê]ù-çkÜtûýéEêÀüýè8#mÜÌçá>=oÜ"ü ß-ù�#Fûýé8��ù�é�# #mß@?�é"�öü ß9ù�#Fûýé��Xß A #mù9ákùBAbÞ�ß9ë�ù~ÝuÞ�ßFê]Ü]çkçkß�Þákßùë�èmü á=û êÌß-ëtÝuèoákÜ�Þù�é�#Nùpü çkß�ê]ùpüýü ç�ákß"Ýoù�Þ�ùpüýü ÜÀüdÞ�ß�èoá=ûýéEÜ]çCAbÞ�ß9ë�ùçkÜED-èEÜÀéuá=û ùpü'ê]ß1#uÜ-÷

î(î�éNM'O 0 )�)

� � ���� ���N( ������� ���# ����($���( #�� ��*@( �����@�$��*

1�É ��6�4;�h�d�54xÉ06©É 7��8�Ä²Ê �d�F���FÄ`�u� ´�^`VD\FQJU�U#cdr`_b\uToa�_bQJr`e�a-^dToa"Q`\F\ucdP�ndVFa]«VFVur+a9^`VfdP]Q`\FVFe�e�Q`P]eN\FQ`P�P]VFe9fµQJr`q�a�Q�TªfdVuP]epQJrdTogb_b­FVFq � Togbg ¨�apQo¨ÌTogbg��'l'_k¥ýV2¥�T#a�QLa9Tog�VFvJ\F^dT2r`S2V�«�_ba9^�U�VFe]¨e9ThS2Ve�_b­FVFe"qdVufdVur`qd_xr`S�Qh£'a9^`VNfdP]Q`\FVFe�e�QJPReu¥E´~^`V�VuU#_be�e�_bQJr�ndc �(VuP�_be 0'gbgbVFq�T2r`q�a9^`V�P]VF\FVuf`a�_bQJrndc �(VuP]e�ep_i­FVFe,_be�qdVFapVuP�U#_xr`VFq°a9^dT2rd�Le�a�Q#a9^`V�\FQ`U�fdc`a9Toa�_bQJrªQh£ºa9^`V�q'Tha9T�e�VFa�eu¥

intersection in two dimensions

distribution 1indices intersection bufferdistribution 2

Î�_bSJcdP]V�� ���PpT2fd^`_b\uTogòPRVufdP]VFe�VurLa9Toa�_bQJrÅQo£"a9^`V�P]VFe�VuT2PR\¯^ÅQo£"a9^`V#_xr2a�VuPRe�VF\Fa�_bQJr¤Qo£"a]«Q¬n`gbQJ\F�o¨\Fw`\Fgb_i\Âqd_iepa9P]_xndc`a�_bQJr`eª£kQJP�T�fdTo_xP+Qo£�fdP]QJ\FVFepe�QJP]eu¥ >,r`V�qd_xU#Vur`e�_bQJr�\uToe�V¤QJrBa9^`V¤giV-£kaul�a]«Qqd_xU#Vur`e�_bQJr¤\uToepV,QJr�a9^`V�PR_iS`^2au¥

����� � ) &%) ���� ����� � ! "*$

´�^`V�n`gbQ`\¯�+e�\uT2rdr`_xr`Sª_be�qdQJr`V,qd_xU#Vur`e�_bQJr¤n2w�qd_xU#Vur`ep_iQ`r ®5e�VFVNÎ�_bSJcdPRV��2³T2r`q¬a-^`V,QE�LVuP]gxT2fL¨f`_xr`Sª_�r`qd_b\FVFe�T2PRV,a9^`V�¼�T2P]apVFe�_xT2r¬fdPRQJq'c`\FaDQo£ºa9^`V,_xrLa�VuP]�oTogbeN\FQJU+fdc`a�VFq�_xr�VuTo\F^ªqd_xU#Vur`e�_bQJr®=a-^`VuP]V#_ie�r`Q�gb_xU#_ba9Toap_iQ`r°_xr�a-^`V�r`cdU#ndVuPDQo£"qd_xU#Vur`ep_iQ`r`eXep\uT2rdr`VFq�T2r`q°a9^`V#\FQJU�f`gbVFvJ_ba]w�_begb_xr`VuT2PD_xr�a9^`V,e-cdU�Qo£ºa9^`V,qd_xU#Vur`ep_iQ`r�ep_i­FVFeD«N^`_bgbV#a9^`V�fdTo\F�2_xr`S�_beDQJn2�`_bQJc`e�gbw�gb_xr`VuT2P�_xrÅa9^`Ve�_b­FV�Qo£òa-^`V�q'Toa9TL³F¥´�^`_beX«QJPp��_be#qdQJr`Vª_xr�VuTo\F^BfdP]QJ\FVFepe�QJP�_xr�QJPRqdVuP,a�Q¤e�Vur`q©q'Tha9T�T2r`q�P]VFe9fµVF\Fa�_b�2VFgbw�a�Q

P]VF\FVF_b�2V#q'Toa-TXTLr`q�e�a�QJP]V,a-^`VuU Toa�a9^`V�PR_iS`^2aNf`gxTo\FV�_xr�gbQJ\uThgºU#VuU#QJPRw2¥'´�^`V,«N^`Q2gbV�\FQJU�U#cL¨r`_b\uToa�_bQJr¤_beNT�fµVuP]e�QJrdThgi_b­FVFq � Togbg˨�a�Qo¨ÌThgig���Toe~_�rÅe�VF\Fa�_bQJr!:'¥öW2¥

��� � ��� !#��$-����.! �'�0&*/

¸�V�fdPRVFe�Vur2a#_xrBa9^`_be#e�VF\Fa�_bQJrBe�VF�LVuP�Tog"Q`f`a�_xU#_b­uToa�_bQJr`e�Qo£Da9^`V¬SLVur`VuP�Tog�TogbS2QJP]_ba9^dU a9^dTha�«V_xU�f`gbVuU#VurLa�VFq(¥�´�^`V 0�P]e�aDQJr`V�P]VFq'c`\FVFe�a-^`V�TLU#QJcdrLaNQo£a�_xU#V�T2r`q�U#VuU#QJPRw�r`VF\FVFe�e9T2P]w¤£kQJPa9^`V�\FQJU�fdc`a9Tha�_bQJr�Qo£�a9^`V�q'Toa9T#e�VFapeZT2r`q¤a9^`V,Q2a9^`VuPReN\FQJr`\FVuPpr�a9^`V�\FQJU�U#cdr`_b\uToa�_bQJr�Qh£"a9^`VP]VFe-c`giap_�r`S�ndc �(VuP]e~qdVufdVur`qd_xr`S�Qo£�a9^`V,a9T2P]SLVFa{^dTLP]qd«�T2P]V�A�>,»ª\uT2fdTLn`_igb_ba�_bVFeu¥

ôöíºî�ôxõ

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

�(�o�6C6;4�6��B� ´�^`V�QJnL�`_iQ`c`e�e�\uT2rdr`_xr`S,e�a9P�Tha�VFS2w,a�VFe�apeºQJr�VuTo\F^#fdP]Q`\FVFe�e�QJP�a9^`V«N^`QLgiV�P�TLr`S2VQo£'_xr`qd_b\FVFeQo£'a9^`VtaÌ«Q�q'Tha9T�qd_be�a-P]_xndc`a�_bQJr`e�a�Q 0�r`qXa-^`V¹_ba�VuU�eºa9^dThaònµVFgiQ`r`S2e�a�QDa9^`V¹fdP]Q`\FVFe�e�Q`P�09XT2r`q�e9^`QJc`gbq§ndV°\FQJU�U#cdr`_b\uToa�VFq�a�Q �;:E¥¹¦c`a�Toeª«¹V¤«�_bgbg,e�VFV�_xr�fdP]QJfµQ2e�_ba�_bQJr :'lta9^`V_xrLa�VuP]e�VF\Fa�_bQJr�Qo£,_xr2apVuP]�ETogbe�®=a9^dToa�P]VufdPRVFe�Vur2aªa9^`V�_xr`qd_b\FVFe�Qo£,a9^`V�q'Toa-T°_ba�VuU#e9³#_xr�T�n`gbQJ\F�\Fw`\Fgb_i\Dqd_be�a9P]_xndc`a�_bQJr�_iet_xr,£�To\Fa"fµVuP]_bQJqd_b\DQo£�fdVuP]_bQ`q�gb\uU²® P�O�l P � O � ³F¥F»`Q'lo_xr`e�a�VuToq#Qo£�T~£�c`gbg'n`gbQJ\F�e�\uTLrdr`_�r`S%lºa-^`V�e�\uT2rdr`_xr`S�TogbS2QJP]_ba9^dU \uT2rÂe�a�QJf°Toe�e�QJQ`r�Toe�_ba#P]VuTo\F^`VFe�a9^`V�\Fw`\Fgb_b\�ndQJcdr`q(¥|°QJP]VFQo�2VuP~a9^`_be�Togbe�Q�P]VFq'c`\FVFe�a-^`VXndQ`cdr`q�QJr�a-^`V�r`VF\FVFepe9T2P]w¤e�a�QJP�ThS2VD£5Q`P�a9^`V#_xrLa�VuP]e�VF\Fa�_bQJrfdToapa�VuP�r`eu¥

��� Æ�6ø� 8(�mÉ�6�É��(�����mÄd�oÄ 4���Ä�¹�oÉ� �C6;4=�o�d�54öÉ�6�� ��r.a9^dToa�\uToe�V2lma9^`V"\FQJU+UXcdr`_b\uToap_iQ`rXThg ¨S2Q`P]_ba9^dU _be"e�_xU�f`gbV2¥d´�^`V�e�_b­FVFetQo£�a9^`VDU#VFe�e9ToSLVFe�a�QZnµVNP]VF\FVF_b�2VFq�T2P]V\FQJU�fdc`apVFq 0�P]e�au¥2´~^`Vur�la9^`V�The�w r`\F^dP]QJr`Q`c`e{PRVF\FVF_i�LVFeNT2P]V,fdQ2epa�VFq�£kQ2gbgbQo«¹VFq�nLw�a9^`V,epVur`qdeu¥d´�^`VuP]V,_be�r`Q¬Toe�e9cdU+f`a�_bQJrQJrÅa9^`V,a9T2PRS2VFa\FQJU�fdc`a�VuP�TLn`_igb_ba]w�a�Q¬P]VF\FVF_b�2V#U#VFe�e9ThS2VFe� ��( �� � ���N¥

��Æ�6(� 8(�mÉ�6�É��(���oÉ� �C6;4=�o�d�54öÉ�6 � �Àr Q`P]qdVuP�apQ Tm�2Q2_bq >,» qdVuThq�gbQ`\¯�Le�q'c`V°a�QBa9^`V0'v`VFq²r`cdU#ndVuPQo£�\FQJU�U#cdr`_b\uToa�_bQJr�ndc �(VuP]e{£5QJPt_xr`e�a9TLr`\FV2l'«VDqdVFe�_bSJr`VFq�T2r�TogbS2QJP]_ba9^dU a9^dToa_be�c`e�_xr`S�T�n`giQ`\F�2_xr`S©P]VF\FVF_b�2V�fdP]Q2apQJ\FQ2gk¥�|°QJP]VFQo�2VuP l�_�rÂQJP]qdVuPDapQ²U#_xr`_xU#_b­FV�a9^`V¬fdP]Q`\FVFe�e�Q`P_bqdgbV¬ap_�U�V2l'VFv`\¯^dTLr`S2VFe�T2P]V#ndc`_igbaul _k¥ýV2¥(TogbgòPRVF\FVF_i�LV#£ cdr`\Fap_iQ`r�\uThgigbe#^dTu�LV�a9^`V#\FQJP�PRVFe9fdQJr`qd_xr`Se�Vur`q�£ cdr`\Fa�_bQJr�\uTogbgbe¹fµQ2e�a�VFq�nµV-£5QJPRV¹Q`PºToa�a9^`Ve9T2U�V"a�_xU#V�®=a9^`_be"P]VuU�QE�LVFeºa9^`V���� �� )=���������,��%����#�%��Toe�e9cdU�f`ap_iQ`rªQJrªa9^`V�a-T2P]S2VFa\FQJU�fdc`apVuP�³F¥´�^`_beXepa9P�Toa�VFSLw2l�_igbgxc`e�a9PpToa�VFq�_xr�Î�_bSJcdPRV¡�dl \uT2rBndVª\FQJU�fdT2P]VFqBa�Q°T�P]Q2gbgb_xr`S�\uToa�VuPpf`_igbgxT2P

Qo£¹fdPRQJ\FVFe�epQJP]eD«N^`VuP]VXThaNT�SL_i�LVur�e�a�Vuf��'lµVuTo\¯^¤fdP]Q`\FVFe�e�QJP � 9 l�®k������� �,³DVFv`\F^dT2r`S2VFeD_iapeq'Toa-T#«�_ba9^�fdP]Q`\FVFe�e�QJP ��� �"!$# 9 #&%('�) � %*!+'F¥

0 1 2 3

7 6 5 4

Step 1 step 2

0 1 27 3 6 5 4

step 3

7 0 1 2

6 5 4 3....8 processors

7 processors0 1 2 36 5 4

0 1 26 5 4 3

6 0 1 25 4 3

....

Î�_bSJcdP]V�� J´�^`VD\uToa�VuP�f`_bgbgxT2P~\FQJU�U#cdr`_b\uToa�_bQJrªU#VFa9^`Q`q�_ie_bgbgxc`e�a9P�ToapVFq�«�_ba9^ªT2r+VF�2Vur�®�J2³tT2r`qT2r}QJqdq§®k�2³DrJcdU#nµVuP�Qh£{fdP]Q`\FVFe�e�QJPReu¥(´�^`V�\FQJU+UXcdr`_b\uToap_iQ`r�Q`\F\ucdP]eXnµVFa]«¹VFVur}a9^`V��2VuP]ap_i\uThgfdTo_xP]eDQo£"fdP]Q`\FVFe�e�QJP]e#®kT#fdP]QJ\FVFepe�QJP�TogbQJr`V�\FQJU�U#cdr`_b\uToa�VFe�«~_ia-^�_ba�e�VFgË£¯³F¥

î(î�éNM'O 0 )�)

J � ���� ���N( ������� ���# ����($���( #�� ��*@( �����@�$��*

1�É ��6�4;�h�d�54xÉ06 2�4>2�Ä ��4�6;4�6 �¿� ´�^`V�f`_xfdVFgb_xr`V¬U#VFa-^`QJq¤a9T2�oVFe�Toqd�oT2r2a-ToS2V,Qo£ºa-^`VXfdQLe]¨e�_xn`_bgb_baÌw�Qh£øqd_b�`_bqd_�r`S+a9^`VN«QJPp�,_xr�e-U�Togbg�cdr`_ba�eu¥d�Àr`e�a�VuToqªQo£(«�To_ba�_xr`S,£kQJP"Togbg%a9^`VN_xrL£kQJP�U�Tha�_bQJr£�P]QJU T2r`QLa9^`VuP�fdP]Q`\FVFe�e�QJPFloVuTh\¯^�fdPRQJ\FVFe�epQJP ��9(P]VF\FVF_b�2VFeTDe9U�Togbg�fdTo\¯�hVFa�Qo£�VFgbVuU#VurLa�eul2T2r`q+_�ra9^`VDe9T2U�V�a�_xU#V,fdTo\¯�Le¹T.e9U�Togbg�fdTo\¯�hVFa"Qo£(VFgbVuU#VurLa�ea�Q�ndVDe�VurLa{TLr`q¬cdrdfdTo\F�2ea9^`V�VFgbVuU#VurLa�e_batyRc`e�aDP]VF\FVF_b�2VFq(¥�´~^`_ieD_beNT2rªQo�2VuP]gxT2fdf`_xr`Sªe�a9P�Tha�VFS2w�\FgbQ2e�V�a�Q#a9^`V�«QJPp�#qdVFe�\uP]_xndVFq°_xr 8Ë��<]¥´�^`_be,QJf`a�_xU#_b­uToa�_bQJr¤_beN_xU�f`gbVuU#VurLa�VFq�«~_ia-^`_�r¤a9^`V�\uToa�VuP�f`_bgbgxT2P�U�VFa9^`QJq¤«N^`VuP]V.£5QJP~VuTo\F^

e�apVufªa9^`VuP]V,T2P]VDe�VF�2VuP�Tog%e�Vur`q�A2P]VF\FVF_b�2V�VFvJ\F^dT2r`S2VFem¥ ´~^`_ie~QE�LVuP]gxT2f¬nµVFa]«¹VFVur+\FQJU�U#cdr`_b\uToa�_bQJrT2r`q�\FQJU�fdc`a9Tha�_bQJrD_xU�fdP]Qo�2VFq,a9^`V�a�_xU#_xr`S2e�QJr,U�Th\¯^`_xr`VFe�«�_ba9^�P�Tha9^`VuP%e�gbQE«Â\FQJU�U#cdr`_b\uToa�_bQJr`e � ¥� ��� � �#�������]�J� �D� *.!��¸�V\FQJr`ep_iqdVuP�a-^`V¹P]VFqd_be�a9PR_�ndc`ap_iQ`r#Qo£�T�U#c`gba�_bqd_xU#Vur`e�_bQJrdTog(T2P�PpTuw~Qo£'e�_b­FV�� "�� � "���!"����,¥´�^`Vªq'Toa9TÅqd_be�a9P]_xndc`a�_bQJr`e¬TLP]V¬qdV$0�r`VFq nLw°q'Toa9T¤n`giQ`\F��e�_b­FVFe � � �� � � ��� T2r`q�T�fdP]Q`\FVFe�e�Q`PSJPR_iq�Qh£ºe�_b­FV � " � � " � � " ���§Toe_xrªe�VF\Fa�_bQJr��µ¥J[ fdP]_xU#V,«�_bgbg(_xr`qd_b\uToa�V�a9^`V�fdT2P�T2U�VFa�VuP]e_xr�a-^`V�a9T2PRS2VFaqd_iepa9P]_xndc`a�_bQJr�¥Î'QJP~a9^`V�e-T2�oV#Qo£"e�_xU�f`gb_b\F_baÌw2l «V�qdVF\FQJU�fµQ2e�V#a9^`V#e�a9c`qdw¤_xr²�ªfdT2P]apeN\FQJPpP]VFe9fdQ`r`qd_�r`S¤a�Q

a9^`V�\FQJrLa9P]_xndc`a�_bQJr�Qh£º_xr`qd_i\FVFe�\FQJU�fdc`a9Tha�_bQJr�T2r`q�q'Toa9T+U#Qo�2VFe�T2r`q�a9P�T2r`eR£5VuP]e ���uÉ2�É���4x�54öÉ�6�� ���@� ����� � � �� �� ����� ����L�&%�� ��� % 9��� �E9 3!� � ��"$#&% � ��� ) )

'���� (*)�&*&*��&���) ��$ �%�'),+*� !�-´�^`V�n`gbQ`\¯�+e�\uT2rdr`_xr`Sª_be�qdQJr`V~£5QJPtVuTo\F^�fdP]Q`\FVFe�e�QJPt_xrªQJP]qdVuPta�Q#e�Vur`q�a-^`V,q'Toa9T�TLr`q�P]VFe9fdVF\-¨a�_b�2VFgbw�a�QªP]VF\FVF_b�2V�a9^`V,q'Tha9T�T2r`q�a�Q+e�a�QJP]VFq�a-^`VuU Toaa9^`VXPR_iS`^2aNf`gxTo\FV�_xr�gbQJ\uThgºU#VuU#QJPRw/.o¥´�^`V�QJnL�`_iQ`c`e�e�\uT2rdr`_xr`S�e�a9P�Tha�VFS2w�_be�a�VFe�ap_�r`S�TogbgD_�r`qd_b\FVFe°Qo£,a9^`V�SLgiQ`ndTogZU+Toa9P]_bv�¥"[,r

VFgbVuU#VurLa9T2P]wBQJfµVuP�Toa�_bQJrÂ_beXa-^`VurBa9^`V¬\FQ`U�fdc`a9Toa�_bQJrÂQo£�a9^`Vª_xr`_iap_�ThgNT2r`q 0�rdTogQE«Dr`VuP]e#Qo£NTS2_b�2Vur}VFgiVuU�Vur2au¥(´~^`_ie#P]VF·dc`_�PRVFeXTªU#QJq'c`gbQ¤QJfdVuP�Tha�_bQJr°Toe,a9^`V#q'Toa9T+qd_be�a9P]_xndc`a�_bQJrÂ_be,\FwJ\Fgb_b\2¥¦c`a�Toe�«V¬PRVufdVuToa#_ba�£5QJP�Toqoy�To\FVur2a�_ba�VuU#eul a9^`_beX\FQ`U�fdc`a9Toa�_bQJrÂ\uT2r�nµV¬qdVF\FQ`U�fdQ2epVFq T2r`qa9PpT2r`e]£kQJP�U#VFq�_xrÅyRc`epaXT+£kVF« Thqdqd_iap_iQ`r`e�TLr`q�\FQJU�fdT2PR_iepQJr`e,£kQJP�Thgigndc`a�a9^`V 0�P]e�a�VFgbVuU#Vur2am¥´�^`_beDe�a9P�Tha�VFS2w�\FQJU+f`giVFv`_ba]w�_ie ���uÉ2�É���4x�54öÉ�610 � ����"�# 3!24365

97 �898 � � 9�:; ûx÷ Ü9÷mÛJõ�í°ß@A ?�ß�Þ�<uçkákù-á=û ß�éoç�ù�é�#>=;ßpü #�?{ÝEùpÞ�ù�üýü ÜÀü�ê]ß-ëtÝuèEákÜÀÞ�ç@ õ(á ù�éuó,á=û ëÜ��2á>=EÜtêÌß-ëtÝuèoákù9á=û ßpé#û ç$#ußpéEÜ ?�û á>= �pü ß��où�ü�ûýé8# û ê]ÜÌçtß@A�á>=EÜtë¹ù-á=Þkû -(�uèoáºßpéuü ó.ü ßFê]ù�ü%ûýé8#Fû ê]Ü]çA ê]ßpÞkÞ�ÜÌç=Ý2ßpé8# ûýé8��ákß�á>=oÜDü ßFê]ù�ü�ÝEùpÞ�á�ß Aòá>=oÜ,ë¹ù-á=Þkû -�BùpÞ�ÜDéEÜ]ê]Ü]çkçkù�Þ�óªákß�ù-ê]êÌÜ]çkç�á>=oÜ #uù-ákù#çkákß�Þ�Ü�#+ûýé¡Ü]ù-ê =

ÝmÞ�ßFêÌÜ]çkçkßpÞ�÷

ôöíºî�ôxõ

����������� ����������������������������! "�$#��&%'�($��)�*+��,�� �� ( ��� �=% %���(������ ������($�B� �?��( �������?%����rÂQJP]qdVuP,apQ�_�U+fdP]Qo�2V�a9^`V�\FQ`U�f`gbVFvJ_ba]w2l�«¹V+a9T2�oV�_xrLa�Q²Th\F\FQJcdr2a�a9^`V¬n`gbQ`\¯��fdTha�a�VuP�r¤Qo£

a9^`V�qd_be�a9P]_xndc`a�_bQJrÅ£kQJPa9^`V�_�r`qd_b\FVFe�fdP]QLSJP]VFe�e�_bQJrB®=qdVFe�\uP]_xnµVFq²_xr :'¥xWu³F¥d´�^`VurÅa9^`V�\FQ`U�f`gbVFvJ_ba]w_be� ���uÉ2�É���4x�54öÉ�6��

� �����"�# 3!2 �� �97 �898 #��

� 9��9���9 % � 9

� �9 � �9�� ������ � �@�!%'�$�������� � %�#�����I)��������I)�� )=�������F���#�� � ���#����������, � � �� #�� �� %'������ � ������ ������� ���@��( � ��( ������ � ! � � ( �=% � �!���� �� ! �� � )������=��%F��6 �@� ��($� �+�,����"� ( ��% �#�%$N���� � #��&%'�($��)�*+��,������( �@� %���*+( �$�&� ( �=% �#�$N���� �� ( ��� �=% %���(�� ���+��%��� � ��($�L�����$�������?%1� ( � ���%��� #!��� ��� �� �@� ��( �+� %'� ( ����� �@� ( �� �� (' �@���� � �@���$�� � ����� ������&% �*)�*+���?�� �@�����������* ) ��(N�*�7)=��������%+�,¸�V�qdVFe�\uPR_�nµV#_�r¤a9^`VD£kQ2gbgbQo«�_xr`S�a9^`V�P]VFq'c`\Fa�_bQJr¤Qo£�a9^`V�ep\uT2rdr`_xr`S�\FQJU�f`gbVFvJ_ba]w�QJn`a9Th_�r`VFq

«�_ba9^�a9^`V°QJf`ap_�U�_i­uTha�_bQJr`e�_xU�f`gbVuU#Vur2apVFq§_xr�QJcdPªTogbS2QJP]_ba9^dU�¥´�^`V�To_xU _be�a�Q©P]VFq'c`\FVÂa9^`Ve�\uTLrdr`VFq�_�rLa�VuP]�oToga�Q°_ba�e�fdVuP]_bQ`qd_i\�fdToapa�VuP�r�Toe�fdP]Q`fdQ2e�VFqB_xrBe�VF\Fa�_bQJr :'¥Ë�d¥º½�VFa�c`e�^dTm�2V�TgbQ`QJ�©ThaXa-^`V�fdP]QJn`gbVuU�_xrBQJr`V�qd_xU#Vur`e�_bQJr�lt«N^`VuP]V�n`gbQ`\¯�Â\FwJ\Fgb_b\J®��o³�U#VuTLr`e¬T¤\FwJ\Fgb_b\°q'Toa9Tqd_be�a9PR_�ndc`ap_iQ`r�nLw�n`gbQ`\¯�LeQo£(e�_b­FV �`¥2¸�V�qdVF\FQ`U�fdQ2epV,a9^`V�\Fw`\Fgi_b\#fdToa�a�VuP�r`eta�Q�a-^`VF_�Pt\uT2r`Q`r`_i\uThg£kQJP�U�®kfdP]QJfµQ2e�_ba�_bQJr :J³~_xr�Q`P]qdVuP�apQ¡nµV�TLn`giV#a�Qª\FQJU�fdc`apV�a9^`VF_xPD_xr2apVuP]e�VF\Fa�_bQJr�®kfdP]Q`fdQ2e�_ba�_bQJrG2³~T2r`q�_ba�eNep_i­FV�®kfdP]QJfµQ2e�_ba�_bQJr¤�2³F¥d´�^`Vurªa9^`V#P]VFe9c`gba�eNT2PRV,S2_b�2Vur�_xr�fdPRQJfdQ2ep_iap_iQ`r`eZ��TLr`qFJd¥-ÅÉ'�F�d�54öÉ�6��/.�®�� ��0E�*1F³ 332547684 3 � %:9 0 %<; � 9:=?> � ;�= 8 � � �@1BA W$<DC²T2r`q!E¡®�� ��0k³!3.�®�� ��0E�uWm³F¥F´�^`Vur�lo_xr#a9^`VQJr`V-¨�qd_xU#Vur`e�_bQJrdTog%\uToe�V2loa9^`Ve�VFa�Qh£�_ba�VuU#e�Qo«Nr`VFq¬nLwXTDfdP]Q`\FVFe�e�QJP�_beT�fdTha�a�VuP�rB.�®�� � � � � �o³Fl�T2r`qBE�®�� � � �,³'_xr.a9^`V"U#QJP]V"fdTLP]a�_b\uc`gxT2P(\uThe�V�Qo£'T\FwJ\Fgb_b\qd_be�a9P]_xndc`a�_bQJr�¥���uÉ2�É���4x�54öÉ�6<F ���@� �,����( %'�$������� �*�1HG � )=������� ������������� �+� �?������($?%�� %� �@�B*+����� �*�1����%$% �@� � " � � ������������� � ����($?%1�*� %'�JI'�"�K�� ������ LNMKOPORQ�S � ��UT.��T � ��V��*V � ) � � ����� �@��( %��W��X®XTYAZT � ³1�&%1� *+����9� ���B� � �@�$#J®HV���V � ³ �@��&[�1��\E¡®XT �*V�³^]_E¡®XT � ��V � ³ 3`E¡®H1������ �®HV���V � ³p³�����K%��aE¡®HT.��V�³b]_E¡®HT � �*V � ³ 3�c ����+��� ( ���*� �&%1� %'� � ��� � � � ���,�$������� �*������I���*+" �+� ��( � d�

î(î�éNM'O 0 )�)

Wu� � ���� ���N( ������� ���# ����($���( #�� ��*@( �����@�$��*

� ��7.�®�� � � � � �o³����# .¡® ��� � ��� � � � ���i³ ) � HG �F)������=� � ���������,� � ������( %+�.�®�� � � � � �o³ 3 �97�� 898 � # E¡®�� % �@� � �,³

�@�� G ���@�������.�®�� � � � � �o³�� .¡® � � � � � � � � � � ³ 3 �� 9 : '��� � 898 � # � 898 � � # �� E¡®�� % �@� � �,³ ]_E¡® � � % ; � � � � � ³

��� � � � ����� � ��,���� �7 �@����� � ' � ��� �B�,�#������,# *@�����,����( %'�$�������!� % � � ��1� (7�&%��B�����������*�?�� ����( �*� � ��($����#!��� �®�� � � ��� � �;³*' %��L �+� ������( %��$�������F�&%1�� *@��,� I� �1�� � %'�� "�����������������?������($?%�� ,���uÉ2�É���4x�54öÉ�6�� ���@��������( %��$�������L� �7HG �L)��������1������������� �+� �?������($?%7�&%*�?��($�,��#��,��� � �?��($�,��#��� �®�� � � � � � � ³ � G��@��( � � � �# � � ��( � �@�B�*� B) ��(��*��� ( ��� �=% %���( % �� � � ���#�� %��($� )=*@����� ���� ������ � �@� %��&%F#���($����� # ��( �� �@� � ( ���*� �*� � ( � �?��%'������� � �����*+����� �*� �?������($?%F������?��($�,��#��,� G �, �� �@� %��� � � ��($����#!��� �®������ � � � � ³1�&%1��� %���?��($�,��#��,�"G �� �F �@� %'�� � �?��($�,��# ,Í�Vur`\FV�«V�\uTLr�c`e�V#a9^dToa�fdP]Q`fdVuP]a]w°a�Q�e9fµVFVFq'cdf�a9^`V#\FQJU+fdc`a9Toa�_bQJr�TLr`q�qdVF\uP]VuToe�V+a9^`V

U#VuU�QJP]w�r`VFVFqdeu¥���uÉ2�É���4x�54öÉ�6�� ���@��( �YG ���,� ) �1� ��%��� " ��� ���� % �� ��F������(������ �*�1������@ �"V �*�1 �@�( ��%�*+������� �?������($ �� ������ � � �@����� %�� ��I�� �@�&� ( ���*�B�*� � ( � � ��%�����,�� � �@�� �@�B( ��%�*@������� �?������($ �&% �@�*+����� �*�B��� ��%� ��" ��������������� �?������($?%*'��*�B�,#�������� � � �?��($�,��#�� W � �� �?��($�,��# ' G � �@��������� ( � � ( ��%������������� �*�1� � ���� �� �?�������� �1 �@�B*+�����&,��r#a9^`Vt£kQ2gbgbQE«~_�r`S%l �öÄJ�Y1���Ä�U�Tov�®��-� �;�  ®�� � � � � � � ³p³F¥E´~^dT2rd�2e�apQNa9^`VDfdP]VF�J_bQJc`efdP]QJfµQo¨

e�_ba�_bQJr`eml'ToeqdVFe�\uP]_xndVFq°_xr!:'¥ �µl2«VN\uTLrªe�a�QJf+a9^`VDe�\uT2rdr`_xr`S¡The¹epQJQJrªToe«VZPRVuTo\¯^ªa-^`VNS2gbQJndThg_xr`qdVFv&1¹nµVF\uT2c`e�VDa9^`V�\FQJr`epa9P�c`\Fa�_bQJr#Qh£øa-^`V�_xr2a�VuPRe�VF\Fa�_bQJrªfdToa�a�VuP�r`e�_be\FQJU�f`gbVFa�VFq(¥L|°QJP]VFQo�2VuP«VX^dTm�2V#T¬nµQJcdr`q°Qo£ ��"�� � QJr�a9^`V#rJcdU#nµVuPDQo£òqdVFep\uP]_xf`a�QJP]e,Qo£"a-^`V�_xr2apVuP]e�VF\Fa�_bQJr°fdToa�apVuP�r®=a-^`_ieD«�_bgbgònµV,_xU�fdQJPRa9T2rLa£5Q`Pa9^`V�P]VF·dc`_xP]VFq�e�apQJP�ToS2Vu³ ¥��r°fdP�To\Fa�_b\FV#a9^`_be�Q`f`a�_xU#_b­uToa�_bQJr¤_be,QJr`gbw�_xr2apVuP]VFe�a�_xr`S�«D^`Vur�«V#^dTu�LV��2VuP]w¤e9U�Togbg�n`gbQJ\F�

e�_b­FVFe#QJP.�2VuP]w�gxT2PRS2V¬U�Toa-P]_b\FVFeu¥([Dgba9^`QJc`SJ^°a-^`VuP]V¬T2P]V+\uToe�VFe�«D^`VuP]V�_ba�_be�U#QJP]V#_xrLa�VuP]VFe�a�_xr`S

ôöíºî�ôxõ

����������� ����������������������������! "�$#��&%'�($��)�*+��,�� �K�a�Q�T2rdThgiw`a�_b\uTogbgbw�qdVFa�VuP�U#_xr`V,a9^`V,fdToa�apVuP�r#£kQ2gbgbQo«�_xr`S¬fdP]QJfµQ2e�_ba�_bQJr!:¡®;£5QJPt_xr`e�a9TLr`\FVD£5QJPH\FwJ\Fgb_b\qd_be�a9PR_�ndc`ap_iQ`r`e�Toe�_xr 8xWm��<k³Fl�S2Vur`VuP�TogbgbwB£5Q`P�n`gbQJ\F�BQo£ZP]VuThe�QJrdT2n`gbV�gbVur`S2a-^���QJcdP#epa9P�Toa�VFSLw²_benµVFa�a�VuPF¥dÎ%QJP_xr`e�a9TLr`\FV2l'e�VFV,Î�_bSJcdPRV1:�«N^`VuPRV�a9^`V,\FQJPpP]VFe9fdQ`r`qd_�r`Sª\FQJU�f`gbVFv`_ba�_bVFeXT2P]V�f`gbQ2apa�VFqToe�T~£�cdr`\Fa�_bQJr�Qo£'a9^`V�Tm�2VuP�ToS2V"n`gbQ`\¯�.e�_b­FV�� � � � ®=a9^`V"ep· cdT2PRV¹P]Q`Q2a�Qo£'a9^`V_xr`_iap_�Thgdqd_iepa9P]_xndc`a�_bQJrn`gbQ`\¯�ªe�_b­FV�a�_xU#VFeNa-^`V 0�rdTog(qd_be�a9P]_xndc`a�_bQJr°n`gbQ`\¯�ªe�_b­FVu³F¥

0.0 10.0 20.0 30.0 40.0 50.00.0

1.0

2.0

3.0

4.0

Î�_bSJcdP]V�: "¼QJU�fdTLP]_be�QJrÂQo£Na9^`V�a]«Q°e�\uT2rdr`_xr`S©epa9P�Toa�VFSL_iVFe#_xrBQJr`V�qd_xU#Vur`e�_bQJr�¥�´~^`V�a�_xU#V®kU�_igbgb_be�VF\FQJr`qde9³t_be"f`gbQ2a�a�VFq�ToS`To_xr`e�aºqd_��(VuP]VurLaòPpT2r`qdQJU n`giQ`\F��ep_i­FVFeN®5a9^`V¹T2n`ep\F_iepe9T,Tov`_ie"PRVufdP]V-¨e�VurLa�ea9^`VNTm�2VuPpToS2V�n`gbQJ\F��ep_i­FVu³ ¥��"Th\¯^+�ETogxc`VD_be¹Q`n`a9To_xr`VFq�£�P]QJU¶a9^`V,U#VuT2r#Qo£(QJr`VDa9^`QJc`e-T2r`q_ba�VuP�Tha�_bQJr`eNQh£òa9^`V,e-T2U#V,e�\uT2rdr`_xr`SªQJr�T�» � �¿»dfdTLP]\�G�«QJPp�2e�a-Toa�_bQJr�¥

��Q2ap_i\FV#a9^dTha�«�_ba9^¤a9^`VXfdPRVF\FVFqd_�r`S¤QJf`a�_xU#_b­uToa�_bQJr`eml�a9^`V�\FQJU+f`giVFv`_ba]w�_be,_xr`qdVufdVur`qdVurLa�Qo£a9^`V#U�Toa9PR_ivªe�_b­FV¡®5VFvJ\FVuf`a£kQJPe-U�Togbg(U�Toa9P]_b\FVFeD«N^`VuP]V � ��gb\uU°®�� � � � � � � ³�³ ���uÉ2�É���4x�54öÉ�6�� ���@� $N�����%�� ��������B�$�� � �������,��B�� ���� #�� �� %'����F�&% �

� .� ��"$# 3 2� 1� � % 1

� � � ��G��@��( �Y1B�&% �+��U�Tov�® �-����� �® � � � � � � � ³p³� ûx÷ Ü9÷� �CAxß�ÞºêÀü ù-çkç=û ê]ù�üdÝoù�Þ�ùpüýü ÜÀü�ê]ß9ëtÝmèEákÜÀÞ�ç

î(î�éNM'O 0 )�)

Wu� � ���� ���N( ������� ���# ����($���( #�� ��*@( �����@�$��*

� ������ � ( �� � ( � � ��%��,������� ' �@�N%�� ���������������(��������&%"( � # *@�$� # �� � ! )�� �@��� ( � � �$�� � *+�����,�

�*� �@� ����������� � ������( ������( %�� ����,�� %+�b,Î�_�rdThgigbw2lda-^`VZP]VFe-c`gia\uT2rªndVDe�a9PpTo_bSJ^2aR£5QJPR«{TLP]qdgbwXVFv`a�Vur`qdVFqªa�Q�U#c`giap_iqd_xU#Vur`ep_iQ`rdTogºT2P�PpTuw`e

���uÉ2�É���4x�54öÉ�6�� � �� ��"$# 3 243 �9

1 9� 9 �09 % 1 9

� �9 � �9:G��@��( �Y1 9 3§U+Tovø® � 9'����� X® ��9���9E� ���9 � �9 ³�³� ������ ����($����$# ��( �� � ( � � ��%��,����� � )��B��# #��,��F �@� �$��%' �, �$�����#�� ��?%����� �b,

��Q2ap_i\FV~a9^dToa�a9^`Ve�\uT2rdr`_xr`S#q'cdP�Toap_iQ`r#_be"SJP]VuTha�gbwXP]VFq'c`\FVFq�n2w�QJcdP�Q`f`a�_xU#_b­uToa�_bQJr�ndc`ata9^`VfdTo\F�L_�r`SªQJr`V�P]VuU+To_xr`eNa9^`V,e-T2U#V2¥

'���� � )����'& � &%+6) ��$-$�*&%� )�.! �'��& ) ��$ �*��) +*��!�-´�^`V�fdTh\¯�L_xr`S�\FQJr`e�_be�a�eQo£ � U#Qo�2V ��Q`fdVuP�Toap_iQ`r`e�a�QXndc`_bgbq²T#ndc �(VuPFl`a9^`V�ndc �(VuPt_be�a9^`Vurªe�VurLau¥Î%Q2gbgiQo«�_xr`S°a9^`V�\FgxToepe�_b\uToggi_xr`VuT2P�a9PpT2r`e]£kVuPDa�_xU#V¬U#Q`qdVFgklº«V#qdVFe�\uP]_xnµV�a9^`V�\FQJU�U#cdr`_b\uToa�_bQJr\FQ2epa�\FQJU�f`gbVFvJ_ba]w,£kQJP�U#c`giap_ ¨�qd_xU#Vur`ep_iQ`rdTog(T2P�P�Tmw`e(_xr#a�VuP�U#e�Qo£�U#Qo�2VT2r`q#a9P�T2r`eR£5VuP a�_xU#VQJr`gbw®=a-^`VuP]V,_beZTogb«�TmwJe � A WDe�a9T2PRa9cdfªqdVFgxTuw`eD_�rÅQJcdPTogbS2QJP]_ba9^dU+³ ���uÉ2�É���4x�54öÉ�6� � ��� ) ) 3!2 365

97 $898 � � 9�09�� �9

: �����F���#I�( �� % ����( � �?��( �������?% �'��( � �?����(B�*� � ( ���$��%$%'��( % � ���#I�$��� � � ( ��� �=% %���("G ���,� ��� �� �@����+�"G �, �F����� �@� �� �@��( % ��� ������ � �,�H����� ����%B �@�� ��( �L�� ) �B%����N��I� �?��( ����( *�%��) � � ���=� �$#F�� � ���)�*��7��(���( �$#�*@�$��%����($�*�� � ����( �@�$��#�� ���@�1�$��($( ��% � ���#��,��L�$��%� � %�� ( � �?��($��,������ ��� �@� �*� B) ��(��*������ ����% � ��(�� ( ��� �=% %���(�' �@� "�&%1�� ������( � �@�� 97 �898 � �B�! � ! �� � ,

ôöíºî�ôxõ

����������� ����������������������������! "�$#��&%'�($��)�*+��,�� ��-'�� � � �0$ � � ��/ �0&�� ) !�� ) ) & / )�&%&*�'& � &*+ � )�����&���� ) ��$-$ �%&*� )� ! ����&>) ��$��

�%�'),+*� ! �')./�Àr�QJPRqdVuP,a�Q�e�a-c`qdw²a-^`V¬P]VFgxToa�_b�2Vª_xU�fdQJPRa9T2r`\FV�Qo£�VuTo\F^�fdT2PRa,Qo£�QJcdP,TogbS2QJP]_ba9^dU ®=e�\uT2rdr`_xr`ST2r`q�fdTo\F�2_xr`S@Ah\FQJU�U#cdr`_b\uToa�_bQJrd³FlL«¹VDfdP]VFe�VurLa"_xr�a-^`V£5QLgigbQo«�_xr`S#a9^`VNnµVu^dTu�`_bQJPtQo£(a9^`VF_xP"P�Tha�_bQ«N^`Vur ��_be"_xr`\uP]VuToep_�r`S%¥J´�^`V�P�Toap_iQ�Qo£�a9^`V\FQJU�f`gbVFv`_iap_iVFeQo£�a9^`Ve�\uT2rdr`_xr`SXTLr`qXa-^`VNU#Qo�2VFeT2r`q¤a9P�T2r`eR£5VuP]e_be� ���uÉ2�É���4x�54öÉ�6����

3 ��� ��"$#� ) �� � 3!2 �� �9

� :��7 9 !��� �� 9 �

3 243 �9

� � � 9� �09 � 9 : 3 � �� 2 3 �9

� 9��9���9 :G��@��( � � 3 �097�&%1 �@� �������H�* B) ��(��*��� ( ���$��%$%'� ( %��� ������ � * %' � �� ���� ��( �� �@�B( ������L� ��� ( � �?��%'���������L���# �� ,

´�^`Vur�gbQ`QJ�2_xr`S�Toaa9^`V�gb_xU#_ba�

���uÉ2�É���4x�54öÉ�6�� � W � � 9���� �@��� 3�� 9!�� ��� �"! �# �� � � �

� ������ � �@� ( � ��,�!$�� � � �_,

[Ne�wdU�f`a�QLa�_b\NfdP]QJfµQ2e�_ba�_bQJr�W2W�U�VuT2r`e"_xr�fdP�Th\Fa�_be�V�a9^dToa"The�e�QJQ`r�Toe ��_begxT2P]S2VVur`QJc`S`^�%ol_Ë£�QJcdPDQJf`a�_xU#_b­uToap_iQ`r`eXT2P]V�T2fdf`gb_bVFq§®=\-£R¥�epVF\Fa�_bQJr :'¥ �2³Fl%a9^`V�P]VFqd_be�a9PR_�ndc`ap_iQ`r�\FQ2epa,_ie#P]QJc`SJ^`gbwVF·dcdTog�a�Q#a]«¹Q�U�VuU#QJP]w�\FQ`f`_iVFe#®=QJr`VD«N^`Vur�a-^`VNq'Toa-T�_bee�VurLa�T2r`q¬Q`r`V,«N^`Vur�_ba_beNP]VF\FVF_b�2VFq'³f`gxc`e,a9^`V,a9PpT2r`e]£kVuP"ap_�U�V�Qo£�a9^`V,_ba�VuU#e�Qo«Nr`VFq¤gbQJ\uTogbgbw2l%_5¥ V2¥'a9^`V,ep\uT2rdr`_xr`S¬_be,r`VFS2gb_iSL_�n`gbV2¥

& ' � � �- ��&(�"$ *��]��$

�Àrªa9^`_be¹epVF\Fa�_bQJr�l`«¹VD«�T2rLa"a�Q�e9^`Qo« a9^`VDV-±¬\F_bVur`\Fw¬Qh£ºQJcdP"ThgiSLQJP]_ba9^dU#e�T2r`qªa9^`VD\FQJP�P]VF\Fa9r`VFepeQo£QJcdP~\FQJU�f`gbVFvJ_ba]w�P]VFe9c`gba�e�ndc`a,Togbe�Q�a�QªqdVuU#QJr`e�a-P�Toa�V�a9^`V�P]VuTog�SJTo_xr¤a9^dToaD_ie,Q`n`a9To_xr`VFq°_�r( �=mû çdû ç�á=ÞkèEÜ ÜÌú-ÜÀé A�ßpÞ�ëù-á=Þkû ê]ÜÌç�ß AJëß1#uÜÀÞ�ù-ákܺç=û*)]Ü�� A�ßpÞ'ûýéoçkákù�éoê]Ü�?=oÜÀé,+�- � û ç�ákÜÀé{á=û ëÜ]ç �uû ���9ÜÀÞ(á>=Eùpéá>=oÜ�éFèEë �2Ü�Þ�ß@AdÝmÞ�ßFêÌÜ]çkçkßpÞ/.�ç]÷

î(î�éNM'O 0 )�)

W�: � ���� ���N( ������� ���# ����($���( #�� ��*@( �����@�$��*

fdP�Th\Fa�_b\FVD£5QJPtgb_xr`VuT2PDTogbS2VundP�T#�oVuP�r`VFg�\FQJU�fdc`a-Toa�_bQJrªc`e�_xr`S�QJcdPP]VFqd_be�a9P]_xndc`a�_bQJr¤P]QJc`a�_xr`VFeu¥µ¸�V\F^`QJQLe�V,£kQJPta9^dToaN�+P]QJc`a�_xr`VFeDQo£ºa9^`V#»d¼[Z½ø[ZO%[N¼�¾¶gb_�ndPpT2P]w da9^`VXU+Toa9P]_bv�UXc`gba�_xf`gbw ®5|²|�³Fl½���£ Th\Fa�QJP]_b­uToa�_bQJrB®k½��D³NTLr`q²a-P]_xT2r`SJc`gxT2PDe�Q2gb�2V�®k´�»d³F¥%��r¤a9^`V�»d¼[,½�[ZO%[N¼~¾ gb_xndP�T2PRw2l�a9^`VP]Q`c`a�_xr`VFeZP�cdr+£kQJP�TLr2wª�J_xP]a-cdTog(SJP]_bq�Qo£"fdP]Q`\FVFe�e�QJPReNT2r`q�T2rLw¬ep_i­FV�Qo£�e�· cdTLP]V�n`gbQJ\F�2em¥

>,cdP,VFvdfdVuP]_xU#VurLa�e�«VuP]VªP�T2r¤QJr¤a9^`V¬¼�PpTuw�´~�oÁ T2r`q�QJr¤a9^`V¬�Àr2apVFg"O�T2P�ToS2Q`r�fdT2P�ThgigbVFg\FQJU+fdc`a�VuP]eu¥L¸�VNc`e�VNT.\FQJU�f`_bgbVFq��2VuP]e�_bQJr�Qo£(a9^`VN»d¼[,½�[,O'[D¼�¾¿TLr`q¬¦½�[N¼�»#gb_xndP�T2PR_iVFe~QJra�Q`f�Qh£¹a9^`V#�2Vur`qdQJP.QJf`a�_xU#_b­FVFq©¦½�[,»°QJr¤a9^`V¬O�TLP�ToS2QJr�l%T2r`q©Tª\FQJU�f`_bgbVFqB�2VuP]e�_bQJr¤Qo£a9^`VP]VFqd_be�a-P]_xndc`a�_bQJr�P]QJc`a�_xr`V�«~_ia-^�Tª�2Vur`qdQJP._�U+f`giVuU�Vur2a9Tha�_bQJr�Qh£¹a9^`V¬½ ��l |²|�l�´�»�l(¦½�[N¼�»T2r`q°¦½�[Z»¤P]QJc`a�_xr`VFe�Q`r�a9^`V�´��hÁ�¥Î'QJPT,SL_i�LVur�rJcdU#ndVuP{Qo£òfdP]Q`\FVFe�e�QJPReZ®]Wu�L³"T2r`q�T�S2_b�2VurªU�Toa9PR_iv+e�_b­FV¡®]Wu�L��: "°Wu�2� :J³Flo«V

0�P]epa�qdVFa�VuP�U#_xr`V�a9^`VZnµVFe�aNn`gbQ`\¯�ªe�_b­FV�T2r`q�S`P]_bq�e9^dT2fdV.£5Q`P"VuTh\¯^ª\FQJU+fdc`a9Toa�_bQJrª�oVuP�r`VFgk¥ >,cdPa�VFepa�ee9^`Qo«�a-^`VNa�_xU#_xr`S2eDToe�T,£�cdr`\Fa�_bQJrªQo£�a9^`VZn`gbQ`\¯�+e�_b­FVFe£kQJPt�J_xP]a-cdTog(fdP]QJ\FVFepe�QJP]eSJP]_bqdeQo£Togbg�e9^dT2fµV �W "©Wm�dld��"IJdl+: " :'l?J�"²�µl�Wm��"�W�fdP]Q`\FVFe�e�QJPRe�SJP]_bqdeu¥%Î'QJPt\FgxT2P]_ba]w2l%a9^`V�n`gbQJ\F�e�_b­FVFe,T2P]V,\F^`Q2e�Vur¤fdQo«VuP"Qh£H�dl%ndc`a�QJcdP�P]VFqd_be�a-P]_xndc`a�_bQJr°P]QJc`a�_xr`VFeNP�cdr¡£5Q`P�T2rLw�e�_b­FVFeu¥¸�V#e9^`Qo«¿_xr°Î�_bSJcdP]VBGªa9^`V�P]VFqd_be�a-P]_xndc`a�_bQJr°VFgxT2f`e�V�ap_�U�V�£kQJP,P�TLr`qdQJU n`gbQ`\¯�¤e�_b­FVFe#T2r`q

P�TLr`qdQJU�SJP]_bq¤e9^dT2fdVFe~QJr�a9^`V�O�TLP�ToS2QJr��m¥

400.0 900.0 1400.0 1900.0matrix size

0.00

0.10

0.20

0.30

0.40

0.50

time

(sec

onds

)

Î�_bSJcdP]V�G `zVFqd_be�a9P]_xndc`a�_bQJrÅVFg�TLf`e�V,a�_xU#V~£5QJPHqd_��øVuPRVur2a�U�Toa-P]_bv�e�_b­FVFeQJr¬PpT2r`qdQJU S`P]_bqde¹Qh£HW=:a�Q¤Wu��fdP]Q`\FVFe�e�QJPRe¹«~_ia-^�P�T2r`qdQ`U n`gbQJ\F�ªe�_b­FVFeu¥

��� Ü�?�ÜÀÞ�Ü�éEß-á�ù��mü Ü"ákß�ÞkèméZákÜ]çkákç?�û á>=�éoß�éDÝ2ß@?�Ü�Þ(ß A O ��Þkû #mç(ßpéNá>=EÜ������ �2Ü]êÌùpèEçkÜ�ß Adú9ÜÀé8#mß�Þ���Û`õºì(ïüýû ëtû ákù-á=û ß�é�ßpé�á>=uû ç�ë¹ù-ê =mûýéEÜ

ôöíºî�ôxõ

����������� ����������������������������! "�$#��&%'�($��)�*+��,�� � �¸�V�a9^`VurªfdP]VFe�VurLa"a9^`VDV-±¬\F_bVur`\Fw�Qo£(QJcdPHe�Q2gxc`a�_bQJr`eul`«�_ba9^ªT,\FQ`U�fdT2P]_be�QJr�Qo£(a9^`V�a�_xU#_xr`S2e

«�_ba9^ÂT2r`q°«�_ba9^`QJc`a�P]VFqd_be�a9PR_�ndc`ap_iQ`r`eXQh£"a9^`V#q'Toa9T�®=e�VFV#a9^`V#\FQJPpP]VFe9fdQ`r`qd_�r`S°TogbS2QJP]_ba9^dU�e,QJrÎ�_bSJcdP]V��L³t£5Q`P"a-^`V���gb_xr`VuT2PDTogbS2VundP�T#\FQ`U�fdc`a9Toa�_bQJr��hVuP�r`VFgbeX®k½���ld|°|�l ´~»d³"Q`rªa9^`V�U#c`giap_ ¨\FQJU+fdc`a�VuP�¥

� � �'É�� 4ö��8� � � �'É���È 4x��8 �mÄdÊ�4=�F�u��4=� ���54xÉ06Á�Toa9T P]VFqd_be�a9P]_xndc`a�_bQJr(®5[�¨�q'Toa-Toqd_be�aulý[~¨ÌndVFe�aR£5QJPp�oVuP�r`VFgx³

¼Q`U�fdc`a9Toa�_bQJr �oVuP�r`VFg�®=[~¨�q'Toa9Toqd_be�a-³ ¼QJU�fdc`a-Toa�_bQJr �hVuP�r`VFg�®=[�¨ÌnµVFe�a]£kQJP��oVuPpr`VFg�³Á�Toa9T P]VFqd_be�a9P]_xndc`a�_bQJr(®5[�¨ÌnµVFe�a]£kQJP��oVuP�r`VFgklý[~¨�q'Toa9Thqd_iepa9³

Î�_bSJcdP]V#� `[NgbS2Q`P]_ba9^dU#e~£5QJPtQJr`V,\FQ`U�fdc`a9Toa�_bQJr��hVuP�r`VFg(«�_ba9^`QJc`aDT2r`q�«�_ba9^�q'Tha9T#P]VFqd_be�a9P]_xndcL¨a�_bQJr�¥

¸�_ba9^�a9^dTha"P]VFe9c`gba�eulo«Va9^`Vur#e�a-c`qdwXT2r�VFvdT2U�f`gbV�«�_ba9^#a-^`V{�Dgb_xr`VuT2PtTogbS2VundP�T,Q`fdVuP�Toap_iQ`r`eVFv`VF\uc`a�VFq#e�VF·dc`Vur2a�_xTogbgbw#_xr,a9^`Vte9T2U#VfdT2P�TogbgbVFg fdPRQ2SJP�T2U�®=e�VFV�a9^`Vt\FQJP�P]VFe9fµQJr`qd_xr`S�TogbS2QJP]_ba9^dU�eQJr}Î�_iS`cdP]V¬�2³F¥(´~^`V¬rJcdU#nµVuP,Qo£e�wJepa�VuU#e�epQ2gb�2VFq�_be�e�VFa�cdfÂa�Q�QJr`V�£5QJcdPRa9^¤Qo£¹a9^`VªU�Toa9P]_bve�_b­FV,_xr+QJP]qdVuP�£kQJPHa9^dToa¹PRQJc`a�_xr`VNapQX^dTm�2VNT2r�VFvJVF\uc`a�_bQJrªap_�U�VN\FQJU+fdT2P�T2n`gbV�a�Q�a9^`V,�,Q2a9^`VuPRe9³F¥zVuU�T2P��.a9^dToa�_Ë£(e�VF�2VuP�Thg'T2P�P�Tmw`eòT2PRV¹_xU�f`gb_bVFq�_xr#a9^`V�\FQ`U�fdc`a9Toa�_bQJr#�oVuPpr`VFg5lLa9^`Vur#e�VF�LVuP�Tog

q'Toa-T,P]VFqd_be�a9P]_xndc`a�_bQJr`e�U#c`e�a"ndV~\uTogbgiVFq©®5_�r�a9^`V|}Tha9P]_bv�|�c`gba�_xf`gbw#£5Q`P�_�r`epa9T2r`\FVu³F¥`Í�Vur`\FV2lLa9^`VP]VFqd_be�a-P]_xndc`a�_bQJr�e�QLg�c`ap_iQ`rB_�r`\Fgxc`qdVFe���\uTogbgbe�a�Q°a9^`VªP]VFqd_be�a9P]_xndc`a�_bQJr�P]QJc`a�_xr`V�®=_k¥ýV2¥���£kQJP.a9^`V|°|��hVuP�r`VFgkl�W#£kQJP�a9^`V�½�� �hVuP�r`VFgkl"�Å£5QJP.a9^`V�´�»��oVuP�r`VFgkl�f`gxc`e¬QJr`Vªa�Q¤S2Q�ndTo\¯�}a�Q°a9^`V_xr`_ba�_xTog�q'Toa9T�qd_be�a9PR_�ndc`ap_iQ`rd³F¥

����� � & &��5&0! ) � � �.����& � ��'��) � $ ) "*�'&%)>,r�a-^`V©��rLa�VFg�O�T2P�ThS2QJr�l�Î�_bSJcdP]V JBe9^`Qo«�e�a9^dToaªa9^`V©nµVFe�a�fdVuPÀ£kQJP�U+T2r`\FV�Qo£#a9^`V�U�Toa9P]_bvU#c`gba�_xf`gbw�_ie~QJn`a9To_xr`VFq�«�_ba9^�T : "!:#�J_xP]a-cdTog�SJP]_bq�«�_ba9^ªa9^`V,n`_bS2S2VFe�aNn`gbQ`\¯�+e�_b­FV¬®k��:.£5Q`P"Q`cdPfdP]Q`n`giVuU+³F¥Î'QJPta9^`V�½�� qdVF\FQ`U�fdQ2ep_iap_iQ`r�lda9^`V�ndVFepaNP]VFe9c`gbaD_ie~QJn`a9To_xr`VFq�«~_ia-^�n`giQ`\F�2eDQo£ºe�_b­FV J�Q`r�T

W "BWu���J_xP]a-cdTog(SJP]_bq¤Qo£òfdP]Q`\FVFe�e�QJPReu¥Î'QJPta9^`V�e�Q2gb�2V,Qo£¹Wm�2��:�P]_bSJ^LaN^dT2r`q�ep_iqdVFemlda9^`V�ndVFepaZP]VFe-c`gia~_ieDQJn`a9Th_�rª«~_ia-^�a9^`V�n`_bS2S2VFepa

n`gbQ`\¯�ªe�_b­FV�®k��:J³tQJrÅa9^`V�Wu� "©W,�`_xP]a9cdTog SJP]_bq�Qo£"fdP]QJ\FVFepe�QJP]eu¥´�^`V,SJP]_bq�e-^dT2fdV#T2r`q�n`gbQJ\F�ªe�_b­FVFe�T2P]V,�2VuPRw¬qd_��(VuP]VurLa£5Q`Pa9^`VFe�V���PRQJc`a�_xr`VFe�®5e�VFV�´ºTLn`giV

Wu³T2r`q�a-^`V,a�_xU#_xr`S2e�qd_��(VuP]Vur`\FVFeZTLP]VZr`QLa{r`VFSLgi_bS2_xn`gbV2¥(´�^`_be�a�Vur`q�apQXqdVuU�QJr`e�a9P�Tha�VNa-^dToaa9^`VP]VFqd_be�a-P]_xndc`a�_bQJr¤_�U+fdP]Qo�2VuU#VurLa�\uT2rªgbVuToq�apQ¬T2rª_xU�fdQ`P]a9T2rLa�SJTo_xrª_xr�VFgxT2f`epV�a�_xU#V2¥¾,r`QE«~_�r`S�a-^`VFe�V�ndVFe�a,\FQJr 0'S`cdP�Toa�_bQJr`eul�«V#P�T2r¤a9^`V#a�VFe�a�eD\FQJP�PRVFe9fdQJr`qd_xr`S¤a�Q¬a-^`V�ThgiSLQo¨

P]_ba9^dU Qo£¹Î�_bSJcdPRV¬�+£5Q`PDa9^`V���rJcdU#VuPR_i\uThg¹�oVuP�r`VFgbe�«�_ba9^�Wu�ªfdP]Q`\FVFe�e�QJP]em¥�¸�V¬f`gbQ2a�a�VFq«�_ba9^

î(î�éNM'O 0 )�)

Wu� � ���� ���N( ������� ���# ����($���( #�� ��*@( �����@�$��*

� ���%É���4x��8� � ���%É ��È 4ö��8 �uÄdÊC4;� �u� 4=� ���54öÉ�6Á�Tha9T P]VFqd_be�a9PR_�ndc`ap_iQ`r(®=[�¨�q'Tha9Toqd_be�aulý[~¨ÌndVFepa]£5Q`P]|°|}³Á�Tha9T P]VFqd_be�a9PR_�ndc`ap_iQ`r(®k¦H¨�q'Toa9Toqd_be�aul˦�¨ÌnµVFe�a]£kQJP]|°|�³

|�Toa9PR_iv U#c`gba�_xf`gb_b\uToa�_bQJr(®=[~¨�q'Toa9Toqd_be�aml ¦H¨�q'Toa9Toqd_be�a9³ |�Toa-P]_bv UXc`gba�_xf`gb_b\uToa�_bQJr(®=[~¨ÌndVFepa]£5Q`P]|°| l˦�¨ÌnµVFe�a]£kQJP]|°|�³��� ( �=%'*+����&%1��% %�* �$# %'���( �$#B�� � ��� Á�Tha9T P]VFqd_be�a9PR_�ndc`ap_iQ`r(®=[�¨ÌnµVFe�a]£kQJP]|°|�ld[~¨ÌndVFe�a]£kQJP�½ �N³½�� £ Th\Fa�QJP]_b­uToa�_bQJr(®5[�¨�q'Toa-Toqd_be�a9³ ½ � £�To\Fa�QJP]_b­uToap_iQ`r(®=[�¨ÌnµVFe�a]£kQJP�½��D³��� ( �=%'*+���% ��( �B� % %�*� �$#B%����( �$#B�, � ��� Á�Tha9T P]VFqd_be�a9PR_�ndc`ap_iQ`r(®=[�¨ÌnµVFe�a]£kQJP�½���l�[~¨ÌndVFe�a]£kQJP�´~»d³��� � �&%1 �@� %'����*+��,� ���($� � ��� Á�Tha9T P]VFqd_be�a9PR_�ndc`ap_iQ`r(®k¼t¨�q'Toa9Thqd_iepaul ¼t¨ÌnµVFe�a]£kQJP�´�»d³´�P]_xT2r`SJc`gxT2P e�QLgi�LVJ®=[�¨�q'Tha9Toqd_be�aul ¼¹¨�q'Toa9Toqd_be�a9³ ´�P]_xT2r`SJc`gxT2P epQ2gb�2VJ®=[~¨ÌndVFe�a]£kQJP�´~»�l ¼t¨ÌndVFepa]£5Q`P�´�»d³��� ( �=%'*+����&%1��% %�* �$# %'���( �$#B�� � ��� Á�Tha9T P]VFqd_be�a9PR_�ndc`ap_iQ`r(®k¼t¨ÌndVFepa]£5Q`P�´�»�l�¼t¨�q'Toa9Toqd_be�a-³

Î�_bSJcdP]V�� J[DgbS2QJP]_ba9^dU�e{c`ep_�r`S+e�VF�2VuP�Tog(r`cdU#VuP]_b\uTog(�oVuP�r`VFg «�_ba9^`QJc`a�T2r`qª«�_ba9^ªq'Toa9T#P]VFqd_be�a9PR_ ¨ndc`a�_bQJr`e

0.0 20.0 40.0 60.0 80.0block size

3.0

3.5

4.0

4.5

5.0

time

(s)

1x162x84x48x216x1

Î�_bSJcdP]V�J |}Tha9P]_bv�UXc`gba�_xf`gbwBVFg�TLf`e�V�a�_xU#V�QJrÂa9^`V¬�Àr2apVFg{O�TLP�ToS2QJr¤«�_ba9^�Wu�¤fdP]QJ\FVFepe�QJP]e�_�rqd_��(VuP]VurLaNSJPR_iq�e-^dT2fdVFe,T2r`q�qd_��(VuP]Vur2aDe�·dcdT2P]V�n`gbQ`\¯�ªe�_b­FVFeu¥

qdQ2ape�QJr 0'SJcdPRVFe�W2W2l2Wu�,TLr`q¬Wu�DTogbgda9^`Va�_xU#_xr`S2e�«�_ba9^`QJc`a�P]VFqd_be�a9PR_�ndc`ap_iQ`r`e¹Q`n`a9To_xr`VFq#«�_ba9^#Togbgqd_��(VuP]VurLa"�`_�PRa9cdTogdSJP]_bqdeT2r`q�n`giQ`\F��ep_i­FVFe\FQJU#n`_xrdToa�_bQJr`eH£5Q`P(e�VF�LVuP�Tog'U�Toa-P]_bv�e�_b­FVFeu¥`´�^`VNndVFepa

ôöíºî�ôxõ

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

0.0 20.0 40.0 60.0 80.0block size

2.0

3.0

4.0

5.0

6.0

7.0

time

(s)

1x162x84x48x216x1

Î�_bSJcdP]V#� d½�� qdVF\FQJU�fµQ2e�_ba�_bQJrÅVFg�TLf`e�V�a�_xU#V�QJrªa9^`V��Àr2a�VFg�O�T2P�ToS2Q`r�«�_ba9^ÂWu�#fdP]Q`\FVFe�e�QJP]e~_�rqd_��(VuP]VurLaNSJPR_iq�e-^dT2fdVFe,T2r`q�qd_��(VuP]Vur2aDe�·dcdT2P]V�n`gbQ`\¯�ªe�_b­FVFeu¥

0.0 20.0 40.0 60.0 80.0block size

0.0

100.0

200.0

300.0

time

(s)

1x162x84x48x216x1

Î�_bSJcdP]V�Wu� '´ºPR_�TLr`SJc`gxT2PN»`QLgi�LVXVFgxT2f`epVXap_�U�V�QJrÅa9^`V���rLa�VFgòO�TLP�ToS2QJr+«�_ba9^BWu��fdPRQJ\FVFe�epQJP]eD_�rqd_��(VuP]VurLaNSJPR_iq�e-^dT2fdVFe,T2r`q�qd_��(VuP]Vur2aDe�·dcdT2P]V�n`gbQ`\¯�ªe�_b­FVFeu¥

î(î�éNM'O 0 )�)

W�J � ���� ���N( ������� ���# ����($���( #�� ��*@( �����@�$��*

T2r`q°«QJP]epa�\uToe�VFeDQo£"a9^`V#e9T2U�V�\FQJU�fdc`a-Toa�_bQJr`eD«�_ba9^¤a9^`VXP]VFqd_be�a-P]_xndc`a�_bQJr`eXT2PRVXP]VufdP]VFepVur2a�VFqToe~\ucdP]�2VFe�QJrÅa9^`V,e9T2U�V,SJP�T2fd^�¥

200.0 400.0 600.0 800.0 1000.0 1200.0matrix size

0.0

1.0

2.0

3.0

4.0

5.0

time

(s)

best case with redistributionworst case with redistributionwithout redistribution

Î�_bSJcdP]VªW2W� `|�Toa9P]_bv�|�c`gba�_xf`gbw�VFgxT2f`e�V�a�_xU#V�Q`rªa9^`V���rLa�VFgºO�T2PpToS2QJr+«�_ba9^°Wu��fdPRQJ\FVFe�epQJP]e£kQJPTogbg(S`P]_bq�e9^dT2fdVFeDT2r`q�n`gbQJ\F�ªe�_b­FVFe�«�_ba9^`QJc`a~q'Toa9T#P]VFqd_be�a9P]_xndc`a�_bQJr`eD\FQJU�fdTLP]VFq�«�_ba9^�P]VFqd_be�a9PR_ ¨ndc`a�_bQJrÅa�Q�a9^`V�nµVFe�aD\FQJr 0'SJcdP�Toap_iQ`r�¥

´�^`V�P]VFe9c`gba�e,T2P]V#VFvJa-P]VuU#VFgbw�S2Q`Q`q²ndVF\uTLc`e�V#a9^`VuP]V#_be,QJr`gbw�e�gb_bSJ^La,qd_ �(VuP]Vur`\FVFe#ndVFa]«VFVura9^`V�ndVFepaXT2r`qB«QJP]e�aD\uToe�Vª«�_ba9^°a9^`VªP]VFqd_be�a9P]_xndc`a�_bQJr`eu¥tÍ�Vur`\FVªa9^`_be�fdVuPÀ£kQJP�U�TLr`\FV�e�a9Tmw`e,_�ra9^`V,e-T2U#V�P�T2r`SLVN\FgbQ2e�V,apQXa-^`V,QJf`a�_xU�Togk¥�>,rªa9^`V,\FQ`r2a9PpT2P]w2lJT+ndToq�qd_be�a9P]_xndc`a�_bQJrÅ\¯^`Q2_b\FV�\uT2rgbVuToqªa�Q�qdQ`cdn`giVDa9^`VDVFvJVF\uc`ap_iQ`r�a�_xU#V{£5QJPt½��§T2r`qªUXc`gba�_xf`gbw�n2w 0'�2V�a-^`VNa�_xU#V~Qo£(a9^`V�e�Q2gb�2VFem¥´(Q�\FQJr`\Fgxc`qdV2ld«VZfdP]VFepVur2at_�rªÎ�_bSJcdPRVXW�:�a9^`V�a�VFepa�e"\FQJPpP]VFe9fdQ`r`qd_�r`S+a�Q�a9^`V,TogbS2QJPR_ia-^dU Qo£

Î�_bSJcdP]V#�#«N^`VuP]V,a-^`V�rJcdU#VuPR_i\uThgò�oVuPpr`VFgie,T2P]V,\F^dTo_xr`VFq�_xrÅa9^`V,e9T2U#V�fdTLP�TogbgbVFgòfdP]QLSJP�T2U�¥´�^`V¬P]VFe9c`gba�e�fdPRQE�LVFe,a9^dToaD_�rBTogbg�\uToe�VFeml�_ia,_be#ndVFa�a�VuPDapQ�c`e�V�a9^`V�P]VFqd_be�a9PR_�ndc`ap_iQ`r`eXa-^dT2r

a�Q¤e�a9Tmw�«~_ia-^°a9^`V�e9TLU#V�QJr`V#£�P]QJU�a-^`V¬ndVFS2_xrdr`_xr`S�a�Q�a-^`V�Vur`q(¥�|²Q`P]VFQo�2VuPFl%a9^`V 0�P]e�a,a]«Q\ucdP]�LVFe�Qo£,a9^`V�a�VFe�a�e�T2PRV�a9^`V�«QJP]e�a�fdQLe�e�_xn`gbV�\uToe�V�«D^`_igbV�c`e�_xr`S©P]VFqd_be�a-P]_xndc`a�_bQJr�ndVF\uT2c`e�V0�P]epa�«VXToe�e-cdU#V,a9^dToaNr`Q`r`V�Qo£�a9^`V�T2P�PpTuw+_ieD_xr¤a9^`V,S2Q`QJq¤qd_iepa9P]_xndc`a�_bQJr°Toa�a9^`V�nµVFS2_xrdr`_xr`S'lT2r`q°e�VF\FQ`r`q(l'«VXPRVFqd_iepa9P]_xndc`a�V#a9^`VXTLP�P�TmwJeDToa�a9^`V#Vur`q�_xr¤QJP]qdVuP~a�QªS2Q¬ndTo\F�ªa�Qªa9^`V�_xr`_ba�_xTogq'Toa-T©P]VufdTLP]a�_ba�_bQJr�¥´�^`VFe�V²�Âe�a�Vuf`e�T2P]V°r`Q2a�U�T2r`q'Toa�Q`P]w©_xr�V-±¬\F_bVurLa¡fdP]QLSJP�T2U�U�_�r`S§®ö£kQJP_xr`e�a9TLr`\FV�_ £Da9^`V¬_xr`_ba�_xTog�qd_be�a-P]_xndc`a�_bQJr�_be#«VFgig\F^`Q2e�VurBT2r`qB_ £Da9^`VuP]Vª_ie�r`Q�Toe�e-cdU�f`a�_bQJrÂQJra9^`Vªq'Toa9Tªqd_be�a-P]_xndc`a�_bQJrBq'cdP]_xr`S°a9^`V¬P]VFe�a,Qh£{a-^`V¬fdP]Q2SJPpT2U�³F¥�´�^`c`e�a9^`V�S`To_xr�QJn`a-To_xr`VFq�£kQJP

ôöíºî�ôxõ

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

0.0 500.0 1000.0 1500.0 2000.0matrix size

0.0

5.0

10.0

15.0

time

(s)

best case with redistributionworst case with redistributionwithout redistribution

Î�_bSJcdP]V¤Wu� (½���qdVF\FQJU�fµQ2e�_ba�_bQJrÂVFgxT2f`e�Vªa�_xU#V�QJr¤a9^`V¬�Àr2apVFg¹O�T2P�ThS2QJr�«~_ia-^�Wu��fdP]Q`\FVFe�e�QJPRe£kQJP,Togbg"S`P]_bq�e9^dT2fµVFeXT2r`q©n`gbQ`\¯�¤e�_b­FVFe#«�_ba9^`QJc`a,q'Tha9T�P]VFqd_be�a9P]_xndc`a�_bQJr`e#\FQJU+fdT2P]VFq�«�_ba9^ÂP]V-¨qd_be�a9PR_�ndc`ap_iQ`r�apQ�a9^`V�ndVFepaN\FQJr 0'S`cdP�Toa�_bQJr�¥

î(î�éNM'O 0 )�)

�2� � ���� ���N( ������� ���# ����($���( #�� ��*@( �����@�$��*

200.0 400.0 600.0 800.0 1000.0 1200.0matrix size

0.0

100.0

200.0

300.0

time

(s)

best case with redistributionworst case with redistributionwithout redistribution

Î�_bSJcdP]V�Wm� d´ºP]_xT2r`S`c`g�TLPN»`Q2gb�2V,VFgxT2f`epV�a�_xU#V,QJrªa-^`VZ�Àr2a�VFg�O�T2P�ToS2Q`r�«�_ba9^°Wm�XfdPRQJ\FVFe�epQJP]e£kQJPTogbg(S`P]_bq�e9^dT2fdVFeDT2r`q�n`gbQJ\F�ªe�_b­FVFe�«�_ba9^`QJc`a~q'Toa9T#P]VFqd_be�a9P]_xndc`a�_bQJr`eD\FQJU�fdTLP]VFq�«�_ba9^�P]VFqd_be�a9PR_ ¨ndc`a�_bQJrÅa�Q�a9^`V�nµVFe�aD\FQJr 0'SJcdP�Toap_iQ`r�¥

ôöíºî�ôxõ

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

200.0 400.0 600.0 800.0 1000.0 1200.0matrix size

0.0

10.0

20.0

30.0

time

(s)

best case with redistributionworst case with redistributionwithout redistributionoptimized case

Î�_bSJcdP]V�W�: "¼QJU�fdTLP]_be�QJr�Qh£�a9^`V�e9c`\F\FVFepe�_bQJr�Qo£�a9^`V²�Br`cdU#VuP]_b\uTog��oVuP�r`VFgbe�QJr a9^`V²�Àr2apVFgO�T2PpToS2QJr«�_ba9^�Wu��fdPRQJ\FVFe�epQJP]e#£kQJP#Togbg�SJP]_bq�e9^dT2fdVFeªT2r`q n`gbQJ\F�Be�_b­FVFe¬«~_ia-^ T2r`q�«�_ba9^`Q`c`aq'Toa-T�P]VFqd_be�a9P]_xndc`a�_bQJr`e�TLr`q�_�rÅa9^`V,QJf`a�_xU#_b­FVFq°\uToe�V2¥

î(î�éNM'O 0 )�)

�2� � ���� ���N( ������� ���# ����($���( #�� ��*@( �����@�$��*

a9^`_be\FQJU�fdc`a-Toa�_bQJr#_be¹n`_bS2SLVuP�£kQJP�a9^`_betQJf`a�_xU#_b­FVFq¬\uThe�V2l2«N^`_b\F^�_beP]VufdP]VFe�VurLa�VFqª_�r�a9^`V�gbQo«¹VFepa\ucdP]�LV�Qo£ºÎ�_bSJcdPRV¬W�:'¥

����� � & & � � -�� ��� � ��'�').� $ ) "%�'&*)Î%QJP(a-^`V{¼~P�Tmw�´��oÁ�lFÎ�_bSJcdP]V,W�G~e9^`Qo«�e�a9^dToa�a9^`V�ndVFe�a"fµVuPÀ£kQJP�U�T2r`\FVQo£'a-^`V{U+Toa9P]_bv�U#c`gba�_xf`giw_be�QJn`a-To_xr`VFq¬«�_ba9^ªT�: " :#�`_�PRa9cdTog'S`P]_bq�«�_ba9^�a-^`VNn`_bS2S2VFe�a�n`gbQJ\F�+e�_b­FV�®k� :,£kQJP�QJcdPfdP]Q`n`giVuU+³Toe~QJrªa9^`VXO�TLP�ToS2QJr�¥

0.0 20.0 40.0 60.0 80.0block size

1.0

2.0

3.0

4.0

time

(s)

1x162x84x48x216x1

Î�_bSJcdP]V�W�G "|�Toa9PR_iv�|�c`gba�_xf`gbw�VFg�TLf`e�V�a�_xU#V°QJr a9^`V�¼�P�Tmw ´��oÁ «�_ba9^ Wu�©fdPRQJ\FVFe�epQJP]eª_�rqd_��(VuP]VurLaNSJPR_iq�e-^dT2fdVFe,T2r`q�qd_��(VuP]Vur2aDe�·dcdT2P]V�n`gbQ`\¯�ªe�_b­FVFeu¥

Î'QJPa-^`VX½��¶qdVF\FQJU�fdQLe�_ba�_bQJr�l�a9^`VXnµVFe�a,P]VFe9c`gbaN_be,QJn`a-To_xr`VFq�«�_ba9^°n`gbQJ\F�LeNQo£"ep_i­FV�Wu��QJrT�W "©Wu���`_�PRa9cdTog(SJPR_iq¤Qo£òfdP]Q`\FVFe�e�Q`P]eu¥Î'QJPta9^`V�e�Q2gb�2V,Qo£¹Wm�2��:�P]_bSJ^LaN^dT2r`q�ep_iqdVFemlda9^`V�ndVFepaZP]VFe-c`gia~_ieDQJn`a9Th_�rª«~_ia-^�a9^`V�n`_bS2S2VFepa

n`gbQ`\¯�ªe�_b­FV�®k��:J³tQJrÅa9^`V�Wu� "©W,�`_xP]a9cdTog SJP]_bq�Qo£"fdP]QJ\FVFepe�QJP]eu¥[NSJTh_�r�lµa9^`V,SJP]_bq�e9^dT2fµVXT2r`q�n`gbQ`\¯�+e�_b­FVFe,T2P]V,�2VuP]w+qd_ �(VuP]VurLa~£5QJPta9^`VFepV���P]QJc`a�_xr`VFeX®5e�VFV

´�T2n`gbV�Wu³Fl e�Q°a9^`V¡PRVFqd_iepa9P]_xndc`a�_bQJr ThgiSLQJP]_ba9^dU \uT2rÂgiVuThq�a�Q²T2rÂ_xU�fdQ`P]a9T2rLa#SJTo_xr�_xrBVFgxT2f`e�Va�_xU#V2¥[NeH£5QJPHa9^`V,O�T2P�ToS2Q`r�lu«VNP�TLr#a9^`VDa�VFe�a�et\FQJP�P]VFe9fµQJr`qd_xr`S#a�Q�a9^`VDTogbS2QJP]_ba9^dU#eQo£(Î�_bSJcdP]VD�

£kQJPa9^`V#��r`cdU#VuP]_b\uTog��oVuP�r`VFgbeN«~_ia-^�Wu�#fdP]Q`\FVFe�e�QJPReu¥d¸�V#f`giQLa�a�VFq�Q`r 0'SJcdP]VFe�W�Jµl(Wu��T2r`q��L�Togbg�a9^`V�a�_xU#_xr`S2e,«�_ba9^`QJc`a,P]VFqd_be�a9PR_�ndc`ap_iQ`r`e�QJn`a9Th_�r`VFq°«�_ba9^°Togbg�qd_��øVuPRVur2aD�J_xP]a-cdTogºS`P]_bqde�T2r`q

ôöíºî�ôxõ

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

0.0 20.0 40.0 60.0 80.0block size

1.0

2.0

3.0

4.0

5.0

time

(s)

1x162x84x48x216x1

Î�_bSJcdP]V°Wu� �½���qdVF\FQJU+fdQ2e�_ba�_bQJrBVFgxT2f`epV�a�_xU#V¬Q`r°a9^`V�¼�P�Tmw�´��oÁ «�_ba9^�Wu�¤fdP]QJ\FVFepe�QJP]e�_�rqd_��(VuP]VurLaNSJPR_iq�e-^dT2fdVFe,T2r`q�qd_��(VuP]Vur2aDe�·dcdT2P]V�n`gbQ`\¯�ªe�_b­FVFeu¥

0.0 20.0 40.0 60.0 80.0block size

0.0

50.0

100.0

150.0

200.0

time

(s)

1x162x84x48x216x1

Î�_bSJcdP]V�Wu� "´ºPR_�TLr`SJc`gxT2P�»`Q2gb�2V°VFgxT2f`e�V°a�_xU#V°QJr�a-^`V}¼~P�Tmw ´��hÁ «�_ba9^ Wu�©fdPRQJ\FVFe�epQJP]eª_�rqd_��(VuP]VurLaNSJPR_iq�e-^dT2fdVFe,T2r`q�qd_��(VuP]Vur2aDe�·dcdT2P]V�n`gbQ`\¯�ªe�_b­FVFeu¥

î(î�éNM'O 0 )�)

��: � ���� ���N( ������� ���# ����($���( #�� ��*@( �����@�$��*

n`gbQ`\¯�°ep_i­FVFe#\FQJU#n`_xrdToa�_bQJr`e�£kQJPDe�VF�2VuPpTog¹U�Tha9P]_bv°e�_b­FVFeu¥º´�^`V�ndVFepaXT2r`q�«QJPRe�a,\uToe�VFe�Qo£�a9^`Ve9TLU#V#\FQJU�fdc`a9Toap_iQ`r`e,«�_ba9^¤a9^`V�P]VFqd_be�a9P]_xndc`a�_bQJr`e#T2P]V�P]VufdP]VFe�VurLa�VFq�Toe,\ucdP]�LVFeNQJr¤a9^`V#e9TLU#VSJPpT2fd^�¥

0.0 500.0 1000.0 1500.0 2000.0matrix size

0.0

5.0

10.0

15.0

time

(s)

best case with redistributionworst case with redistributionwithout redistribution

Î�_bSJcdP]VªW�J d|�Toa-P]_bv¬|�c`gba�_xf`gbw�VFgxT2f`e�V�a�_xU#V�QJrªa-^`VX¼�P�Tmw¬´~�oÁ «�_ba9^�Wm��fdP]Q`\FVFe�e�QJP]e{£5QJP~TogbgSJPR_iq°e9^dTLfdVFe,T2r`q²n`gbQ`\¯�Åe�_b­FVFe,«�_ba9^`QJc`a�q'Toa-T¬P]VFqd_be�a-P]_xndc`a�_bQJr`e,\FQJU�fdT2PRVFq�«�_ba9^¤P]VFqd_be�a9P]_xndcL¨a�_bQJrÅa�Q#a9^`VXnµVFe�a�\FQJr 0'S`cdP�Toa�_bQJr�¥

Í�VuP]V�ToS`To_xr�l�a9^`V�c`e�V#Qo£{P]VFqd_be�a-P]_xndc`a�_bQJr©ThgigbQo« a�Q�^dTm�2V�T2r¤VFg�TLf`e�V�a�_xU#V�TogxU#Q2epaN_xr`qdV-¨fµVur`qdVur2a�Qh£Na9^`V�_xr`_ba�_xTog�qd_be�a9PR_�ndc`ap_iQ`r T2r`qB�2VuP]w°\FgbQ2e�Vª£�P]QJU a9^`V¬Q`f`a�_xU�Togk¥�¦¹The�_b\uTogbgiw2lHa9^`VP]VFqd_be�a-P]_xndc`a�_bQJr#_be"c`e�VFgbVFe�e�QJr`gbw�«D^`Vur�a-^`V¹U�Toa-P]_bv,_ie"Thg�PRVuToqdw�_xr�a9^`V"Q`f`a�_xU�Tog`\FQJr 0'SJcdP�Toap_iQ`r�¥´(Q�\FQJr`\Fgxc`qdV2l�«V�fdP]VFe�VurLa�_xrÂÎ�_bSJcdP]V¬�dW#a-^`V�a�VFe�a�e,\FQ`P�P]VFe9fµQJr`qd_xr`S²apQ�a9^`V¬TogbS2QJPR_ia-^dU

_xr°Î�_bSJcdP]V��ª«N^`VuPRVXa-^`V�r`cdU#VuP]_b\uTog"�hVuP�r`VFgbe�T2P]V#\¯^dTh_�r`VFqÂ_�r}a9^`V�e9TLU#V�fdT2P�TogbgbVFg"fdP]Q2SJPpT2U�¥Í~VuP]V�ToSJTh_�r�lHa9^`VuP]Vª_ieªr`Q�e�_xr`S2gbV�q'Toa-T�qd_be�a-P]_xndc`a�_bQJr�a9^dTha�fµVuP�U#_ba�a�Q�P]VuTh\¯^Ba9^`VªQJf`a�_xU�ThgfµVuPÀ£5Q`P�U�T2r`\FV#QJn`a9Th_�r`VFq�nLw�a9^`VXPRVFqd_iepa9P]_xndc`a�_bQJr�nµVFaÌ«VFVur¤VuTo\F^�QJfµVuP�Toa�_bQJr�¥�´�^`V�P]VFe9c`gba�eDQo£a9^`V~QJf`a�_xU#_b­FVFq¬fdP]Q2S`P�T2U�U#_xr`S,«�_ba9^`Q`c`a�_�r`_ba�_xTog�T2r`q 0�rdTog'PRVFqd_iepa9P]_xndc`a�_bQJr`e�T2P]V�Togbe�Q�f`gbQ2a�apVFq(¥

ôöíºî�ôxõ

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

0.0 500.0 1000.0 1500.0 2000.0matrix size

0.0

5.0

10.0

15.0

time

(s)

best case with redistributionworst case with redistributionwithout redistribution

Î�_bSJcdP]V¤Wu� �½���qdVF\FQJU�fdQLe�_ba�_bQJr�VFgxT2f`epV�a�_xU#V�QJr¤a9^`Vª¼�P�Tmw�´��oÁ�«�_ba9^�Wm�¬fdP]Q`\FVFe�e�Q`P]e,£kQJPTogbg(S`P]_bq�e9^dT2fdVFeDT2r`q�n`gbQJ\F�ªe�_b­FVFe�«�_ba9^`QJc`a~q'Toa9T#P]VFqd_be�a9P]_xndc`a�_bQJr`eD\FQJU�fdTLP]VFq�«�_ba9^�P]VFqd_be�a9PR_ ¨ndc`a�_bQJrÅa�Q�a9^`V�nµVFe�aD\FQJr 0'SJcdP�Toap_iQ`r�¥

O�T2P�ToSLQJr ´��oÁ¾DVuP�r`VFg ¦¦» ¦ ��» ¦¹¦» ¦ ��»|°| ��: : " : � : : "�:½�� J W "�Wu� Wm� W "BWu�´�» ��: Wu��"�W � : Wu� "�W

´�T2n`gbVXW 2¦"VFepa"¦"gbQ`\¯�+»`_i­FV#®k¦¦¹»d³tT2r`q¬¦tVFe�a ��P]_bq¬»d^dT2fdV�®k¦ ��»d³ £kQJP�a9^`V~qd_ �(VuP]VurLa"fdT2P�ThgigbVFg\FQJU+fdc`a�VuP]e�Qo£�QJcdPapVFe�a«N^`_bgiV�P�cdrdr`_xr`Sªa9^`V,|�Toa9P]_bvª|�c`giap_�f`gbw�®=|°|}³ lJ½�� qdVF\FQJU+fdQ2e�_ba�_bQJr®k½ �N³�TLr`q²´�P]_xT2r`SJc`gxT2PD»`Q2gb�2VFeX®k´�»d³{\FQJU�fdc`a9Toap_iQ`r��oVuP�r`VFgbeu¥

î(î�éNM'O 0 )�)

�2� � ���� ���N( ������� ���# ����($���( #�� ��*@( �����@�$��*

0.0 500.0 1000.0 1500.0 2000.0matrix size

0.0

100.0

200.0

300.0

time

(s)

best case with redistributionworst case with redistributionwithout redistribution

Î�_bSJcdP]V#�2� d´�P]_xT2r`SJc`gxT2PD»`Q2gb�2V�VFgxT2f`e�V�a�_xU#V,QJrªa9^`V#¼�P�Tmw¬´��hÁ¿«�_ba9^ÂWu�#fdP]Q`\FVFe�e�QJP]e{£5QJP~TogbgSJPR_iq°e9^dTLfdVFe,T2r`q²n`gbQ`\¯�Åe�_b­FVFe,«�_ba9^`QJc`a�q'Toa-T¬P]VFqd_be�a-P]_xndc`a�_bQJr`e,\FQJU�fdT2PRVFq�«�_ba9^¤P]VFqd_be�a9P]_xndcL¨a�_bQJrÅa�Q#a9^`VXnµVFe�a�\FQJr 0'S`cdP�Toa�_bQJr�¥

ôöíºî�ôxõ

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

200.0 400.0 600.0 800.0 1000.0 1200.0matrix size

0.0

5.0

10.0

15.0

time

(s)

best case with redistributionworst case with redistributionwithout redistributionoptimized case

Î�_bSJcdP]V°�dW� "¼{QJU�fdT2P]_be�Q`r�Qo£,a9^`V�e-c`\F\FVFe�e�_bQJr�Qo£,a9^`V²�BrJcdU�VuP]_b\uTogZ�hVuP�r`VFgbe¬QJr a9^`V²¼~P�Tmw´��hÁ�«~_ia-^�Wu��fdP]Q`\FVFe�e�QJPReD£5QJP,ThgigtSJP]_bq�e9^dTLfdVFeXTLr`q©n`gbQJ\F�¤e�_b­FVFe�«�_ba9^�TLr`q�«�_ba9^`QJc`a�q'Toa9TP]VFqd_be�a-P]_xndc`a�_bQJr`e�T2r`q�_xrÅa9^`V,QJf`a�_xU#_b­FVFq�\uThe�V2¥

î(î�éNM'O 0 )�)

��J � ���� ���N( ������� ���# ����($���( #�� ��*@( �����@�$��*

� ���.����-*�$��p�,

�Àr a-^`V�S2Vur`VuP�Tog�\uThe�V2l"a-^`V�P]VFqd_be�a9P]_xndc`a�_bQJr�Qo£�q'Toa-T°_be¡c`e�V-£�c`g�a�QÂ_�U+fdP]Qo�2V�a9^`V�V-±ª\F_iVur`\FwQo£XfdTLP�TogbgbVFg�gb_xr`VuT2P�TogbS2VundP�T�P]Q`c`a�_xr`VFeu¥¹¦c`a�a�QBVur`e9cdP]V�TÂSJTo_xr�QJr�a-^`V²VFgxT2f`epV²ap_�U�V2la9^`VP]VFqd_be�a-P]_xndc`a�_bQJr¤Qo£òq'Tha9T�^dToe�apQ�nµV��2VuPRw¬V-±¬\F_bVurLau¥

>,cdP\FQJU�f`gbVFv`_ia]w�T2rdTogbw`e�_be�e9^`Qo«�ea9^dToaa9^`VDe�\uT2rdr`_xr`Sª_ieDr`VFS2gb_iSL_�n`gbV�£kQJPT2P�P�Tmw`e"Qo£º\FQ`U�¨U#Q`r�e�_b­FV2¥N´�^`Vur «�_ba9^�QJcdPªQJf`ap_�U�_i­uTha�_bQJr`eul�_ba¤_ie¤e9cL±¬\F_bVurLa�a�Q �`r`Qo« a9^`V°qd_iepa9P]_xndc`a�_bQJrfdT2PpT2U#VFa�VuP]e#QJr`gbw�Toa�P�cdrLa�_xU#V�T2r`q�a9^`VuP]V�_be¬r`QBU#QJP]Vª\FQJr`e�a9PpTo_xr2a�e�T2nµQJc`a�fdP]Qo�J_bqd_xr`S�Togbgqd_be�a9PR_�ndc`ap_iQ`r²fdTLP�T2U#VFa�VuPReNToa\FQJU�f`_bgbV-¨�a�_xU#V2¥´(VFe�apeX«VuP]V�P�cdrÂQJrBT²¼~P�Tmw}´~�oÁ T2r`q�QJrBT2rB��rLa�VFg�O�T2P�ToSLQJr�¥�[Ngbg�a-^`V¬VFvdfdVuP]_xU#VurLa�e

\FQJPpP]QJndQ`P�Toa�Vt�2VuP]w�«VFgbgdQJcdP�VFvdfdVF\Fa9Tha�_bQJr`euloa9^`V\FQJU�fdc`a-Toa�_bQJr#Qo£'a-^`V¹q'Tha9TDe�VFa�e�_be¹r`VFS2gb_bS2_xn`gbV\FQJU+fdT2P]VFq�apQXa-^`V,\FQJU�U#cdr`_b\uToa�_bQJr�TLr`q²fdTo\F�2_xr`S'l%T2r`q�a9^`V�SLgiQ`ndTogºP]VFqd_be�a9PR_�ndc`ap_iQ`r�P]QJc`ap_�r`VVFv`VF\uc`a�_bQJr¤a�_xU#V,_beN�LVuP]w¬SLQJQ`q(¥´�^`VFe�VXPRVFe9c`gba�e,e9^`Qo«§a9^dToaDa9^`V#P]VFqd_be�a9P]_xndc`a�_bQJr¤Qo£"q'Toa-T#«�_ba9^¤QJcdPDTogbS2QJP]_ba9^dU#eD\uT2rÅV-±�¨

\F_bVurLa�gbw²fdVuPÌ£5QJPpU�_xr²fdPpTo\Fa�_b\FV�S2_b�`_xr`S¡cdfÅa�QLJ2����SJTh_�r¤T2r`q²The�e9cdP]_xr`S�QJrÅa9^`V�Tm�2VuP�ToSLV,a9^dToaa9^`V¤\FQJU�fdc`a9Tha�_bQJrBa�_xU#V�epa9TmwJe#�2VuP]w�\FgbQ2epV�a�Q°a9^`V¤QJf`a�_xU�Tog�VF�LVur�«�_ba9^�ndToq�_xr`_iap_�Thg,q'Toa9Tqd_be�a9PR_�ndc`ap_iQ`r�\F^`Q2_b\FVFe�®ö£5Q`P�ndQ2a-^�n`giQ`\F�ªe�_b­FVFeZT2r`q�S`P]_bq�e9^dT2fµVu³F¥Î�P]QJU QJcdP�P]VFe-c`giapeulNT 0�P]e�a�_xU�fdP]Qo�2VuU�Vur2a�e9^`Q`c`giq ndV�a9^`V°_xrLa�VFSJP�Toap_iQ`r�Qo£�a9^`VFe�VBP]V-¨

qd_be�a9PR_�ndc`ap_iQ`r P]QJc`ap_�r`VFe#_xrBa9^`V�gb_xr`VuT2P�ThgiSLVundP�T°�oVuP�r`VFga9^`VuU#e�VFgb�2VFe�®knµV-£5Q`P]V¡TLr`q Tm£ka�VuP.a9^`V\FQJU+fdc`a9Toa�_bQJrd³tToe�e�Q`QJr�Toe�a9^`VFw�fdP]Qo�`_bqdVNT,SJTh_�r�lL_k¥ýV2¥o«N^`Vur+a9^`VNU�Toa-P]_bv�e�_b­FVD_ien`_bS2S2VuP�a-^dT2rT#SL_i�LVur�fdQL_�rLaDqdVufdVur`qd_xr`S�Qo£�a9^`V�f`gxToa]£kQJP�U�¥|²Q`P]VFQo�2VuPFl�QJcdPªP]VFe9c`gba�e�T2P]V�Vur`\FQ`cdP�ToS2_xr`S£5QJP+a9^`VÅ£ P]VF·dc`VurLa¡c`e�V¤Qo£�a9^`_be�P]VFqd_be�a9P]_xndcL¨

a�_bQJrÂgb_xndP�T2P]w�P]QJc`a�_xr`VFe#_xr�VFvdf`gb_b\F_ia�fdT2P�ThgigbVFgNfdPRQ2SJP�T2U+U#_xr`S'l(U�Toe�apVuP Aoe�gxTm�2V�e�\F^`VuU#VFe#QJP._�r\FQJU+f`_igbVFq°\FQ`qdVFeNS2Vur`VuPpToa�VFq�nLw¡ÍDO�Î�\FQJU+f`_igbVuP]em¥

>,cdP"TogbS2QJPR_ia-^dU#e"T2P]V_xU�f`gbVuU#VurLa�VFq�«�_ba9^`_xr#a9^`VN»d¼{[Z½�[,O%[N¼�¾�gb_xndP�T2PRw¬T2r`q�T2P]VNTh\F\FVFe]¨e�_xn`gbVt£5Q`P�a�VFe�a(T2r`q#c`e�V�QJrDa9^`V�� . e�_ba�V��������� ��������������������� �!�"� #�$�%���%�&%�$�'� ��(�&)*��+,�-����.�� ¥

� �0/u��&(�{.���t$

8öW$< ��¥[DSJP�Tm«�Togkl"[#¥�»dc`e�e9U+T2r�l�T2r`q§�'¥�»dThgiap­2¥ ¼QJU�f`_bgbVuP�T2r`q P�cdr2ap_�U�V�e9cdfdfdQ`P]a�£kQJPe�a9P�c`\Fa-cdP]VFq�T2r`q�n`gbQJ\F�ªe�a9Ppc`\Fa9cdP]VFq²TLfdf`gi_b\uToap_iQ`r`eu¥¹fdThS2VFe7G2��J�1 G�J2�dl%Wu�2�2�µ¥

8Ë��<X»�¥¹O�¥t[ZU+T2P�Toe�_xr`SJ^`V°T2r`q�|�¥"»�¥¹½ T2U�¥¿¼{QJU�U#cdr`_b\uToa�_bQJr�QJf`a�_xU#_b­uToa�_bQJr�T2r`q�\FQJqdVS2Vur`VuP�Toap_iQ`r¤£5Q`P,qd_be�a9P]_xndc`a�VFq U�VuU#QJP]w�U�To\F^`_xr`VFeu¥°��r6���� ����( ����$�B�� � ( �(�+( �� ����������+*@� �+��#���%�� �@����# � � ��� �����������'lL[Ngxndc`·dc`VuP]· c`V2l �~|�l2�Jcdr`V#Wu�2�2�µ¥o[N¼| »d���D¨O"½�[ ��¥

ôöíºî�ôxõ

����������� ����������������������������! "�$#��&%'�($��)�*+��,�� ���8Ë��<X»�¥�¼~^dToa�a�VuP=y]VFV2ld�'¥�z,¥��,_bgxndVuP]aml�Î�¥d�'¥ ��¥�½�QJr`S'l�z,¥�»`\¯^dP]VF_xnµVuPFl(T2r`q²»�¥�Í�¥�´ Vur`S'¥ �.V-¨r`VuP�Toa�_xr`S#gbQJ\uThg�Toqdq'P]VFepe�VFe�T2r`q¬\FQ`U�U#cdr`_i\uTha�_bQJr�epVFa�et£5QJPHq'Toa9Tm¨ÌfdTLP�TogbgbVFg�fdP]Q2S`P�T2U#eu¥���r� � � ��%���*� 9�� �"($�,���� � ����%7���# � ( ������,�$���*��� � ( � �,����� � ( �(�+( �� ����`lJ»dT2r+qd_bVFS2Q'ld¼[#l|�Tuw¤Wu�2�2�d¥`[N¼{| »d����O"½�[ ��¥

8 :�<��'¥"¼�^`QL_5lH�'¥ý�'¥�Á,Q`r`SJT2P�P�TµlºT2r`q�Á�¥ý¸ ¥�¸©Togx�oVuP ¥ ´~^`V�Á,VFe�_bSJrBQo£�»`\uTogxT2n`gbV²»`Qh£5a]«�T2P]V½�_xndP�T2P]_bVFeª£kQJP+Á,_iepa9P]_xndc`a�VFq�|°VuU#QJP]w�¼QJr`\ucdP�PRVur2a�¼QJU�fdc`apVuP]eu¥§��r �'¥ý�'¥tÁ,QJr`SJTLP�P�TT2r`q�¦N¥�´ QJcdP�T2r`\F^`VuT2c�l�VFqd_ba�QJP]eml �"�����( ��& ����%B���# � ����� % � � ( �N��( ���,����� � ���,����� $"����� � *+��,��Jl'fdThS2VFeN� 1�W�Gd¥��HgiepVF�J_bVuPFl�Wu�2�2�µ¥

8 G�<XO�¥ò¼~P]QJQ`�2e¬TLr`q z,¥"Í�¥òO�VuPpP]Q2a�au¥B½(T2r`SJcdThS2V�\FQJr`e�a9Ppc`\Fa�£kQJP#q'Tha9T°fdT2P]a�_ba�_bQJr`_xr`S�T2r`qqd_be�a9P]_xndc`a�_bQJr�¥d´ VF\¯^dr`_b\uTogdP]VufdQ`P]aulFqdVuf`a�Qo£'¼,¥ »�¥xlF´�^`V��,c`VFVur �ýe �Dr`_b�,Qo£'¦"VFgË£�Toe�aulo¦tVFgË£ Toepa¦´���� � ��l���Q`P]a9^`VuP�r��ÀP]VFgxT2r`q(lºWm�2��:'¥

8Ë��<�Î�¥dÁ,VFe9fdP]VF­#T2r`q²¦D¥�´(Q`cdP�T2r`\F^`VuT2c�¥�½ >,¼�¼�» �½�Qo« >D�2VuPp^`VuToq�¼QJU�U#cdr`_b\uToa�_bQJr¤T2r`q¼�U�fdc`a9Tha�_bQJr©»dcdndPRQJc`a�_xr`VFeu¥ � *@�*+( ���������( ������� ���� � *+���( � ��%��� %Fl�Wu� �L�2� 1 ��J :'lWu�2��:%¥

8Ë��<��'¥ý�'¥�Á,QJr`S`T2P�P�Tdl�z,¥��"T2rBÁ,V �,VF_ yRr�ltT2r`q z,¥ ¼,¥�¸§^dTogbVFw2¥©´{«¹Q¤Á,_xU#Vur`e�_bQJrdTogN¦Toep_i\½�_xr`VuT2P�[NgbS2VundPpT�¼QJU�U#cdr`_b\uToa�_bQJrB»dcdndfdP]Q2S`P�T2U#eu¥��ÀrB�'¥ý�'¥ Á,QJr`SJT2P�PpT�T2r`q ¦D¥�´ QJcL¨P�T2r`\F^`VuT2c�l�VFqd_ba�QJP]eml �N ����( ��& ����%1���# � ��� �K% � ��( �N��( ���,����� � ��������� $"�!���� �� *@�����`lfdToS2VFe�Wm� 1 �2�d¥���gbe�VF�`_bVuPFl�»`Vuf`apVuUXnµVuP�Wm�2�2�d¥

8 J�<X»�¥(Í�_xP�TLrdT2r`q'T2r`_kl�¾#¥º¾DVurdr`VFqdw2l��'¥�|°VFgbgiQ`PÀ¨Ì¼�P�cdU�U�VFw2lºT2r`q�[#¥º»`VFa-^`_5¥ª¼QJU�f`_bgxToa�_bQJra�VF\F^dr`_i·dc`VFe£kQJPn`gbQJ\F�o¨�\Fw`\Fgb_b\Xqd_be�a-P]_xndc`a�_bQJr`eu¥"´ VF\¯^dr`_b\uTog�zVufdQJPRaN´�z���G G2�dW-¨Ì»�l`¼�z�O"¼.lz_b\FV �Dr`_b��¥xl�Í~QJc`e�a�QJr�ld´�� �2�2�L��Gdl`|�T2P]\F^�Wu�L��Gd¥

8Ë��<�[X¥o»`VFa9^`_d¾#¥E¾DVurdr`VFqdw2l ��¥ ��VFqdVFg y��oQo�J_b\2¥d[�gb_xr`VuT2PÀ¨�a�_xU#VDTogbS2QJP]_ba9^dU £5Q`P(\FQJU+fdc`a�_xr`S,a9^`VU#VuU#QJPRw�To\F\FVFe�e�e�VF· c`Vur`\FV~_�r�q'Toa-Tm¨ÌfdT2P�TogbgbVFgdfdP]Q2SJP�TLU#eu¥2��r �N( ������9� ���=%"���# � ( �������� � �*�� ��( ����������� ( � �@( �� �,��Jlm�2Q2gxcdU#VN�2�~Qo£ � �� � W��� � ��� ��� � W�� � � lFfdToS2VFe�Wu�2��1�W2W2W2l»dT2rLa9T#¦¹TLP�ndT2P�Tdld¼[#ld�Jc`gbw�Wu�L��Gd¥`[N¼| O"PRVFe�eu¥

8xWm��<��'¥d|�¥'»`a�_b\F^dr`Q2a9^�ldÁ�¥�> � ÍNThgigxT2PRQJr�l�TLr`q�´,¥'z,¥ ��P]Q2e�em¥ �,Vur`VuP�Tha�_xr`S�\FQJU�U#cdr`_b\uToa�_bQJr`e£5Q`P(T2P�P�TmwDe�a9ToapVuU#Vur2ape� -qdVFe�_bSJr�_�U+f`giVuU�Vur2a9Tha�_bQJr#T2r`q�VF�oTogxcdToa�_bQJr�¥ � � � �tlu�dW� xW�G2� 1�W�GL�dlWu�2��:%¥

8xWLW$<Xz,¥�´�^dT2�`cdPFl�[X¥�¼�^`Q`c`q'^dT2P]w2l�T2r`q ��¥(Î%Qov�¥°z�cdrLa�_xU#V²T2PpP�Tmw}PRVFqd_iepa9P]_xndc`a�_bQJr�_xr�^dfL£fdP]Q2SJPpT2U#eu¥��Àr � �$�����@)������ � ����� �"��(�����( ����$� ���� �� *@����� ��� ����( ���� �m¥(� � � ��l�|�TmwWu�2��:%¥

î(î�éNM'O 0 )�)

�2� � ���� ���N( ������� ���# ����($���( #�� ��*@( �����@�$��*

��p� �~&(� � �.�p�"$

�������1 � �������Å_ie�T�Ot^�¥ýÁ ¼�T2r`qd_bq'Toa�V�T2r`q The�e�_be�a9T2rLa¬fdP]Qh£5VFe�epQJP�Toa�a-^`V � �D»L¨Ì½�w2Q`r�lÎ�P�T2r`\FV2¥'Í~VXP]VF\FVF_b�2VFq°^`_be,| ¥Ë»�¥JqdVFS`P]VFVFeN_xr¤¼QJU�fdc`a�VuP~»`\F_iVur`\FV�£�P]QJU � �D»L¨Ì½�w2Q`r�_xr�Wu�L��:'l^`V�Togbe�Q,P]VF\FVF_b�2VFq�T�|�¥ »�¥mqdVFSJP]VFVt_xr�|�Toa-^`VuU�Toa�_b\Fe�_�rªWu�L��Gd¥uÍ~_beòP]VFepVuT2P]\F^,_xr2apVuP]VFe�a�e�_xr`\Fgxc`qdVFefdT2PpTogbgiVFg(T2r`qªqd_be�a9P]_xndc`a�VFq�TogbS2QJPR_ia-^dU#eulJfdTLP�TogbgbVFg�Q`fdVuP�Toap_�r`S�e�w`e�a�VuU#eul`T2r`q�VurL�J_xP]Q`rdU#Vur2at£kQJPfdT2PpTogbgiVFg�fdP]Q2S`P�T2U�U#_xr`S'¥� � � -¤� ����� ��� �ª�&-�1����N����_beXTÅe�Vur`_bQJP#P]VFe�VuT2P]\F^�Toepe�QJ\F_xToapV�Qo£N¼ �Dz�»�Toa�a9^`V½(�ÀO _xr � �D»L¨Ì½�w2Q`r�l�Î�P�TLr`\FV2¥òÍ~V�P]VF\FVF_b�2VFq�^`_be¬ÍDT2n`_bgb_ia-Toa�_bQJrBqdVFSJP]VFVª£�P]QJU �Nr`_b�2VuPRe�_baÌwBQo£½�wLQJrª_xr°Wu�2��:'l2^`_beNOt^�¥ýÁ�¥2qdVFS`P]VFV,_xr�¼QJU�fdc`a�VuPt»`\F_bVur`\FV,£�P]QJU �Nr`_b�2VuPRe�_baÌwªQo£ ��PRVur`QJn`gbV,_�rWu� J2�d¥uÍ~V"«�ToeºT~P]VFe�VuT2P]\F^�The�e�Q`\F_�Tha�V¹Toa a9^`V �Dr`_b�2VuP]e�_ba]w,Qo£'´(Vurdr`VFe�epVFV2l2¾,r`QEv`�`_igbgbV£�P]QJU Wm�2�2�a�QªWu�2��:%¥EÍ~_be¹U�ThyÌQJP _xr2apVuP]VFe�a�eT2P]VNfdT2PpTogbgiVFg�T2r`q�qd_be�a9PR_�ndc`apVFq¡TogbS2Q`P]_ba9^dU#euloVurL�J_xP]Q`rdU#Vur2at£kQJPfdT2PpTogbgiVFg�fdP]Q2S`P�T2U�U#_xr`S'¥dÍ~V�_beNT�U�VuUXnµVuP�Qh£º_bVFVFV�TLr`q�To\uU�¥

ôöíºî�ôxõ

Unité de recherche INRIA Lorraine, Technopôle de Nancy-Brabois, Campus scientifique,615 rue du Jardin Botanique, BP 101, 54600 VILLERS LÈS NANCY

Unité de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES CedexUnité de recherche INRIA Rhône-Alpes, 46 avenue Félix Viallet, 38031 GRENOBLE Cedex 1

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

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

ISSN 0249-6399

top related