groups.csail.mit.edugroups.csail.mit.edu/commit/papers/2000/Barua-PhD.pdf · è ú %&' "î å ()...

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int a,b,c;

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int *p, *q;

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(a) (b)

int a int c

int b

bank 1bank 0

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ordering constraint

*p = . . .

*q = . . .

dynamic message

p = &A[x+1]p = &A[x]

*p = . . .

(b)

bank 0 bank 1

*q = . . .(a)

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XWý�þ!¡Tw$¢Y{rW��¥KVI]e ��K�SuI�þ!ý/þ#KbXT[�SuK�eRpJSuI�þ!�Wí ��à Q�e�ü�\S'þ�ý�OR\POR[$��K�üLK�QPORMNs¸ORv�þmef\tSuI�þmM#e�U�þ4üL\,dPvZ^XWý�þ�¡Tw$¢Y{ÆOf�_wy�$í �zà ORKVK�ü�vf\]K»O�X]\�ü�î�XWþJ\�XTl�WOþý#gYM�OZ[L[�þnUcOjÚLÛ1�bÜRÝÆ�ÅÛ¸ÞPßÅà �_Ý6ÞPá#�@â_�_ã_gZSueRþ�ORMNI�ORW]KkS'ýbORMnSeRW�Â/þ#M_SNü�\@SbI�þ�QWýbefv�ýbORl�w � \mORW]KkS'ýuOZMnS~UPOZSNO�efW�Â/þnMnSNüLKiþ�ü$SuI�þý~O¥KVSuORMNsf^OR[�[Le�M�OZS�þ#UKìZO"ý�üLORW][�þJUWþ#M#[LO¸^ýbOZS'ü�ef\Tü�\�SuI�þ�QWýbefv�ýbORl�goe�ý�O�v�ýbefXTQ!eRp�UT��\PORl�üLMzefWYÂ/þ#MnSbKJM�ý/þ�OZS�þ#U�W�mSuIWþ�KuORl�þzI�þ#ORQo^�l�þ#l@e�ýk�OZ[L[Le�M�O¸S'üLeR\�M�OZ[L[]K/ü�S'þ.ü�\cSbI�þ�Q�ýbeRv�ýuORl�w9ÙwüLvfXWý�þ�¡Tw$¢Y{�WP�yKVI]e ��KiSbI�þ�ORW]KkS'ýuOZMnS~UPOZSNO�efWYÂ/þ#MnSbK~l�O"ýVs"þ#U�.ü$SuI|ܸàNà_�$äRÞ�M#efl@l1þn\SuK#g��.ü�SbImSuI�þ�[Le�M�OZS�üLef\@K�þ_SJ\YX]l�W%þ�ýbK�pre"ý~SuIWþzefW�Â/þnMnSuK�[�üLKkS'þnU!OR[LeR\]vfK�ü�U�þ�SuI�þMnefl@l1þn\YS#w � \dþ#\S�ü�ý�þcO ý�ýuO��cüLKzMnef\]K/üLU�þý�þ#UºO�K/üL\]vf[�þ�efW�Â/þnMnS#g�W]XTSAþ#ORMNIºd�þ#[LUdü�\�O�KVS�ýbX]M_SuX�ý/þ(üLKMnef\]K/üLU�þý�þ#U�OjK�þnQPO"ýuO¸S'þ¥efW�Â/þ#M_S�wiÙwü�\POR[L[$�fg�Q�e"üL\S'þý�OZ\POR[���K/üLK�OR\]\TeRSNOZS�þ#K.þ�OZMNI�l�þ#l@e�ýV�Tý�þ_pnþ�ý�þn\]M�þü�\]KVS�ýbX]M_S'ü�ef\,�.ü$SuI�O¤ÚLÛ1�bÜRÝ��ÅÛ¸ÞPßÅà �_ÝzÚå�Là_Ý�g»O�[�ü�KVS�eRp�[Le�M�OZS�üLef\o^`K�þ_S�\YX]l�WOþýbKjM#e"ý�ý�þnKbQ�ef\]U�ü�\]v©Sue�SuI�þeRW�Â/þ#M_SuK~SbIPOZSiSuI�þ�l�þ#l@e�ýV�Kü�\]KVS�ýbX]M_S'ü�ef\�M#OR\Tý/þnpnþý#wC÷�\�d]vfX�ý�þ�¡Tw$¢Y{�W�� gLþ�ORMNI�l�þ#l@e�ýk�KüL\]KkS'ýVX]MnS�üLef\ü�KyOR\]\TeRSNOZS�þ#U��.ü$SuIKü�SbK�[Le�M�O¸S'üLeR\o^�K/þnS�[�üLKkS�gRKVI]e ��\mOZKJãN�ræ�Mnefl�l�þ#\SbK#wyÙPe�ý9K�üLl@Q][�üLM�ü$S��fg¸[Le�M�OZS�üLef\�^�K�þ_S\YX]l�WOþýbK�O ý�þ�KbI]e ��\�ef\T[��jpre�ýiefW�Â/þnMnSuKiSuIPO¸S�IPO�ì"þzQ�ýbeRv�ýuORl;[LeORUPçZKVSbe�ý�þnKiSbe�SuIWþ#l�"$ü�\1ý/þ�OR[�ü�S��@OR[�[eRW�Â/þ#M_SuK�O"ý�þ¥ORKVK�ü�vf\�þ#U�KbXTMNI�\YX]l�W%þ�ýVK#wè?éTê6é�ëiì�í�ë~îVï�ðTê�ñóò»ïzò»ô�õ�é]îé �ORQ]K�X]K/þ#K6Q�e"üL\S'þý�OR\POR[$��K�üLK�pre�ý�SuI�ý/þ�þyQ]XWýbQ�efK�þnK�¹6l1ü�\�üLl�üL�#OZS'ü�ef\�eRpoU�þ#Q%þ#\]UWþ#\]MþHþnU]v�þ#Kng þîYX�ü�ìZOR[�þ#\]Mþ_^Mn[aORKVK9XT\�ü�dPM#OZS'ü�ef\!{�xi}��z�_gZOR\]U�KbeRp¼S���O"ý�þ~K�þý�üaOZ[�e"ýbU�þý�üL\Tvj{Å�T�]ö��_wZÙwü�ýVKVS�g¸Q�e�üL\S'þýBOZ\POR[���K/üLKeüLK�X]K�þnUSbe�l�üL\�ü�l1ü���þJl�þ#l@e�ýV�^�UWþ#Q%þ#\]U�þn\]M�þAþnU]v�þnK#wy÷^ü�SbI�\]eQü�\Tpre�ýbl�OZS�üLef\�gYOZ[L[�l1þnl�e�ýk��ý�þnpnþý�þn\]M�þnK�O"ý�þQ�eRS�þ#\S'üLOR[L[$�,UWþ#Q%þ#\]U�þn\S�g�OZ\]U�I�þ#\]Mþcþ_ì�þýV��QPO"ü�ý�eRp�l1þnl�e"ýV�<üL\]KkS'ýbXTMnS'ü�ef\]KjeRp���I�ü�MNI�O¸Sc[�þ�OZKVSeR\�þ(üL\]KkS'ýbXTMnS'ü�ef\@üLK�O�KkSue�ý/þmO"ý/þ�Mnef\]\�þnMnS�þ#UtW�,O�UWþ#Q%þ#\]U�þn\]M�þTþ#U]v�þZw�÷^ü�SbI,Q�e�üL\S�þ�ý¥OR\POR[$��K�üLKngITeø�Hþì"þ�ýng~l@e�ý�þ�Q�ý�þnM�ü�K�þ!üL\opre�ýbl�OZS'ü�ef\@üLK�O�ìZO"ü�[aORW][�þRw � QPO"ü�ý�eRp�l�þ#l@e�ýV�@üL\TKVS'ýVX]MnS�üLef\TK�IPO�ì�þ�OUWþ#Q%þ#\]U�þn\]M�þ1þ#U]v"þ@W%þnS��Hþ�þn\¤SuIWþ#lvü�p~OR\TU,ef\T[��&ü$p~SbI�þ(üL\S'þýbK/þ#MnS�üLef\ºeRpiSuI�þü�ý�[Le�M�O¸S'üLeR\o^�K/þnSz[�üLKkSuKü�Kz\]ef\�^�þ#l@QTS��Pù_g6OR\]U,OZS�[�þ�ORKkS�ef\�þjeRpiSuI�þ(ü�\]KVS�ýbX]M_S'üLeR\]KAüLK�O!KVSbe�ý�þZwcÙwüLvRX�ý�þj¡ow�¢Y{rW���KbITeø��K�SuIWüLKýVX][�þcpre�ý�üL\]K/þ�ýkS'üL\Tv�U�þnQOþn\]U�þn\]M�þ!þnU]v�þ#K¹(þ#U]v�þnK#g�KVI]e ��\t�.ü�SbI¤U]eRSVS'þnUqO"ý�ýVe ��K#gyO"ý/þmQ�ý/þ#K/þ#\SjW%þ_^S��Hþ�þn\!l1þnl�e"ýV�(üL\TKVS'ýVX]MnS�üLef\TK~ef\T[��c��I�þ#\�SbI�þGü�\S'þ�ýVK�þnMnS'ü�ef\�eRphSuI�þ�ý�þnpnþý�þn\]M�þnU�[�eoM#OZS'ü�ef\o^`K�þnS»[�üLKkSuKü�KJ\]ef\o^Vþ#l@QTS��fw

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�� á_âwè�Ù��"Ù9×Fè}Ù9×FØ9Ù/Ô�� Ù9Õ��OÙ/Ù9×RÖ|â`âVÖ'ß�â`Ù���Ù9â`Ù9× Ø�Ù/Ô����mÙJØ/Ö|×Rè�á��"Ù�ÕhÕhÙ/â�����áÓê×�Õ`Ù9âeÖ_×FÖ|ç�ß�ÔbÓ�Ôeè�á�Ù�ÔS×}á_Õwè�Ó�ÔnÕ`Óê×�� �}Ó�Ô"!� Ù9Õ��OÙ/Ù9×PâhÙ���Ù9â`Ù9×FØ9Ù/ÔzÕháAè�Ó$#"Ù/âhÙ/×�ÕiÙ9ç�Ù9à-Ù9×�Õ`ÔxÓ�×KÖ|×QÖ|â`â`Ö'ß��7ÔbáGÕ%! Ö�Õiâ`Ù���Ù9â`Ù9× Ø�Ù&�FÖ|Ó�â`ÔCÔ%�FØ'!QÖ_Ô)(+*gÛ�,%Ö|×Fè-(.* Ü/,OÖ_âhÙÖ|×FÖ_çêß109Ù�èAÕ`á2�"Ù�è�Ù��"Ù9× è�Ù9×�Õ�� á_âiÖ|â`âVÖ'ßGâ`Ù���Ù/âhÙ/×FØ�Ù�Ô��43�Ö5�FÔ��LÓ�×PÖè}è�ÓêÕhÓ�á_×4�4� ÔbÙ�ÔwÕhâVÖ_è�ÓêÕhÓ�á_×FÖ_ç:Ö|â`â`Ö'ßAè�Ù�� Ù/×Fè�Ù9× Ø�ÙÖ|×FÖ_çêß�ÔhÓ�ÔOÕhá-á���Õ`Ö_Óê×26F×}Ù/â7�_âVÖ|Ó�×}Ù/è�è�Ù��"Ù9× è�Ù9×FØ9ÙCÓê×8��á_â`à-Ö|ÕhÓ�á_×9* Ý�ã�,:�

Page 55: groups.csail.mit.edugroups.csail.mit.edu/commit/papers/2000/Barua-PhD.pdf · è ú %&' "î å () Pÿ çbárâ áAü íbárâbã %& AèMñ èMë *& AèMñfë ÿmèMþrå ( é}ñ çAç

Î�Ï�òoÏ�Ì�ÑcÒ�Ó Ô�ÉzÕ6Ì»Ö�Ç�ÌiõuÇ�Õ�É�×T×�Ò9Ö�Ó�óBÓ_ÇBÉ�Ë�Ó_ØJÖ<;�Ì�Ë9È�Ø�= ¿f¿

malloc.y (6) malloc.z(7)(c)

(d)

*q=33f.y=11 *r=44 p->x=22

malloc.z(7)

pf->y=55

malloc.y (6)malloc.x (5)f.x (1)ctr (4)f.z (3)f.y (2)

pf->y = 55

pf->y=55

malloc.x (5)ctr (4)f.z (3)f.y (2) f.x (1)

f.y=11 *q=33 p->x=22*r=44

void fn (int cond) {

int x, y, z;struct foo {

};

struct foo f; int ctr; struct foo *pf, *p; int *q, *r;

pf = (struct foo *) malloc (sizeof (struct foo));

p = cond ? &f : pf;

q = cond ? &f.y : &f.z;

r = cond ? &f.x : &ctr;

}

// ref : 2 f.y = 11;

p->x = 22;

*q = 33;

*r = 44; pf->y = 55;

void fn (int cond) {

int x, y, z;struct foo {

};

struct foo f; int ctr; struct foo *pf, *p; int *q, *r;

pf = (struct foo *) malloc (sizeof (struct foo));

p = cond ? &f : pf;

q = cond ? &f.y : &f.z;

r = cond ? &f.x : &ctr;

}

f.y = 11;

p->x = 22;

*q = 33;

*r = 44; pf->y = 55;

// ref : 1,4// ref : 6

// assign : 1,2,3

// assign : 5,6,7

// assign : 4

// ref : 1,5

// ref : 2,3

(a) (b)

Equiv. class 2Equiv. class 1 Equiv. class 4Equiv. class 3

(e)

f.x (1)

malloc.x(5)

ctr (4)

malloc.y(6)f.y (2)

f.z (3)

malloc.z(7)

Bank 0 Bank 1 Bank 2 Bank 3

p = cond ? &f : pfp->x = 22r = cond ? &f.x : &ctr*r = 44

q = cond? &f.y : &f.z*q = 33

f.y = 11

PE 0 PE 1 PE 2 PE 3

ÙwüLvfXWý�þ�¡Tw�¢"¹?>A@�¼�¿AÈ$Å�É�Ç�Ë7¾ÊJÃ�»7Ï�É|Í ÎLÃCB�¼�Å�É|»0Ä'Éu�bÄ�Å�¼�Ç�Ç�Î0»LÃED$Ä�¼�½�Ã�¾}»¥ÆGFU¼�HJI`»LÃ�½�Ã�¼�ÅxÈLÌ�¾�Ï�Ì9¼�¿(ÆKF�ÑLHAºHÁ�½�É�ÌÈ�¾}Ã�» ½�É�Ìi¼�»L¼�Å�À�Ç�Ã�Ç5M"Ç�ËL¾_ÊJÃ�»LÏGÅ�¾�Ä�¼�½�Ã�¾}»f�bÇ�É�½i» ÎL¿�Ñ�É�Ì9Çx¼�»LÂQÂ7É�È�É|»LÂ7É|»LÄ'É-É|Â7Ï�É|Ç�FUÂL¾�½�½�É|Â�Å�Ã�»LÉ|Ç�H/ÆFUÄ H)N�É|¿z�¾�Ì9Àd¾�ÑPOVÉ|Ä'½�Ço¼�»LÂ@Ã?»LÇ�½�Ì9ÎLÄ'½�Ã�¾}»LÇ\Ì�É�ÈLÌ�É|Ç�É|»F½�É|Â<¼�ÇRÑ0à �nÈ$¼�Ì�½�Ã�½�ÉKÏ�Ì9¼�È0Ë¥ÆQFUÂRHTSw¾}»L»7É|Ä'½�É|Â<Ä'¾}¿AÈ�¾}»7É|»F½�Ç�¾�ÁÑ$à �nÈ0¼�Ì�½�Ã�½�É�Ï�Ì9¼�È0Ë�¿R¼�Ì��É|Â�¼�Ç�É|Í"ÎLÃUB�¼�Å�É|»LÄ'É�Ä�Å?¼�Ç�Ç�É|Ç9F:>VSxÇ�H/Æ(ÐJË7É�Ì�É�¼�Ì�É-WX>SxÇZY\[n�]M_^a`8M[�b1McWdM_ea`8Mf[�ga`8M[ih�`}Æ&F�É�HpºJÁ�½�É�ÌeÄ'¾�Â7ÉCÏ�É|»7É�Ì9¼�½�Ã�¾}»:Æ7j�>lkmjeÌ�¾�Ä'É|Ç�Ç�Ã�»7Ï-É|Å�É|¿RÉ|»F½�Æ7>e¼�Ä/ËT>S�Ã�Ç*¿R¼�ÈLÈ�É|ÂG½�¾�¼�Ç�Ã�»LÏ}Å�ÉxÑ0¼�»�on½�ËLÉGÄ'¾�Â7É\Ã�ÇHÂ0Ã�Ç�½�Ì9Ã�Ñ$Î7½�É|Â(¼�¿A¾}»7ÏP½�Ë7É\ÂLÃEp�É�Ì�É|» ½+j�>zÇNÊJËLÃ�Å�É�É|»0Ç�Î7Ì9Ã�»LÏR½�ËL¼�½NÌ�É�Á�É�Ì9É|»LÄ'ÉGÃ?»LÇ�½�Ì9ÎLÄ'½�Ã�¾}»LÇJ¼�Ì�ÉÅ�¾"Ä�¼�Å:½�¾�½�Ë7É.Ñ0¼�»�P½�Ë7É�ÀQ¼�Ä�Ä'É|Ç�Ç�Æ

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Bank3Bank2 Bank 1Bank 0

PE 2 PE 3PE 1PE 0

(c)

A[i+1] = A[i+1] + B[i+2] A[i+2] = A[i+2] + B[i+3] A[i+3] = A[i+3] + B[i+4]

for i = 0 to 99 step 4 do A[i] = A[i] + B[i+1]

(d) i’ = 0for i = 0 to 99 step 4 do

i’ = i’ + 1

00

0

1

2

3

1

1

2

3

2

3

A [i’] = A [i’] + B [i’] A [i’] = A [i’] + B [i’] A [i’] = A [i’] + B [i’] A [i’] = A [i’] + B [i’+1]

Code

Data

A[i] = A[i] + B[i+1]

B[ ]

(a)for i = 0 to 99 do

A[ ]

A[i] = A[i] + B[i+1]

(b)for i = 0 to 99 do

A [ ] A [ ] A [ ] A [ ]

B [ ]B [ ]B [ ]B [ ]

0 1 2 3

3210

A [ ] A [ ] A [ ] A [ ]

B [ ]B [ ]B [ ]B [ ]

0 1 2 3

3210

A [ ] A [ ] A [ ] A [ ]

B [ ]B [ ]B [ ]B [ ]

0 1 2 3

3210

Edge from memory access to possible bankScalar dependence edge (register communication)

A [i’]=A [i’] +tmp1 A [i’]=A [i’] +tmp2 A [i’]=A [i’] +tmp3 A [i’]=A [i’] +tmp00

tmp0 = B [i’+1] tmp1 = B [i’] tmp2 = B [i’] tmp3 = B [i’]

0

1 2 3

33221 10

(e)

A [ ]

B [ ]

0

0

A [ ]

B [ ]

2

2

A [ ]

B [ ]

1

1

A [ ]

B [ ]

3

3

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A [ ]0 1A [ ]

A[i+x+3] = . . .

A [ ]

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A[i+x+1] = . . .

A [ ]

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3

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for i = 0 to 99 step 4 do

Non-disambiguated (dynamic) access requestNon-disambiguated (dynamic) access acknowlegdement

Code

Data

for i = 0 to 99 do(a)

A[ ]

A[i+x] = . . .

(b) A[i + x] = . . .

A[i + x + 1] = . . .

A[i + x + 2] = . . .

A[i + x + 3] = . . .

0 1 2 3A [ ] A [ ] A [ ] A [ ]

(c)

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Dependence edge

A[B[i]] += . . .

A[B[i+1]] += . . . A[B[i+1]] += . . .

A[B[i]] += . . .

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loopsforloops towhileConvert

unstructuredBuild structured control-flow from

Dismantle aggregate-object copies

Forward-propagate to array indices

Pointer analysis

Handling array reshapes

Handling library calls with pointerarguments

Strip-mine distributed arrays

Equivalence-class unification

computation

Affine-access detection

Dynamic-access generation

Communication generation

Memory-dependence analysis

Epoch detection and handling

Instruction partitioning

Thread generation

Raw executable

Sequential C or FORTRAN program

Unroll loops, producing mod-region

Infer mod values using mod-regionannotes

annotes

creation sites to global address

Replace accesses disambiguated with modulo unrolling to use local address

Data and instruction placement

Modulo unrolling unroll-factor

Strength-reduce mod and divoperators

Convert distributed-array address-

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. . .

A[2*i + 3] = 11; A[2*i + 4] = 22;

for i = . . .

int x;int x;

for i = . . . x = 2*i + 3; . . .

A[x] = 11; A[x+1] = 22;

(a) (b)

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(a) (b)for i = x to 99 do A[i] = i

int new_lb;

if (x % 4 == 0) new_lb = xelse

for i = x to (new_lb-1) do A[i] = i

for i = new_lb to 99 step 4 do

A[i+1] = i+1 A[i] = i

A[i+3] = i+3 A[i+2] = i+2

new_lb = (x div 4)*4 + 4

//mod-region i % 4 == 0

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ø�×ÙØ~×ÛÚKÜ Ý�Þ�ß5û*ÜÀÚáà6â.ã4ä�ß5û,ÜNåçæ'ûÀâ}èéÞ@ê�ÚëÚKÜcú³à�å.ßìä Ý,ß^â.ê í�í

(a)

A[i] = ifor i = 0 to 99 doint A[100]

(b) int A[100]

int A[25][4]

for i = 0 to 99 step 4 do

int A[25][4](c) (d)

A[i div 4][i%4] = i A[(i+1) div 4][(i+1)%4] = i+1

A[(i+2) div 4][(i+2)%4] = i+2 A[(i+3) div 4][(i+3)%4] = i+3

int A[25][4]int A[25][4] (f)(e)

A[i+1] = i+1 A[i] = i

A[i+3] = i+3 A[i+2] = i+2

for i = 0 to 99 step 4 do

i’ = 0for i = 0 to 99 step 4 do

A[i’][i%4] = i A[i’][(i+1)%4] = i+1 A[i’][(i+2)%4] = i+2 A[i’][(i+3)%4] = i+3

i’ = i’ + 1

i’ = 0for i = 0 to 99 step 4 do

A[i’][0] = i A[i’][1] = i+1 A[i’][2] = i+2 A[i’][3] = i+3

i’ = i’ + 1

i’ = 0for i = 0 to 99 step 4 do

i’ = i’ + 1

// bank 0// bank 1// bank 2// bank 3

//mod-region i % 4 == 0

//mod-region i % 4 == 0 //mod-region i % 4 == 0

//mod-region i % 4 == 0//mod-region i % 4 == 0

0 A [i’] = i A [i’] = i+1

A [i’] = i+3 A [i’] = i+2

1

2

3

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Ø {�{ à�ó�Þ@ä Ý�ÜNå�ø�×dã¼Þ@äcúzßCã!äcû*ÜÀã4ÜNê�Ý�Þ Ý,ß^â6ê¤��H¸ô©���«8�"¤�¢���£�¤�¢h£/­@¢¨�Ô"�H¢¾õ�ÕÀö"¯z÷(ø�ª/¤�©Û¯±¥·�¤Â�¨ Tª/�8£���©��"�H�Û�T£��h�s§�ª/«2m�¬�(¥À¢��¨�¨�Gm²Ãy«8��¤��¤��¿ª(¤ª/�¨�¡ª¦½�ùhúû>/üÓú Ø { ÿ ü�­�£(�6¯ýt oh°Tª(¤Bh²¢nl� �ª(¤����H¢P�¡ ��uª(���¡ª¦½8�¡£hùhúû>/üÓú ÿ ×/üyú o(üþ��ÿNª1l� ¼ª(�¨�¨ª¦½&���H­��"���"¤MlH�u��¢lH£��¨���"¢_m�£�¤�©���¤��(¬½z«8£~©��#�T�"©�þ�­�£(���s§�ª/«2m�¬�(¥Zùhú 'þüÓú � ü���¢�l� Tª(¤����H© �¡£�ùhú 'þüÓú � ©��#¶Mo(üÓú � «�£²©�o(üþ� ü �¨�¨�=m~ë8��¤���¤��u¬��"ª(©�¢À�¡£:�¨ ��@¢¡ª/«����¨�"¢���¬�Pª(¢·¸. ��H¤G�¡ �� ¬µª(¢C�·©��«��H¤�¢¨�£�¤&£/­ �¡ ��Kª/�¨�¡ª¦½8��¢XmTª(©�©��"©G�¡£��¡ ��¤��^§²�� ����( ��"�<«:��¬�¨�Gm�¬��£(­�¯±¥À­�£�¬¬�£¹¸ �"©?°�½1¢C�¡�¨�=m~Ãy«8��¤���¤��������"¤Bl"�z¢��¨�¨�Gm²Ãy«8��¤��¤��vlH£�¤Ml"�Sm��¡��ª(¬�¬#½mZ�"�C­�£��¨«8¢YmTª/©�©���¤��Mþ�mTª(©�©���¤��G�¨ ��h¬µª(¢C�@©��«��H¤�¢¨�£�¤¿��¢ �¨�"�q����¨�H©®­�£(� �¡ ���m�ª(©�©��¤���£)m��¨��«8��Ô¦ªb�¡��£(¤©��"¢FlH�¨�°��H©!�¤¼¢��%l^�¡��£(¤/o��³o������������^×�� o � l¦� ¢¨ �£¹¸.¢K�¡ ��h�¨�"¢���¬�¨¢K£(­À¢��¨�¨�Gm²Ãy«8��¤��¤��&ª/�¨�¡ª¦½4ùë­�£�� �¨ ���l"£²©��h��¤ �������¨��×~�³o � °��

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ø�×ÙØ~×ÛÚKÜ Ý�Þ�ß5û*ÜÀÚáà6â.ã4ä�ß5û,ÜNåçæ'ûÀâ}èéÞ@ê�ÚëÚKÜcú³à�å.ßìä Ý,ß^â.ê Ø { Ø

issue_write( ((addr_y>>2)&3), ((addr_y>>4)<<2), 22, 3) // turnstile 0

block_for_write() // tile 2block_for_write() // tile 3

// turnstile 0issue_write( ((addr_x>>2)&3), ((addr_x>>4)<<2), 11, 2)

A[y] = 22

A[x] = 11

A[y div 4][y%4] = 22 A[y div 4][y%4] = 22

A[x div 4][x%4] = 11

int A[100]

int A[25][4]

int A[25][4]

(c)

(a)

int A[25][4]

A[x div 4][x%4] = 11

local_addr_x = &A[x div 4]

addr_y = (local_addr_y<<2) | ((y%4)<<2)local_addr_y = &A[y div 4]

addr_x = (local_addr_x<<2) | ((x%4)<<2)

(b)

(e)

(d)addr_x = (local_addr_x<<2) | ((x%4)<<2)

addr_y = (local_addr_y<<2) | ((y%4)<<2)

*addr_x = 11

*addr_y = 22

local_addr_y = &A[y div 4]

local_addr_x = &A[x div 4]

int A[25][4]

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