P P P 0 1 2 3 4 5 6 7 8 9 10 12 14 11 13 15 P .edu

9SV8V5(`$EC?A@C8AC0WT8%Af)@1?E?)TCC&AT0bF5X91I1?A'6aA'5AfE2@%E0aC%SCA8A?8@b%?YS9G(?5P1'(CE@%EAE8A4?C)c9)S5CIT5A0?908YI@E51dIP1@7H62SCHD2831T?3IAA7UC4H459b@AT6TH8EU9&@Q56IASTI88'56@HC@05@7IIHT51@PT58A859@b8P@98EA6@PC9aA8P4ES8@07E57'Eb8!PAW9AE15QB9E)%DUA8I@F1c59CQ'8ADRC%C7'"95FA8WACAA9#(UA)A6CTA8I17e)@E7'5U058%5AE1A6@@@862PEAaC8PE@9@P7%eE8@A9'8@ASHC5EAR5@CPSEAA99

gggmkkqmhhhneiiienihhxuipxoeopvqsrvvewfwwrxeysstrslrxtxyyyounfysxjwperrezexkq|qwwqvyvwwsxqsvdxrose|yynrtwqyxrswetovfiulssqfv{swxxdttfxistnsegxuepvwserjyfh

P0

P1

P2

P3

0 1 2 3 4 5 6 7 8 9 10 12 14 11 13 15

Column-wise block striping

0 4 8 12 1 5 9 13 2 6 10 14 3 7 11 15

Row-wise cyclic striping

P0 P1 P2 P3

P0

P1

P2

P3

(0,0) (0,1) (0,2) (0,3) (0,4) (0,5) (0,6) (0,7)

P0

P1

P2

P3

(1,0) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (1,7)

P4

P5

P6

P7

P8

P9

P 10

P 11

(6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) (6,7)

P 12

P 13

P 14

P 15

(7,0) (7.1) (7,2) (7,3) (7,4) (7,5) (7,6) (7,7)

Block-checkerboard partitioning

(0,0) (0,4) (0,1) (0,5) (0,2) (0,6) (0,3) (0,7)

P0

P1

P2

P3

(4,0) (4,4) (4,1) (4,5) (4,2) (4,6) (4,3) (4,7)

(1,0) (1,4)

P4

P5

P6

P7

(5,0) (5,4)

(2,0) (2,4)

P8

P9

P 10

P 11

(6,0) (6,4)

(3,0) (3,4)

P 12

P 13

P 14

P 15

(7,0) (7,4)

Cyclic-checkerboard partitioning

?)????0?!?'4?"51?26#??#3$!%??!?457$89@!4$&987'("&!$ $DDAdE(??eb1$#f2?&"R!#''!$$'S(!1!$%T1"!$&$#U#!&02V&!1&&!W#(U!"$(!?&'X3''$?1$FH0?aY!IE(&&QPbG&&$2?&#H!&&?g?$a'?$$&?&2!?0#a'01&(&'?B0c&`2"0!$!C!$"E"F&C"!CF hipqrstqiuvwxvrprxuxuhyv ded def gheee giejeed kmnoprsqltruvwstruxwpqs

(0,0)

P0

(0,1)

P1

(0,2)

P2

(0,3)

P3

(1,0)

P4

(1,1)

P5

(1,2)

P6

(1,3)

P7

(2,0)

P8

(2,1)

P9

(2,2)

P 10

(2,3)

P 11

(3,0)

P 12

(3,1)

P 13

(3,2)

P 14

(3,3)

P 15

Communication step in transposition

(0,1)

P0

(1,0)

P1

(2,0)

P2

(3,0)

P3

(0,1)

P4

(1,1)

P5

(2,1)

P6

(3,1)

P7

(0,2)

P8

(1,2)

P9

(2,2)

P 10

(3,2)

P 11

(0,3)

P 12

(1,3)

P 13

(2,3)

P 14

(3,3)

P 15

Final configuration

yz{|}~|z}{}| ????? ???????????????????????????????? ???????????????????????

(0,0) (0,1) (0,2) (0,3) (0,4) (0,5) (0,6) (0,7) (1,0) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (2,0) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (2,7) (3,0) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (3,7) (4,0) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (4,7) (5,0) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (5,7) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) (6,7) (7,0) (7.1) (7,2) (7,3) (7,4) (7,5) (7,6) (7,7)

Divide matrix into four blocks

(0,0) (0,1) (0,2) (0,3) (4,0) (4,1) (4,2) (4,3) (1,0) (1,1) (1,2) (1,3) (5,0) (5,1) (5,2) (5,3) (2,0) (2,1) (2,2) (2,3) (6,0) (6,1) (6,2) (6,3) (3,0) (3,1) (3,2) (3,3) (7,0) (7,1) (7,2) (7,3) (0,4) (0,5) (0,6) (0,7) (4,4) (4,5) (4,6) (4,7) (1,4) (1,5) (1,6) (1,7) (5,4) (5,5) (5,6) (5,7) (2,4) (2,5) (2,6) (2,7) (6,4) (6,5) (6,6) (6,7) (3,4) (3,5) (3,6) (3,7) (7,4) (7,5) (7,6) (7,7)

Each block divided into four smaller blocks

(0,0) (0,1) (0,2) (0,3) (0,4) (0,5) (0,6) (0,7)

P0

P1

P2

P3

(1,0) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (1,7)

(2,0) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (2,7)

P4

P5

P6

P7

(3,0) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (3,7)

(4,0) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (4,7)

P8

P9

P 10

P 11

(5,0) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (5,7)

(6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) (6,7)

P 12

P 13

P 14

P 15

(7,0) (7.1) (7,2) (7,3) (7,4) (7,5) (7,6) (7,7)

Communication pattern in first step

(0,0) (0,1) (0,2) (0,3) (4,0) (4,1) (4,2) (4,3)

P0

P1

P2

P3

(1,0) (1,1) (1,2) (1,3) (5,0) (5,1) (5,2) (5,3)

(2,0) (2,1) (2,2) (2,3) (6,0) (6,1) (6,2) (6,3)

P4

P5

P6

P7

(3,0) (3,1) (3,2) (3,3) (7,0) (7,1) (7,2) (7,3)

(0,4) (0,5) (0,6) (0,7) (4,4) (4,5) (4,6) (4,7)

P8

P9

P 10

P 11

(1,4) (1,5) (1,6) (1,7) (5,4) (5,5) (5,6) (5,7)

(2,4) (2,5) (2,6) (2,7) (6,4) (6,5) (6,6) (6,7)

P 12

P 13

P 14

P 15

(3,4) (3,5) (3,6) (3,7) (7,4) (7,5) (7,6) (7,7)

Communication pattern in second step

(0,0) (0,1) (2,0) (2,1) (4,0) (4,1) (6,0) (6,1) (1,0) (1,1) (3,0) (3,1) (5,0) (5,1) (7,0) (7,1) (0,2) (0,3) (2,2) (2,3) (4,2) (4,3) (6,2) (6,3) (1,2) (1,3) (3,2) (3,3) (5,2) (5,3) (7,2) (7,3) (0,4) (0,5) (2,4) (2,5) (4,4) (4,5) (6,4) (6,5) (1,4) (1,5) (3,4) (3,5) (5,4) (5,5) (7,4) (7,5) (0,6) (0,7) (2,6) (2,7) (4,6) (4,7) (6,6) (6,7) (1,6) (1,7) (3,6) (3,7) (5,6) (5,7) (7,6) (7,7)

Last subdivision and transposition

(0,0) (1,0) (2,0) (3,0) (4,0) (5,0) (6,0) (7,0) (0,1) (1,1) (2,1) (3,1) (4,1) (5,1) (6,1) (7,1) (0,2) (1,2) (2,2) (3,2) (4,2) (5,2) (6,2) (7,2) (0,3) (1,3) (2,3) (3,3) (4,3) (5,3) (6,3) (7,3) (0,4) (1,4) (2,4) (3,4) (4,4) (5,4) (6,4) (7,4) (0,5) (1,5) (2,5) (3,5) (4,5) (5,5) (6,5) (7,5) (0,6) (1,6) (2,6) (3,6) (4,6) (5,6) (6,6) (7,6) (0,7) (1,7) (2,7) (3,7) (4,7) (5,7) (6,7) (7,7)

Final configuration

????????????????????? %)B(!7"9(#&)$$%!%&&01'C)"%1%8&'()&30($$"7!"!9)$019%1$!'%0011"2)(%1!'0$(13$4105@!0#)(36%A!!6& D$0%1"B')(3%1(1"!3!04$%AD$!&B!(3A1E9)((6&$%($!0)6%F)()&$"71B8((1&%$(1!$"0%G)'H&1(I8$P(3HB4)"@!(929(('06 Q&!!'$%%160!F74%1D9B)(A7!)&&)BR(7(&&'"'02%(19S(1%TB"'%6%B&!'FB UVWGXPHY`abPcdeY`fGPHYXd`gPhipP

P0 P1 P2 P3

Transposition with striped partitioning

Processors Matrix A P0 P1

Vector x

0 1

n/p

Pp-1

p-1

P0

0

P1

1

Pp-1

p-1

????????????????? !"#$% &'()0 1%2#%3$4567#3$4!% 8'9@ ???ABCC?DE????F 1"!73#7%Q6$%4R7"1166G4P$7515$PT435HI6316S573G3%5H%HH%16%#56!1!$%!5$365$$H44I!13$10"74#S"$#5%$$7%4"037$G%45$1U6P "!5#175$6$644"%1664W75#X$934P@$3Y4Q!5a%3VUH415161#Pb!$1$!P%55!6$44S137W51HX$`P591"Y$$Q4W!$X59$65YT6

Processors Matrix A P0 P1

Vector y

0 1

n/p

Pp-1

p-1

a 11

a 12

a 13

.....

a 1,p-1

P0

x0

x1

x2

.....

xp-1

P1

0

1

p-1

0

p-1

0

p-1

Pp-1

0

p-1

"?)$'5A%#$?(5#0?0!?'#&?$$('?0!0#($$'?'#9!G$2?5Y!1$%`'#!0?6a%0"$'##?!&#?H?$?&I"b!$%P'##&QS1%R@$$#H??%B5I?&2C2SP?D?Q7!$?HRT?$?$??8#Ua?#'E$S$'F%3d$?H#T1VUG5U??#%?V'$4cW#'$X$!?%V#"?!''!(#0& ????????????????????????

?????????? ?? ??????

??????? ??????????

q{yexyzvyf|ghyqr}|iq~lyrpgmy}ygyqnstyyrr|pqohqsqq|itd}uerfvqpxgwftfffdugewthfqzfpiyfytjqgdhpryiknpjhryudprdyrfprhvgyhw lnppijrp

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 ................
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