P M 2 1 Ideal Distributed-Memory Parallel Computer P+M P+M - PDF document

Hypercubic Networks Charles E. Leiserson 6.895 Theory of Parallel Systems November 2, 2003 1 “Ideal” Parallel Computer P M P M P M … … P M 2 1

“Ideal” Distributed-Memory Parallel Computer P+M P+M P+M P+M P+M P+M P+M P+M 3 Interconnection Networks linear array binary tree 2D mesh 4 2

Hypercube d = 0 d = 1 d = 2 N = 1 N = 2 N = 4 d = 3 d = 4 5 N = 8 N = 16 Hypercube 110 111 010 011 100 101 000 001 6 3

Cube-Connected Cycles 7 Butterfly (FFT) Network 0 1 2 0 000 000 001 001 010 010 011 011 100 100 101 101 110 110 111 111 8 4

Butterflies 9 Decomposing a Butterfly 10 5

Decomposing a Butterfly 11 Decomposing a Butterfly 12 6

Decomposing a Butterfly 13 Decomposing a Butterfly II 14 7

Decomposing a Butterfly II 15 Decomposing a Butterfly II 16 8

Decomposing a Butterfly II 17 Decomposing a Butterfly II 18 9

Decomposing a Butterfly II 19 Decomposing a Butterfly II 20 10

Routing on a Butterfly 0 1 2 0 000 000 001 001 010 010 011 011 100 100 101 101 110 110 111 111 21 Tree in Butterfly 0 1 2 0 000 000 001 001 010 010 011 011 100 100 101 101 110 110 111 111 22 11

Tree in Butterfly 0 1 2 0 000 000 001 001 010 010 011 011 100 100 101 101 110 110 111 111 23 Beneš Network 24 12

Decomposing a Beneš Network n /2 Beneš n /2 Beneš 25 Routing on a Beneš Network 5 0 0 1 7 2 4 3 6 4 2 5 3 6 1 7 26 13

Routing on a Beneš Network 5 0 0 1 7 2 4 3 6 4 2 5 3 6 1 7 27 Routing on a Beneš Network 5 0 0 1 7 2 4 3 6 4 2 5 3 6 1 7 28 14

Routing on a Beneš Network 5 0 0 1 7 2 4 3 6 4 2 5 3 6 1 7 29 Routing on a Beneš Network 5 0 0 1 7 2 4 3 6 4 2 5 3 6 1 7 30 15

Routing on a Beneš Network 5 0 0 1 7 2 4 3 6 4 2 5 3 6 1 7 31 Routing on a Beneš Network 5 0 0 1 7 2 4 3 6 4 2 5 3 6 1 7 32 16

Routing on a Beneš Network 5 0 0 1 7 2 4 3 6 4 2 5 3 6 1 7 33 Routing on a Beneš Network 5 0 0 1 7 2 4 3 6 4 2 5 3 6 1 7 34 17

Routing on a Beneš Network 5 0 0 1 7 2 4 3 6 4 2 5 3 6 1 7 35 18