Hypercubic Networks

1
Hypercubic Networks

• Charles E. Leiserson
• 6.895 Theory of Parallel Systems
• November 12, 2003

2
Ideal Parallel Computer
3
Ideal Distributed-Memory Parallel Computer
PM
PM
PM
PM
PM
PM
PM
PM
4
Interconnection Networks
linear array
binary tree
2D mesh
5
Hypercube
d 0 N 1
d 1 N 2
d 2 N 4
d 3 N 8
d 4 N 16
6
Hypercube
110
111
010
011
100
101
000
001
7
Cube-Connected Cycles
8
Butterfly (FFT) Network
0
1
2
0
9
Butterflies
10
Decomposing a Butterfly
11
Decomposing a Butterfly
12
Decomposing a Butterfly
13
Decomposing a Butterfly
14
Decomposing a Butterfly II
15
Decomposing a Butterfly II
16
Decomposing a Butterfly II
17
Decomposing a Butterfly II
18
Decomposing a Butterfly II
19
Decomposing a Butterfly II
20
Decomposing a Butterfly II
21
Routing on a Butterfly
0
1
2
0
000
000
001
001
010
010
011
011
100
100
101
101
110
110
111
111
22
Tree in Butterfly
0
1
2
0
23
Tree in Butterfly
0
1
2
0
24
Beneš Network
25
Decomposing a Beneš Network
n/2 Beneš
n/2 Beneš
26
Routing on a Beneš Network
27
Routing on a Beneš Network
28
Routing on a Beneš Network
29
Routing on a Beneš Network
30
Routing on a Beneš Network
31
Routing on a Beneš Network
32
Routing on a Beneš Network
33
Routing on a Beneš Network
34
Routing on a Beneš Network
35
Routing on a Beneš Network
36
Shuffle-Exchange Network
110
111
010
011
100
101
000
001
37
Shuffle-Exchange Network
38
Perfect Shuffle
39
Perfect Out-Shuffle
40
Perfect In-Shuffle
41
In-Shuffle Out-Shuffle Exchange
42
SE ! DeBruijn
43
SE ! DeBruijn
010
011
110
111
000
001
100
101
44
SE ! DeBruijn
010
011
110
111
000
001
100
101
45
SE ! DeBruijn
010
011
110
111
000
001
100
101
46
DeBruijn Network
1
1
010
01
110
11
000
00
0
1
1
0
100
10
0
0
47
DeBruijn Sequence
48
Complex Plane Diagram of SE