1. Static Interconnection Networks
Miodrag Bolic

2. Linear Arrays and Rings
Ring arranged to use short wires
Linear Array
Asymmetric network
Degree d=2
Diameter D=N-1
Bisection bandwidth: b=1
Allows for using different sections of the channel by different sources concurrently.
Ring
d=2
D=N-1 for unidirectional ring or for bidirectional ring
Difference between the linear array and the bus
Ring: unidirectional and bidirectional

3. Ring Fully Connected Topology Chordal ring
Needs N(N-1)/2 links to connect N processor nodes.
Example
N=16 -> 136 connections.
N=1,024 -> 524,288 connections
D=1
d=N-1
Chordal ring
N=16, d=3 -> D=5

4. Multidimensional Meshes and Tori
3D Cube
2D Grid
Mesh
Popular topology, particularly for SIMD architectures since they match many data parallel applications (eg image processing, weather forecasting).
Illiac IV, Goodyear MPP, CM-2, Intel Paragon
Asymmetric
d= 2k except at boundary nodes.
k-dimensional mesh has N=nk nodes.
Torus
Mesh with looping connections at the boundaries to provide symmetry.
K is the dimension
Ring D=k*(n-1), B=n
Torus: d=4, D= 2 * ring diameter, B=2n

5. Trees Diameter and ave distance logarithmic Fixed degree
k-ary tree, height d = logk N
address specified d-vector of radix k coordinates describing path down from root
Fixed degree
Route up to common ancestor and down
Bisection BW?
Binary tree
Number of nodes 2^k-1
Leaves, roots

6. Trees (cont.) Fat tree Star The channel width increases as we go up
Solves bottleneck problem toward the root
Star
Two level tree with d=N-1, D=2
Centralized supervisor node

7. Hypercubes Each PE is connected to (d = log N) other PEs d = log N
Binary labels of neighbor PEs differ in only one bit
A d-dimensional hypercube can be partitioned into two (d-1)-dimensional hypercubes
The distance between Pi and Pj in a hypercube: the number of bit positions in which i and j differ (ie. the Hamming distance)
Example:
10011  = 11010
Distance between PE11 and PE9 is 3
: select any bit position, those PEs which have 0's at that bit position will make up one partition and others will form the second partition
In an n-dimensional hypercube, n separate paths always exist between two nodes. This makes the hypercube highly fault
tolerant
Degree increases linearly with dimension, which limits scalability.
100
110
000
010
111
101
001
011
0-D
1-D
2-D
3-D
4-D
5-D
*From Parallel Computer Architectures; A Hardware/Software approach, D. E. Culler

8. Hypercube routing functions
Example
Consider 4D hypercube (n=4)
Source address s = 0110 and destination address d = 1101
Direction bits r = 0110 1101 = 1011
1. Route from 0110 to 0111 because r = 1011
2. Route from 0111 to 0101 because r = 1011
3. Skip dimension 3 because r = 1011
4. Route from 0101 to 1101 because r = 1011

9. k-ary n-cubes
Rings, meshes, torii and hypercubes are special cases of a general topology called a k-ary n-cube
Has n dimensions with k nodes along each dimension
An n processor ring is a n-ary 1-cube
An nxn mesh is a n-ary 2-cube (without end-around connections)
An n-dimensional hypercube is a 2-ary n-cube
N=kn
Routing distance is minimized for topologies with higher dimension
Cost is lowest for lower dimension. Scalability is also greatest and VLSI layout is easiest.

10. Cube-connected cycle d=3 D=2k-1+ Example N=8
We can use the 2CCC network

12. References
Advanced Computer Architecture and Parallel Processing, by Hesham El-Rewini and Mostafa Abd-El-Barr, John Wiley and Sons, 2005.
Advanced Computer Architecture Parallelism, Scalability, Programmability, by  K. Hwang, McGraw-Hill 1993.