1

Static Interconnection Networks

CEG 4131 Computer Architecture III

Miodrag Bolic

2

Linear Arrays and Rings

• 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

Linear Array

Ring

Ring arranged to use short wires

2/ND

3

Ring

• Fully Connected Topology– 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– Example

• N=16, d=3 -> D=5

4

Multidimensional Meshes and Tori

• 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.

2D Grid 3D Cube

5

Trees

• Diameter and ave distance logarithmic– 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?

6

Trees (cont.)

• Fat tree – 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 01001 = 11010

• Distance between PE11 and PE9 is 3

0-D 1-D 2-D 3-D 4-D 5-D

001 011

000 010

100 110

111101

*From Parallel Computer Architectures; A Hardware/Software approach, D. E. Culler

8

Hypercube routing functions

• ExampleConsider 4D hypercube (n=4)Source address s = 0110 and destination address d = 1101Direction bits r = 0110 1101 = 10111. Route from 0110 to 0111 because r = 10112. Route from 0111 to 0101 because r = 10113. Skip dimension 3 because r = 10114. 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

2/k

11

12

References

1. Advanced Computer Architecture and Parallel Processing, by Hesham El-Rewini and Mostafa Abd-El-Barr, John Wiley and Sons, 2005.

2. Advanced Computer Architecture Parallelism, Scalability, Programmability, by  K. Hwang, McGraw-Hill 1993.