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## PowerPoint Slideshow about 'Static Interconnection Networks' - helki

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Ring

Ring arranged to use short wires

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

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

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.

3D Cube

2D Grid

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?

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

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

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

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

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.

Cube-connected cycle

- d=3
- D=2k-1+
- Example N=8
- We can use the 2CCC network

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.

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