Static Interconnection Networks

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Static Interconnection Networks. CEG 4131 Computer Architecture III Miodrag Bolic. Linear Array. 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

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### Static Interconnection Networks

CEG 4131 Computer Architecture III

Miodrag Bolic

Linear Array

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)

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.