Static Interconnection Networks

1
Static Interconnection Networks

• Miodrag Bolic

2
Linear Arrays and Rings

• Linear Array
• Asymmetric network
• Degree d2
• Diameter DN-1
• Bisection bandwidth b1
• Allows for using different sections of the
  channel by different sources concurrently.
• Ring
• d2
• DN-1 for unidirectional ring or for
  bidirectional ring

3
Ring

• Fully Connected Topology
• Needs N(N-1)/2 links to connect N processor
  nodes.
• Example
• N16 -gt 136 connections.
• N1,024 -gt 524,288 connections
• D1
• dN-1
• Chordal ring
• Example
• N16, d3 -gt D5

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 Nnk nodes.
• Torus
• Mesh with looping connections at the boundaries
  to provide symmetry.

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 dN-1, D2
• 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

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
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Hypercube routing functions

• Example
• Consider 4D hypercube (n4)
• 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
• Nkn
• 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

• d3
• D2k-1
• Example N8
• We can use the 2CCC network

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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.