Static interconnection networks
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Static Interconnection Networks. 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

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Static interconnection networks

Static Interconnection Networks

Miodrag Bolic


Linear arrays and rings

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


Static interconnection networks

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

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

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

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

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

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

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

Cube-connected cycle

  • d=3

  • D=2k-1+

  • Example N=8

    • We can use the 2CCC network


References

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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