Transcript Static Interconnection Networks CEG 4131 Computer Architecture III Miodrag Bolic Linear Arrays and Rings Linear Array Ring Ring arranged to use short wires • Linear Array – – – – – • Asymmetric network Degree d=2 Diameter D=N-1 Bisection.

Static Interconnection Networks
CEG 4131 Computer Architecture III
Miodrag Bolic
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Linear Arrays and Rings
Linear Array
Ring
Ring arranged to use short wires
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Linear Array
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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 D  N / 2 for bidirectional ring
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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
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Multidimensional Meshes and Tori
2D Grid
3D Cube
• 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.
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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?
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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
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Hypercubes
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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
000
110
010
111
101
001
0-D
1-D
2-D
011
3-D
4-D
5-D
*From Parallel Computer Architectures; A Hardware/Software approach, D. E. Culler7
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
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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.
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Cube-connected cycle
• d=3
• D=2k-1+ k / 2
• Example N=8
– 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-ElBarr, John Wiley and Sons, 2005.
2. Advanced Computer Architecture Parallelism,
Scalability, Programmability, by K. Hwang, McGraw-Hill
1993.
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