Static Interconnection Networks

1. 1 Static Interconnection Networks CEG 4131 Computer Architecture III Miodrag Bolic

2. 2 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 Difference between the linear array and the bus Ring: unidirectional and bidirectionalDifference between the linear array and the bus Ring: unidirectional and bidirectional

3. 3 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

4. 4 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. K is the dimension Ring D=k*(n-1), B=n Torus: d=4, D= 2 * ring diameter, B=2nK is the dimension Ring D=k*(n-1), B=n Torus: d=4, D= 2 * ring diameter, B=2n

5. 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? Binary tree Number of nodes 2^k-1 Leaves, rootsBinary tree Number of nodes 2^k-1 Leaves, roots

6. 6 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

7. 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 : select any bit position, those PEs which have 0's at that bit position will make up one partition and others will form the second partition In an n-dimensional hypercube, n separate paths always exist between two nodes. This makes the hypercube highly fault tolerant Degree increases linearly with dimension, which limits scalability. : select any bit position, those PEs which have 0's at that bit position will make up one partition and others will form the second partition In an n-dimensional hypercube, n separate paths always exist between two nodes. This makes the hypercube highly fault tolerant Degree increases linearly with dimension, which limits scalability.

8. 8 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

9. 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 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.

10. 10 Cube-connected cycle d=3 D=2k-1+ Example N=8 We can use the 2CCC network

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12. 12 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.