1 Static Interconnection Networks CEG 4131 Computer Architecture III Miodrag Bolic

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

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 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=n k nodes. Torus –Mesh with looping connections at the boundaries to provide symmetry. 2D Grid 3D Cube

5 Trees Diameter and ave distance logarithmic –k-ary tree, height d = log k 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 d=N-1, D=2 –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:  = Distance between PE11 and PE9 is 3 0-D1-D2-D3-D 4-D 5-D *From Parallel Computer Architectures; A Hardware/Software approach, D. E. Culler

8 Hypercube routing functions Example Consider 4D hypercube (n=4) Source address s = 0110 and destination address d = 1101 Direction bits r = 0110  1101 = Route from 0110 to 0111 because r = Route from 0111 to 0101 because r = Skip dimension 3 because r = 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 N=k n 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 d=3 D=2k-1+ Example N=8 –We can use the 2CCC network

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12 References 1.Advanced Computer Architecture and Parallel Processing, by Hesham El-Rewini and Mostafa Abd-El- Barr, John Wiley and Sons, Advanced Computer Architecture Parallelism, Scalability, Programmability, by K. Hwang, McGraw-Hill 1993.