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Static Interconnection Networks CEG 4131 Computer Architecture III Miodrag Bolic 1 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 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 2 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 3 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. 4 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? 5 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 6 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 3 100 000 110 010 111 101 001 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 8 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. 9 Cube-connected cycle • d=3 • D=2k-1+ k / 2 • Example N=8 – We can use the 2CCC network 10 11 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. 12