1. Static Interconnection Networks Miodrag Bolic

2. 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

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=nk nodes. • Torus • Mesh with looping connections at the boundaries to provide symmetry. 3D Cube 2D Grid

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?

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: • 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

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

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.