**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

**11. **11

**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.