1. Interconnection topologies
CS 433
Laxmikant Kale
University of Illinois at Urbana-Champaign
Department of Computer Science

2. Distributed memory m/cs
We discussed the communication interface (Ch 7)
PE0
PE1
PEp
Mem0
Mem0
Memp
Interconnection Network
Now, let us talk about the network topology

3. Topology requirements
Must connect each processor to all others
I.e. it should be possible to send “messages” between any pair of processors
Desirable:
Intermediate routing, if needed, should happen without interfering with computation on intermediate nodes.
Example: ring (of theoretical interest only).

4. Topology metrics What makes one topology better than other?
What should we measure?
What can we measure?
C: the number of connections (degree) per node
How densely is the graph connected?
Metric: Diameter of the graph
Others?
How much bandwidth is available?
Store-and-forward vs. circuit switched
Number of independent wires
Bisection bandwidth
Divide the set of processors in two equal partitions
How much data can flow from one partition to the other (per unit time)?
Take the minimum of this quantity over all possible partitionings
(If all processors were to exchange data with random others, ..)

5. Ring: just to check our definitions
Let B be the bandwidth of each link
C=2, D= P/2, bisection BW: 2B
If every processor wanted to send b bytes to every other….

6. Grids
2D and 3D grids

7. Hypercubes Definition: (recursive) To make a hypercube of degree k,
Take 2 hypercubes of degree (k-1)
Connect the cooresponding processors from each set (with 1 extra link)
Base case: one processor
Number of processors must be a power of 2
C: log(P) =
Diameter: log(P)
Neighbors?
Number processors, 0 to P-1, and consider their binary representation
My neighbor in dimension A is obtained by flipping my A’th bit.

8. K-ary n-cube Disadvantage of hypercube?
Wire lengths? May span the entire machine
Leads to reduced bandwidth per link
Number of connections per node increases with the size of the machine
K-ary ncube allows one to seek intermediate points between grids and hypercubes.
N-dimensional “hyper” cube, but number of processors along each dimension is k (instead of 2 of binary hypercube)

9. Writing parallel programs
Programming model
How should a programmer view the parallel machine?
Sequential programming: von Neumann model
Parallel programming models:
Shared memory (Shared address space) model
Message passing model
Shared Objects model