Download lecture 7: interconnection networks – ii

CS405 Parallel Computing
BS-Computer Science (SSUET)
LECTURE 7: INTERCONNECTION NETWORKS – II
HYPERCUBE NETWORKS:
Constructing an N-Dimensional Hypercube

For dimension N+1, just create two hypercube of dimension N, and connect analogous vertices.
N
Nodes:
2N
Links per Node: N
Longest Path: N
Dimensions:
3
2 3=8
3
3
3-D Hypercube
Node Numbering

Each node must differ from an adjacent node by at most one binary digit.
Three-Dimensional Hypercube
Adjacent nodes differ by one binary digit.
Example in Node Numbering
Question: In a four-dimensional hypercube, what nodes connect to node 12?
Answer: 4, 8, 13, and 14
Solution:
First Convert to Binary:
Node 12 = 1100 in Binary
Then apply the exclusive-OR (XOR) operation with a "1" digit in each of the four positions
Lecturer: Engr. Muhammad Nadeem
Page 1
CS405 Parallel Computing
1100
1100
1100
1100
XOR
XOR
XOR
XOR
0001
0010
0100
1000
=
=
=
=
1101
1110
1000
0100
BS-Computer Science (SSUET)
or 13
or 14
or 8
or 4
The resulting values are the nodes that should link directly to node 12.
Mesh Networks:
1. Mesh network is a type of networking where each node must not only capture and disseminate its
own data, but also serve as a relay for other nodes, that is, it must collaborate to propagate the data in
the network.
2. A mesh network can be designed using a flooding technique or a routing technique.
3. When using a routing technique, the message is propagated along a path, by hopping from node to
node until the destination is reached.
Lecturer: Engr. Muhammad Nadeem
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CS405 Parallel Computing
BS-Computer Science (SSUET)
4. To ensure all its paths' availability, a routing network must allow for continuous connections and
reconfiguration around broken or blocked paths, using self-healing algorithms.
5. The self-healing capability enables a routing based network to operate when one node breaks down or
a connection goes bad. As a result, the network is typically quite reliable, as there is often more than
one path between a source and a destination in the network.
BINARY TREE NETWORKS:
BUTTERFLY:
A Butterfly network is made of (n + 1)*2n processors organized into n+1 rows, each containing 2n
processors.
Lecturer: Engr. Muhammad Nadeem
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CS405 Parallel Computing
BS-Computer Science (SSUET)

Rows are labeled 0 … n. Each processor has four connections to other processors (except processors
in top and bottom row).

Processor P(r, j), i.e. processor number j in row r is connected to P(r-1, j) and P(r-1, m)

Where m is obtained by inverting the rth significant bit in the binary representation of j.
PYRAMID
A pyramid consists of (4d+1 – 1)/3 processors organized in d+1 levels so as:



Levels are numbered from d down to 0
There is 1 processor at level d
Every level below d has four times the number of processors than the level immediately above it.
Note the connections between processors.
Lecturer: Engr. Muhammad Nadeem
Page 4
CS405 Parallel Computing
BS-Computer Science (SSUET)
Pyramid interconnection can be seen as generalization of the ring – binary tree network, or as a way of
combining meshes and trees.
STAR NETWORK:

A generalized star interconnection network has the property that for a given integer N, we have
exactly N! Processors.

Each processor is labeled with the permutation to which it corresponds.

Two processors Pi and Pj are connected if the label I can be transformed into label j by switching
the first label symbol of I with a symbol of j (excluding 1st symbol of j)

Below we have a star network for N=4, i.e. a network of 4! = 24 processors. Example: Processors
labeled 2134 and 3124 are connected with two links.
NOTE: The whole idea is to make each node a center node of a small star!
Lecturer: Engr. Muhammad Nadeem
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CS405 Parallel Computing
BS-Computer Science (SSUET)
Metrics of Interconnection Network:
Degree:
No. of links per node.
Diameter:
Longest distance between two nodes in the network.
Bisection:
Min no. of wire cuts to divide the network in 2 halves.
Cost:
No. of links and switches.
Lecturer: Engr. Muhammad Nadeem
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