Topology Embedding: Techniques and Metrics for Mapping Networks, Slides of Parallel Computing and Programming

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Topology Embedding

Topologies Embedding Issues

•^

Performance Metric– Dilation– Congestion– Expansion

•^

Embedding Sparser networks in Denser Networks– Linear Array into a Mesh– Linear Array into a Hypercube– Mesh into a Hypercube– Simultaneous Multiple topology embedding

•^

Embedding Denser Networks in Sparser Networks– Mesh into a Linear Array– Hypercube into a Mesh– Hypercube in Linear Array

Mapping Techniques for Graphs: Metrics •^

When mapping a graph

G(V,E)

into

G’(V’,E’),

the

following metrics are important:

•^

The maximum number of edges mapped onto any edgein^

E’

is called the

congestion

of the mapping.

•^

The maximum number of links in

E’

that any edge in

E

is

mapped onto is called the

dilation

of the mapping.

•^

The ratio of the number of nodes in the set

V’

to that in

set

V

is called the

expansion

of the mapping.

Embedding a Linear Array

into a Hypercube using Gray Code ( a) A three-bit reflected Gray code ring; and (b) its embedding into a

three-dimensional hypercube.

1−bit Gray code

2−bit Gray code

3−bit Gray code

3−D hypercube

8−processor ring

Reflectalong thisline

(a) 110

010 000

011 001

111 101

(b) 100

Embedding a Mesh into a Hypercube using K-map (a) A 4

×^ 4 mesh illustrating the mapping of mesh nodes to the nodes in a four-dimensionalhypercube; and (b) a 2

×^ 4 mesh embedded into a three-dimensional hypercube.

Once again, the congestion, dilation, and expansion of the mapping is 1.

Processors in a column haveidentical two least−significant bits

Processors in a row have identicaltwo most−significant bits

(a)

(b)

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Embedding a Mesh into a Linear Array

•^

Since a mesh has more edges than a linear array, wewill not have an optimal congestion/dilation mapping.

•^

We first examine the mapping of a linear array into amesh and then invert this mapping.

•^

This gives us an optimal mapping (in terms ofcongestion).

Embedding a Hypercube into a 2-D Mesh •^

Each

node subcube of the hypercube is mapped to

a^

node row of the mesh.

•^

This is done by inverting the linear-array to hypercubemapping.

•^

Embedding a Hypercube into a Array This can be shown to be an optimal mapping.

Embedding a Hypercube into a 2-D Mesh: Example

Embedding a hypercube into a 2-D mesh.

P = 32 (b)

P = 16 (a)

P= 64