MAPPING OF ONE MODEL INTO OTHER MODEL

Submitted by:-RATIKA AGARWALM.Tech.-CSE

• Relationship between two networksG(V,E) G’(V’,E’)

• Graph G, G’• Vertices V, V’• Edges E, E’

Embedding:

MAP

E E’

V V’

Process Mapping (Eg.- 2-d Mesh)

Underlying Architecture

Processes and their interactions

Mapping of Processes to nodes

G(V,E) G’(V’,E’)Communication structure of an algorithm differs

from the interconnection architecture of the intended parallel machine.

Number of processes required by the algorithm exceeds the number of processors available.

Need of Mapping:

Types of network topologies: (Continued..)

Linear Arrays

2-d Mesh

Binary Tree

Types of network topologies: (Hypercube)

0-d Hypercube

1-d Hypercube

2-d Hypercube

3-d Hypercube

4-d Hypercube

Mapping techniques for graphs:

Topology embedding:Embed a communication pattern into a given interconnection topology. Linear Array in a Hypercube . 2-D Mesh in a Hypercube. Linear Array in a 2-D Mesh. 2-D Mesh in a Linear Array.

And more…

Metrics used for Embedding:

DILATION

CONGESTION

EXPANSION

Expansion =No.of Nodes in V’ = |V’| / |V|No.of Nodes in V

Where ( |V|<|V’| )

Expansion

Congestion

Congestion of k: ‘k’ Es to one E’

k>1

Dilation

Dilation of k: One E to ‘k’ E’s.

MAPPING

Linear Array/Ring into Hypercube

Node ‘i’

Linear Array 2d nodes d-dimesional Hypercube

G(i, d) : Gray code of ith node of Array.

RGC : Binary Reflected Gray Code.

Node G(i,d)

MAP

G(i,x+1) G(i,x) for i < 2x

2x + G[ (2x+1 –1) -i, x] for i >= 2x

G(0,1) = 0G(1,1) = 1

• 2d = 23= 8, i.e., 8 nodes in ring and hypercube.

Example: Say d = 3

Node ‘i’ in Ring

NodeG(i, d) in

hypercube (x+1=3)

(x+1=2) (x+1=1)

G(0,1)=0

G(1,1)=1Binary

Notation

0 (000) G(0,3)= G(0,2)= G(0,1)= 0= 0 000

1 (001) G(1,3)= G(1,2)= G(1,1)= 1= 1 001

2 (010) G(2,3)= G(2,2)= 2+ G(1,1)= 2+1= 3 011

3 (011) G(3,3)= G(3,2)= 2+ G(0,1)= 2+0= 2 010

4 (100) G(4,3)= 22 +G(3,2)= 4+ 2+ G(0,1)= 4+2+0= 6 110

5 (101) G(5,3)= 22 +G(2,2)= 4+ 2+ G(1,1)= 4+2+1= 7 111

6 (110) G(6,3)= 22 +G(1,2)= 4+ G(1,1)= 4+1= 5 101

7 (111) G(7,3)= 22 +G(0,2)= 4+ G(0,1)= 4+0= 4 100

Gray Coding

0 0 0 0 00

1 0 1 0 01

1 1 0 11

1 0 0 10

1 10

1 11

1 01

1 00

1 bit RGC 2 bit RGC 3 bit RGC

G(0,1) = 0G(1,1) = 1

Congestion=1Dilation=1

<Mapped Model>

01234567

ALTERNATIVE METHOD

Gray Coding

0 0 0 0 00

1 0 1 0 01

1 1 0 11

1 0 0 10

1 10

1 11

1 01

1 00

1 bit RGC 2 bit RGC 3 bit RGC

ALTERNATIVE METHOD

G(0,1) = 0G(1,1) = 1

Congestion=1Dilation=1

<Mapped Model>

01234567

2-D Mesh into Hypercube

2r*2s nodes of Mesh (r+s) dimension Hypercube of 2r+s nodesMAP

Node (i, j) Node G(i, r) || G(j, s)

‘||’ is concatenation of two Gray codes

Wraparound Mesh

MAP

Mesh Node Hypercube Node

Binary Notation

(0,0) G(0,1)||G(0,2) 0||0 0||00 000

(0,1) G(0,1)||G(1,2) 0||1 0||01 001

(0,2) G(0,1)||G(2,2) 0||3 0||11 011

(0,3) G(0,1)||G(3,2) 0||2 0||10 010

(1,0) G(1,1)||G(0,2) 1||0 1||00 100

(1,1) G(1,1)||G(1,2) 1||1 1||01 101

(1,2) G(1,1)||G(2,2) 1||3 1||11 111

(1,3) G(1,1)||G(3,2) 1||2 1||10 110

Example for 2*4 mesh…

Dilation=Congestion =1

00 00

11 00

01 00

10 01

10 00

10 11

10 10

Same row of Mesh ‘r’ identical MSBs in mapped node Same column of Mesh ‘s’ identical LSBs in mapped node

Linear Array into 2-D Mesh

Congestion=1

16 node linear array 2-d Mesh MAP

Solid lines: links in arrayNormal lines: links in the mesh

2-D Mesh into Linear Array

Inversion of the previous mapping.

Congestion =5

Solid lines: links in arrayNormal lines: links in the mesh

Congestion =5

Congestion =5

Summary

Applications of Embedding: To minimize the communication overhead of a

parallel algorithm. Laying out circuits on chips.

Reasons for embedding: An algorithm can run on many architecture. Algorithm mapped on interconnection network to

achieve maximum parallelism.

THANK YOU