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# Interconnection Networks - PowerPoint PPT Presentation

Interconnection Networks. Lecture 5 : January 29 th 2007 Prof. Chung-Kuan Cheng University of California San Diego Transcribed by: Jason Thurkettle. Topics. Graph Construction. Project phase 1: See Z9 IBM Journal Research and Development Jan 2007. Hyper Cube (HC).

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• Lecture 5 : January 29th 2007

Prof. Chung-Kuan Cheng

University of California San Diego

Transcribed by: Jason Thurkettle

• Graph Construction

• Project phase 1: See Z9 IBM Journal Research and Development

Jan 2007

• From A -> B:

Divide Q3 into 2 opposed Q2’s.

Note that there are two paths to any point on a Q2.

Now connect the Q2’s.

• How do you change a hypercube to improve metrics?

• Generalized Hypercube

• Toroidal Mesh Hypercube

• Crossed Hypercube

• Folded Hypercube

• Cube Connected Hypercube

• Q(d1, d2,…,dn) = Kd1 x Kd2 x … Kdn

Note: Kd is the complete graph of the degree derived. Cliques

• 1) d1+d2+…+dn=n Regular

• 2) diameter = Dimension = n

• 3) Connectivity

Toroidal Mesh Hypercube C(d1,d2,…,dn)

• x = x1,x2,…,xn & y = y1,y2,…,yn : are linked

iff

or C(d1,d2,…,dn) = Cd1xCd2x…xCdn

where Cdi is a undirected cycle

• 1) 2n regular

• 2) diameter =

• 3) connectivity = 2n

• 4) # nodes =

• x = x1,x2,…,xn & y = y1,y2,…,yn : are linked

iff

• a) xn…xj+1 = yn…yj+1

• b) xj ≠ yj

• c) xj-1 = yj-1 if j is even

• d) x2i,x2i-1 ~ y2i,y2i-1

e.g. x1x2 ~ y1y2 : {(00,00),(10,10),(01,11),(11,01)}

• 1) 2n vertices, n2n-1 edges

• 2) diameter

• 3) connectivity n

i.e.

• 1) 2n vertices (n+1)2n-1 edges

• 2) n+1 regular

• 3) diameter

• 4) connectivity n+1

• (x,i), (y,j) are linked iff

1) x=y, |i-j|= 1 mod n or

2) i=j, |xi-yi| = 1

Note: 1 & 2 refer to a cycle and not a hypercube.

• 1) n2n vertices – 3n2n-1 edges

• 2) 3 regular

• 3) Diameter=

• Def 1: d-ary sequence of length n

• Def 2: iterated line digraphs

• B(d,1) = Kd+ B(d,n)=Ln-1 (Kd+) n≥2

Note: Kd+ is a complete d vertex graph

• Def 3: V = {0,1,…,dn-1}

E={(x,y),y=xd+β mod dn, β=0,1,…d-1}

• 1) dn vertices, dn+1 edges

• 2) d regular

• 3) diameter = n