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Interconnection Networks

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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

- 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

Start with a Hypercube Qn:

Add edges (x,y) if

i.e.

linking the longest distance pairs

- 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

- B(d,1) = Kd+ B(d,n)=Ln-1 (Kd+) n≥2
- 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