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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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interconnection networks
Interconnection Networks
  • Lecture 5 : January 29th 2007

Prof. Chung-Kuan Cheng

University of California San Diego

Transcribed by: Jason Thurkettle

topics
Topics
  • Graph Construction
  • Project phase 1: See Z9 IBM Journal Research and Development

Jan 2007

hyper cube hc
Hyper Cube (HC)
  • 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?
hypercube variations
Hypercube Variations
  • Generalized Hypercube
  • Toroidal Mesh Hypercube
  • Crossed Hypercube
  • Folded Hypercube
  • Cube Connected Hypercube
generalized hypercube
Generalized 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 d 1 d 2 d n
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 =
crossed cube hypercube cq v e
Crossed Cube Hypercube CQ(V,E)
  • 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
folded hypercube fq v e
Folded Hypercube FQ(V,E)

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
cube connected cycle hypercube ccc n
Cube Connected Cycle HypercubeCCC(n)
  • (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=
de bruijn network 1946 b d n
De Bruijn Network 1946 B(d,n)
  • 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
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