Interconnection networks
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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

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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


Crossed cube hypercube continued

Crossed Cube Hypercube – continued


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


Debruijn networks continued

DeBruijn Networks: Continued


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