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Understanding Hypercubes and Interconnection Networks: Lecture 5 Overview

In Lecture 5 on January 29, 2007, Prof. Chung-Kuan Cheng from UC San Diego discusses the intricate design and construction of hypercubes and various interconnection network topologies. Key topics include graph constructions, the generalized hypercube, toroidal mesh hypercube, crossed hypercube, and folded hypercube. The lecture also covers important metrics such as regularity, diameter, and connectivity of these networks. This session is part of a broader project based on IBM Journal Research and Development, aiming to enhance network performance through innovative design strategies.

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Understanding Hypercubes and Interconnection Networks: Lecture 5 Overview

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  1. Interconnection Networks • Lecture 5 : January 29th 2007 Prof. Chung-Kuan Cheng University of California San Diego Transcribed by: Jason Thurkettle

  2. Topics • Graph Construction • Project phase 1: See Z9 IBM Journal Research and Development Jan 2007

  3. 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?

  4. Hypercube Variations • Generalized Hypercube • Toroidal Mesh Hypercube • Crossed Hypercube • Folded Hypercube • Cube Connected Hypercube

  5. 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

  6. 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 =

  7. 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

  8. Crossed Cube Hypercube – continued

  9. 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

  10. 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=

  11. 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

  12. DeBruijn Networks: Continued

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