Understanding Hypercube Variations in Network Interconnection

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