1. Interconnection Networks • Lecture 6a : January 31th 2007 Prof. Chung-Kuan Cheng University of California San Diego Transcribed by: Carmelo Kintana

2. Uniqueness of Shortest Path P: x= x1x2…xn y= y1y2…yn Given two nodes, can we always find a shortest path and is that shortest path unique? x1x2…xn->x2x3…xny1->x3x4…y1y2 ->…->xny1…yn-1->y1y2…yn

3. There are d-1 vertex disjoint paths of length at most n+1 Proof: Case n=1: B(d,1)=Kd+. True Suppose n ≥ 2: B(d,n-1)= Kd+. True B(d,n)= L(B(d,n-1)) Vertex x=(u,u') y=(v,v') These 2 vertices are in B(d,n-1) Let u≠v: There are d-1 disjoint paths (u',v) of length at most n in B(d,n-1)

4. Comparison

5. De Bruijn Undirected Graph UB(d,n) UB(d,n) is obtained from B(d,n) by deleting the orientation of the edges and omitting multiple edges and loops 1. Minimum Degree δ=2d-2 Maximum Degree Δ=2d 2. Diameter = n 3. Connectivity = 2d-2

6. Generalized DeBruijn Diagraph BG(d,n), d ≥ 2, n ≥ d V= {0,1,…,n-1} E={(x,y), y=xd + r mod n, r=0,1,…d-1} • d regular • Strongly Connected • (i,j) walk of length m iff j= idm+r1dm-1+r2dm-2+…+rmd mod n 4. Diameter = k┌logdn┐

7. Comparison

8. Kautz NetworkK(d,n) Definition 1: (d+1)ary sequence of length n V={x1x2…xn, xi Є{0,1,…,d}, xi≠xi+1 for all i} Vertex x1x2…xn x2x3…xn-1αare connected for all αЄ{0,1,…,d} Definition 2: Iterated Line Digraphs K(d,1) = Kd+1, K(d,n)=Ln-1(Kd+1), n≥2

9. Kautz Network (continued…)K(d,n) Definition 3: Arithmetic Method V={0,1,…, dn+dn-1-1} E={(x,y), y=-(xd+α)mod dn+1+dn-1, α =1,…d}

10. Kautz Network Properties 1. dn+dn-1 vertices, dn+1+dnedges 2. d regular 3. Diameter = n 4. Connectivity = d 5. d disjoint paths for a given x and y. 1 path ≤n, d-n paths ≤ n+1, 1 path ≤ n+2

11. Kautz Undirected GraphUK(d,n) UK(d,n) is obtained from K(d,n) by deleting the direction orientation of all edges and omitting multiple edges and loops 1. Minimum Degree δ=2d-1 Maximum Degree Δ=2d 2. Diameter = n 3. Connectivity = 2d-1

12. Generalized Kautz DigraphKG(d,n) V={0,1,..,n-1} E={(i,j), j=-(id+r) mod n, r=1,…,d} • d regular • Strongly Connected • (i,j) walk of length m iff j= i(-d)m+r1(-d)m-1+…+rm-1 mod n 4. Diameter k = ┌logdn┐-1 = p iff n= dp + dp-q, q is odd, q ≤ p = ┌logdn┐ otherwise