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

Uniqueness of Shortest Path P:x= x 1 x 2 …x n y= y 1 y 2 …y n Given two nodes, can we always find a shortest path and is that shortest path unique? x 1 x 2 …x n ->x 2 x 3 …x n y 1 ->x 3 x 4 …y 1 y 2 ->…->x n y 1 …y n-1 ->y 1 y 2 …y n

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)

Comparison NodesDiameterEdges #edges #nodes Hypercube2n2n nn 2 n-1 n/2 De Bruijndndn nd n+1 d

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

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} 1.d regular 2.Strongly Connected 3.(i,j) walk of length m iff j= id m +r 1 d m-1 +r 2 d m-2 +…+r m d mod n 4. Diameter = k ┌ log d n ┐

Comparison NodesDiameterEdges #edges #nodes Hypercube2n2n nn 2 n-1 n/2 De Bruijndndn nd n+1 d Generalizeddndn nd n+1 d

Kautz Network K(d,n) Definition 1: (d+1)ary sequence of length n V={x 1 x 2 …x n, x i Є {0,1,…,d}, x i ≠x i+1 for all i} Vertex x 1 x 2 …x n x 2 x 3 …x n-1 α are connected for all α Є {0,1,…,d} Definition 2: Iterated Line Digraphs K(d,1) = K d+1, K(d,n)=L n-1 (K d+1 ), n≥2

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

Kautz Network Properties 1. d n +d n-1 vertices, d n+1 +d n edges 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

Kautz Undirected Graph UK(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

Generalized Kautz Digraph K G (d,n) V={0,1,..,n-1} E={(i,j), j=-(id+r) mod n, r=1,…,d} 1.d regular 2.Strongly Connected 3.(i,j) walk of length m iff j= i(-d) m +r 1 (-d) m-1 +…+r m-1 mod n 4. Diameter k = ┌ log d n ┐ -1 = p iff n= d p + d p-q, q is odd, q ≤ p = ┌ log d n ┐ otherwise