1. Minimal Fault Diameter for Highly Resilient Product Networks Khaled Day, Abdel-Elah Al-Ayyoub IEEE Trans. On Parallel and Distributed Systems 2000 vol. 11, no 9 Speaker: Y-Chuang Chen

2. Abstract Fault tolerance of Cartesian Product. A method for building containers (a set of node- disjoint paths) between any two nodes of a product network. Show the best achieve fault diameter. highly resilient and minimal fault diameter => Cartesian product => still highly resilient and minimal fault diameter

3. Some definitions Node-connectivity, diameter, fault-diameter

4. Definition 1: Cartesian product The Cartesian product G = G 1 G 2 of two graphs G 1 =(V 1,E 1 ) and G 2 =(V 2,E 2 ) is the graph G=(V,E), where the set of nodes V and the set of edges E are given by: 1. V = {  x,y  | x  V 1 and y  V 2 }, and 2. For u =  x u,y u  and v =  x v,y v  in V, (u,v)  E iff (x u,y v )  E 1 and y u =y v, or (y u,y v )  E 2 and x u =x v.

5. K Definition 2: container K container K A container K between two nodes u and v in a graph G is a set of node-disjoint paths joining nodes u and v. width The width of K is the number of paths on K. maximum width In a regular graph G, a container is of maximum width if its width is equal to the degree of G.

6. Lemma 1: some known results Let u =  x u,y u  and v =  x v,y v  be two nodes in G 1 G 2. The following properties holds: 1. G 1 G 2 is isomorphic to G 2 G 1. 2. |G 1 G 2 | = |G 1 |  |G 2 | 3. D(G 1 G 2 ) = D(G 1 ) +D(G 2 ). 4.  (G 1 G 2 )   (G 1 ) +  (G 2 ).

7. Maximum-Width Containers in Products Networks They show how to construct a maximum- width container between any two nodes in G 1 G 2.

8. Some notations D(G): the diameter of a graph G. L(K): the length of a longest path in the container K. l(K): the length of a shortest path in the container K. D f (G): fault diameter of G; the maximum diameter of any subgraph of G obtained by deleting less than  (G) nodes from G.

9. Lemma 2. Container Let u =  x u,y u  and v =  x v,y v  be two nodes in G 1 G 2 such x u  x v and y u  y v. If there is a container K 1 of width w 1 between x u and x v in G 1 and a container K 2 of width w 2 between y u and y v in G 2, then there is in G 1 G 2 a container K of width w 1 + w 2 between u and v. Furthermore, L(K) = max{L(K 1 )+l(K 2 ), l(K 1 )+L(K 2 )} and l(K)=l(K 1 )+l(K 2 ).

11. Lemma 3. Let u =  x u,y u  and v =  x v,y v  be two nodes in G 1 G 2 such x u  x v and y u =y v. If there is a container K 1 of width w 1 between x u and x v in G 1 and if deg G2 (y u ) =  2, then there is in G 1 G 2 a container K of width w 1 +  2 between u=  x u,y u  and v=  x v,y v . Furthermore, L(K) = max{L(K 1 ), l(K 1 )+2} and l(K)=l(K 1 ). Notice that since G 1 G 2 = G 2 G 1, a similar result holds for x u =x v and y u  y v.

12. Theorem 1. If G 1 and G 2 are regular each with a maximum- width container between every pair of nodes, then there exists a maximum-width container between every pair of nodes in G 1 G 2. Furthermore, if each path in a G 1 (resp. G 2 ) container has length at most L 1 (resp. L 2 ) and at least one path is of length at most l 1 (resp. l 2 ), then every path in corresponding G 1 G 2 container has length at most max(l 1 +L 2, L 1 +l 2, l 1 +2, l 2 +2) and at least one path has length l 1 +l 2.

13. High Fault Resilience and Minimal Fault Diameter A number of interconnection networks are known to have excellent fault diameter (equal to diameter plus one). Eg. Hypercube, star graph, k-ary n-cube, cube-connected cycles, generalized hypercubes.

14. High Fault Resilience and Minimal Fault Diameter highly resilient Definition of highly resilient: A regular graph G is highly resilient if between any two nodes of G, there exists a deg(G)-wide container such that every path in the container is of length at most D(G)+1 and at least one path is of length at most D(G).

15. Some results If G 1 is regular, G 2 is regular,  (G 1 ) = deg(G 1 ), and  (G 2 ) = deg(G 2 ), then G 1 G 2 is regular and  (G 1 G 2 ) = deg(G 1 G 2 ). The fault diameter D f (G) of any regular connected graph G of diameter D(G)>2 and of node-connectivity  (G) = deg(G) satisfies D f (G)  D(G)+1.

16. Some results (cont.) If G is regular and highly resilient, then  (G) = deg(G). If, in addition, D(G) > 2, then D f (G) = D(G) +1. (By definition of highly resilient) If G 1 and G 2 are highly resilient regular graphs, then G 1 G 2 is highly resilient.

17. Theorem 2. Highly Resilience If G 1 and G 2 are highly resilient regular graphs such that D(G 1 ) + D(G 2 ) > 2, then G 1 G 2 is highly resilient and D f (G 1 G 2 ) = D(G 1 G 2 ) + 1.

18. Corollary If each of G 1,G 2,…,G n is a hypercube, a k- ary n-cube, a star graph, a cube-connected cycles, or a generalized hypercube, then the product G 1 G 2 …G n is highly resilience and has minimal fault diameter provided that D(G 1 ) + D(G 2 ) + … + D(G n ) > 2.