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Minimal Fault Diameter for Highly Resilient Product Networks

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. Abstract . Fault tolerance of Cartesian Product.

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Minimal Fault Diameter for Highly Resilient Product Networks

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  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 = G1G2 of two graphs G1=(V1,E1) and G2=(V2,E2) is the graph G=(V,E), where the set of nodes V and the set of edges E are given by: • V = { x,y | xV1 andyV2 }, and • For u = xu,yu and v = xv,yv in V, (u,v)E iff (xu,yv)E1 and yu=yv, or (yu,yv)E2 and xu=xv.

  5. Definition 2: 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. • The width of K is the number of paths on K. • 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 = xu,yu and v = xv,yv be two nodes in G1G2. The following properties holds: • G1G2 is isomorphic to G2G1. • |G1G2| = |G1|  |G2| • D(G1G2) = D(G1) +D(G2). •  (G1G2)  (G1) + (G2).

  7. Maximum-Width Containers in Products Networks • They show how to construct a maximum-width container between any two nodes in G1G2.

  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. • Df(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 = xu,yu and v = xv,yv be two nodes in G1G2 such xuxv and yuyv. If there is a container K1 of width w1 between xu and xv in G1 and a container K2 of width w2 between yu and yv in G2, then there is in G1G2 a container K of width w1 + w2 between u and v. Furthermore, L(K) = max{L(K1)+l(K2), l(K1)+L(K2)} and l(K)=l(K1)+l(K2).

  10. Lemma 3. • Let u = xu,yu and v = xv,yv be two nodes in G1G2 such xuxv and yu=yv. If there is a container K1 of width w1 between xu and xv in G1 and if degG2(yu) = 2, then there is in G1G2 a container K of width w1 + 2 between u= xu,yu and v= xv,yv. Furthermore, L(K) = max{L(K1), l(K1)+2} and l(K)=l(K1). • Notice that since G1G2 = G2G1, a similar result holds for xu=xv and yuyv.

  11. Theorem 1. • If G1 and G2 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 G1G2. Furthermore, if each path in a G1 (resp. G2) container has length at most L1 (resp. L2) and at least one path is of length at most l1(resp. l2), then every path in corresponding G1G2 container has length at most max(l1+L2, L1+l2, l1+2, l2+2) and at least one path has length l1+l2.

  12. 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.

  13. High Fault Resilience and Minimal Fault Diameter • 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).

  14. Some results • If G1 is regular, G2 is regular, (G1) = deg(G1), and (G2) = deg(G2), then G1G2 is regular and (G1G2) = deg(G1G2). • The fault diameter Df(G) of any regular connected graph G of diameter D(G)>2 and of node-connectivity (G) = deg(G) satisfies Df(G)  D(G)+1.

  15. Some results (cont.) • If G is regular and highly resilient, then (G) = deg(G). If, in addition, D(G) > 2, then Df(G) = D(G) +1. (By definition of highly resilient) • If G1 and G2 are highly resilient regular graphs, then G1G2 is highly resilient.

  16. Theorem 2. Highly Resilience • If G1 and G2 are highly resilient regular graphs such that D(G1) + D(G2) > 2, then G1G2 is highly resilient and Df(G1G2) = D(G1G2) + 1.

  17. Corollary • If each of G1,G2,…,Gn is a hypercube, a k-ary n-cube, a star graph, a cube-connected cycles, or a generalized hypercube, then the product G1G2…Gn is highly resilience and has minimal fault diameter provided that D(G1) + D(G2) + … + D(Gn) > 2.

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