Edge fault-diameter of Cartesian graph bundles Iztok Bani c, Rija - PowerPoint PPT Presentation

Edge fault-diameter of Cartesian graph bundles Iztok Baniˇ c, Rija Erveš, Janez Žerovnik FME, University of Maribor, Smetanova 17, Maribor 2000, Slovenia. and FCE, University of Maribor, Smetanova 17, Maribor 2000, Slovenia. and Institute of Mathematics, Physics and Mechanics, Jadranska 19, Ljubljana. iztok.banic@uni-mb.si, rija.erves@uni-mb.si, janez.zerovnik@imfm.uni-lj.si 7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 1 / 12

Introduction • Introduction Design of large interconnection networks • Previous work on (edge) fault-diameters Usual constraints: • Cartesian product of graphs • Cartesian graph bundle • each processor can be connected to a limited number of • Example 1 • Example 2 other processors • Edge fault-diameter • Results 1 • Results 2 • the delays in communication must not be too long • Example 3 • Future Work Extensively studied network topologies in this context include graph products and bundles. • an interconnection network should be fault-tolerant (some nodes or links are faulty) The (edge) fault-diameter has been determined for many important networks. 7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 2 / 12

Previous work on (edge) fault-diameters • Introduction • M. Krishnamoortthy, B. Krishnamurty: Fault diameter of • Previous work on (edge) fault-diameters interconnection networks (1987) • Cartesian product of graphs • Cartesian graph bundle • K. Day, A. Al-Ayyoub: Minimal fault diameter for highly • Example 1 • Example 2 resilient product networks (2000) • Edge fault-diameter • Results 1 • Results 2 • M. Xu, J.-M. Xu, X.-M. Hou: Fault diameter of Cartesian • Example 3 • Future Work product graphs (2005) • Baniˇ c, Žerovnik: Fault-diameter of Cartesian graph bundles (2006), Edge fault-diameter of Cartesian product of graphs (2007), Fault-diameter of Cartesian product of graphs (2008) 7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 3 / 12

Cartesian product of graphs • Introduction Definition 2. Let G 1 and G 2 be graphs. The Cartesian product of • Previous work on (edge) fault-diameters graphs G 1 and G 2 , G = G 1 � G 2 , is defined on the vertex set • Cartesian product of graphs V ( G 1 ) × V ( G 2 ) . Vertices ( u 1 , v 1 ) and ( u 2 , v 2 ) are adjacent if • Cartesian graph bundle either u 1 u 2 ∈ E ( G 1 ) and v 1 = v 2 or v 1 v 2 ∈ E ( G 2 ) and u 1 = u 2 . • Example 1 • Example 2 • Edge fault-diameter • Results 1 • Results 2 • Example 3 • Future Work 7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 4 / 12

Cartesian graph bundle • Introduction Definition 3. Let B and F be graphs. A graph G is a Cartesian • Previous work on (edge) fault-diameters graph bundle with fibre F over the base graph B if there is a graph • Cartesian product of graphs map p : G → B such that for each vertex v ∈ V ( B ) , p − 1 ( { v } ) is • Cartesian graph bundle isomorphic to F , and for each edge e = uv ∈ E ( B ) , p − 1 ( { e } ) is • Example 1 • Example 2 isomorphic to F � K 2 . • Edge fault-diameter • Results 1 • Results 2 • The mapping p is also called the projection (of the bundle G • Example 3 • Future Work to its base B ). 7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 5 / 12

Cartesian graph bundle • Introduction Definition 3. Let B and F be graphs. A graph G is a Cartesian • Previous work on (edge) fault-diameters graph bundle with fibre F over the base graph B if there is a graph • Cartesian product of graphs map p : G → B such that for each vertex v ∈ V ( B ) , p − 1 ( { v } ) is • Cartesian graph bundle isomorphic to F , and for each edge e = uv ∈ E ( B ) , p − 1 ( { e } ) is • Example 1 • Example 2 isomorphic to F � K 2 . • Edge fault-diameter • Results 1 • Results 2 • The mapping p is also called the projection (of the bundle G • Example 3 • Future Work to its base B ). • We say an edge e ∈ E ( G ) is degenerate if p ( e ) is a vertex. Otherwise we call it nondegenerate . • Note that each edge e = uv ∈ E ( B ) naturally induces an isomorphism ϕ e : p − 1 ( { u } ) → p − 1 ( { v } ) between two fibres. 7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 5 / 12

Example 1 • Introduction • Previous work on (edge) fault-diameters • Cartesian product of graphs • Cartesian graph bundle • Example 1 • Example 2 • Edge fault-diameter • Results 1 • Results 2 • Example 3 • Future Work Figure 1: Nonisomorphic bundles Let F = K 2 and B = C 3 . On Figure 1 we see two nonisomorphic bundles with fibre F over the base graph B . Informally, one can say that bundles are "twisted products". 7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 6 / 12

Example 2 • Introduction • Previous work on (edge) fault-diameters • Cartesian product of graphs • Cartesian graph bundle �� �� � � � � • Example 1 • Example 2 • Edge fault-diameter • Results 1 Figure 2: Twisted torus: Cartesian graph bundle with fibre C 4 over • Results 2 • Example 3 base C 4 . • Future Work It is less known that graph bundles also appear as computer topologies. A well known example is the twisted torus on Figure 2. Cartesian graph bundle with fibre C 4 over base C 4 is the ILIAC IV architecture. 7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 7 / 12

Edge fault-diameter • Introduction Definition 4. The edge-connectivity of a graph G , λ ( G ) , is the • Previous work on (edge) fault-diameters minimum cardinality over all edge-separating sets in G . A graph G • Cartesian product of graphs is said to be k -edge connected , if λ ( G ) ≥ k . • Cartesian graph bundle • Example 1 • Example 2 • Edge fault-diameter • Results 1 • Results 2 • Example 3 • Future Work 7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 8 / 12

Edge fault-diameter • Introduction Definition 4. The edge-connectivity of a graph G , λ ( G ) , is the • Previous work on (edge) fault-diameters minimum cardinality over all edge-separating sets in G . A graph G • Cartesian product of graphs is said to be k -edge connected , if λ ( G ) ≥ k . • Cartesian graph bundle • Example 1 • Example 2 • Edge fault-diameter • Results 1 Definition 5. Let G be a k -edge connected graph and 0 ≤ a < k . • Results 2 Then we define the a -edge fault-diameter of G as • Example 3 • Future Work ¯ D a ( G ) = max { d ( G \ X ) | X ⊆ E ( G ) , | X | = a } . • Note that ¯ D a ( G ) is the largest diameter among subgraphs of G with a edges deleted, hence ¯ D 0 ( G ) is just the diameter of G , d ( G ) . 7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 8 / 12

Results 1 • Introduction Theorem 1. Let F and B be k F -edge connected and k B -edge • Previous work on (edge) fault-diameters connected graphs respectively, and G a Cartesian graph bundle • Cartesian product of graphs with fibre F over the base graph B . Let λ ( G ) be the • Cartesian graph bundle edge-connectivity of G . Then λ ( G ) ≥ k F + k B . • Example 1 • Example 2 • Edge fault-diameter • Results 1 • Results 2 • Example 3 • Future Work 7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 9 / 12

Results 1 • Introduction Theorem 1. Let F and B be k F -edge connected and k B -edge • Previous work on (edge) fault-diameters connected graphs respectively, and G a Cartesian graph bundle • Cartesian product of graphs with fibre F over the base graph B . Let λ ( G ) be the • Cartesian graph bundle edge-connectivity of G . Then λ ( G ) ≥ k F + k B . • Example 1 • Example 2 • Edge fault-diameter • Results 1 • Results 2 Corollary 2. Let G 1 and G 2 be k 1 and k 2 -edge connected graphs, • Example 3 • Future Work respectively. Then the Cartesian product G 1 � G 2 is at least ( k 1 + k 2 ) -edge connected. 7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 9 / 12

Results 2 • Introduction Theorem 3. Let F and B be k F -edge connected and k B -edge • Previous work on (edge) fault-diameters connected graphs respectively, 0 ≤ a < k F , 0 ≤ b < k B , and G • Cartesian product of graphs a Cartesian bundle with fibre F over the base graph B . Then • Cartesian graph bundle • Example 1 D a + b +1 ( G ) ≤ ¯ ¯ D a ( F ) + ¯ • Example 2 D b ( B ) + 1 . • Edge fault-diameter • Results 1 • Results 2 • Example 3 • Future Work 7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 10 / 12

Example 3 • Introduction Let G = K 2 � K 2 , see Figure 3. G is a graph bundle with fiber • Previous work on (edge) fault-diameters F = K 2 over the base graph B = K 2 . Then for a = b = 0 we • Cartesian product of graphs have • Cartesian graph bundle ¯ D a + b +1 ( G ) = 3 , • Example 1 • Example 2 D b ( B ) + ¯ ¯ • Edge fault-diameter D a ( F ) + 1 = 1 + 1 + 1 = 3 . • Results 1 • Results 2 • Example 3 • Future Work Figure 3: G = K 2 � K 2 with one faulty link. 7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 11 / 12