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

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

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Edge fault-diameter of Cartesian graph bundles Iztok Bani c, Rija Erve, Janez erovnik FME, University of Maribor, Smetanova 17, Maribor 2000, Slovenia. and FCE, University of Maribor, Smetanova 17, Maribor 2000, Slovenia. and

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7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 1 / 12

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

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SLIDE 2

Introduction

  • 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

7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 2 / 12

Design of large interconnection networks Usual constraints:

  • each processor can be connected to a limited number of
  • ther processors
  • the delays in communication must not be too long

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.

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SLIDE 3

Previous work on (edge) fault-diameters

  • 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

7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 3 / 12

  • M. Krishnamoortthy, B. Krishnamurty: Fault diameter of

interconnection networks (1987)

  • K. Day, A. Al-Ayyoub: Minimal fault diameter for highly

resilient product networks (2000)

  • M. Xu, J.-M. Xu, X.-M. Hou: Fault diameter of Cartesian

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)

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SLIDE 4

Cartesian product of graphs

  • 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

7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 4 / 12

Definition 2. Let G1 and G2 be graphs. The Cartesian product of graphs G1 and G2, G = G1G2, is defined on the vertex set

V (G1) × V (G2). Vertices (u1, v1) and (u2, v2) are adjacent if

either u1u2 ∈ E(G1) and v1 = v2 or v1v2 ∈ E(G2) and u1 = u2.

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Cartesian graph bundle

  • 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

7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 5 / 12

Definition 3. Let B and F be graphs. A graph G is a Cartesian graph bundle with fibre F over the base graph B if there is a graph map p : G → B such that for each vertex v ∈ V (B), p−1({v}) is isomorphic to F , and for each edge e = uv ∈ E(B), p−1({e}) is isomorphic to FK2.

  • The mapping p is also called the projection (of the bundle G

to its base B).

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Cartesian graph bundle

  • 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

7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 5 / 12

Definition 3. Let B and F be graphs. A graph G is a Cartesian graph bundle with fibre F over the base graph B if there is a graph map p : G → B such that for each vertex v ∈ V (B), p−1({v}) is isomorphic to F , and for each edge e = uv ∈ E(B), p−1({e}) is isomorphic to FK2.

  • The mapping p is also called the projection (of the bundle G

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 ϕe : p−1({u}) → p−1({v}) between two fibres.

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SLIDE 7

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

7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 6 / 12

Figure 1: Nonisomorphic bundles Let F = K2 and B = C3. 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".

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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
  • Results 2
  • Example 3
  • Future Work

7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 7 / 12

  • Figure 2: Twisted torus: Cartesian graph bundle with fibre C4 over

base C4. 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 C4 over base C4 is the ILIAC IV architecture.

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SLIDE 9

Edge fault-diameter

  • 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

7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 8 / 12

Definition 4. The edge-connectivity of a graph G, λ(G), is the minimum cardinality over all edge-separating sets in G. A graph G is said to be k-edge connected, if λ(G) ≥ k.

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SLIDE 10

Edge fault-diameter

  • 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

7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 8 / 12

Definition 4. The edge-connectivity of a graph G, λ(G), is the minimum cardinality over all edge-separating sets in G. A graph G is said to be k-edge connected, if λ(G) ≥ k. Definition 5. Let G be a k-edge connected graph and 0 ≤ a < k. Then we define the a-edge fault-diameter of G as

¯ Da(G) = max {d(G \ X) | X ⊆ E(G), |X| = a}.

  • Note that ¯

Da(G) is the largest diameter among subgraphs of G with a edges deleted, hence ¯ D0(G) is just the diameter of G, d(G).

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SLIDE 11

Results 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

7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 9 / 12

Theorem 1. Let F and B be kF -edge connected and kB-edge connected graphs respectively, and G a Cartesian graph bundle with fibre F over the base graph B. Let λ(G) be the edge-connectivity of G. Then λ(G) ≥ kF + kB.

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SLIDE 12

Results 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

7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 9 / 12

Theorem 1. Let F and B be kF -edge connected and kB-edge connected graphs respectively, and G a Cartesian graph bundle with fibre F over the base graph B. Let λ(G) be the edge-connectivity of G. Then λ(G) ≥ kF + kB. Corollary 2. Let G1 and G2 be k1 and k2-edge connected graphs,

  • respectively. Then the Cartesian product G1G2 is at least

(k1 + k2)-edge connected.

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SLIDE 13

Results 2

  • 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

7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 10 / 12

Theorem 3. Let F and B be kF -edge connected and kB-edge connected graphs respectively, 0 ≤ a < kF , 0 ≤ b < kB, and G a Cartesian bundle with fibre F over the base graph B. Then

¯ Da+b+1(G) ≤ ¯ Da(F) + ¯ Db(B) + 1.

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SLIDE 14

Example 3

  • 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

7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 11 / 12

Let G = K2K2, see Figure 3. G is a graph bundle with fiber

F = K2 over the base graph B = K2. Then for a = b = 0 we

have

¯ Da+b+1(G) = 3, ¯ Db(B) + ¯ Da(F) + 1 = 1 + 1 + 1 = 3.

Figure 3: G = K2K2 with one faulty link.

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SLIDE 15

Future Work

  • 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

7th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, May 13-15, 2008, Gargnano 12 / 12

  • Let G be a k connected graph and 0 < a < k. Then

¯ Da(G) ≤ Da(G) + 1

  • Mixed fault-diameter of G, D(m,n)(G).

Let G be a k connected graph and 0 < a < k, m + n = a. Then

D(m,n)(G) ≤ D(m−l,n+l)(G), l < m

and

¯ Da(G) ≤ D(m,n)(G) ≤ Da(G) + 1, m = 0