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Paired 2-disjoint path covers of Bcube under the partitioned edge fault model

Qingqiong Cai, Wenjing Zhang

TL;DR

This work addresses the problem of embedding paired $2$-disjoint path covers ($2$-DPC) in BCube under partitioned edge faults. It develops the $l$-PEF fault model and proves that for all $n\ge 4$, $k\ge 0$, the BC$_{n,k}$ graph remains Hamiltonian-connected and supports paired $2$-DPC for any two fault-free source-sink pairs when fault counts satisfy $\sum_{i=1}^k f^i \le \sum_{i=1}^k n^i - 5k$ or $\sum_{i=1}^k f^i \le \sum_{i=1}^k \left\lceil \frac{n^i - 1}{2} \right\rceil (n - 1) - k n$. This establishes exponential fault tolerance for $2$-DPC embedding in BCube and demonstrates the efficacy of the $PEF$ framework in enhancing large-scale edge fault resilience in DCN topologies.

Abstract

BCube network, as a typical distributed data center network topology, has significant advantages in fault tolerance, load balancing, and efficient routing due to its unique hierarchical structure. In terms of efficient routing, paired many-to-many m-disjoint path cover (m-DPC) plays an important role in message passing. To explore the capability of BCube in constructing paired many-to-many m-DPCs, this paper investigates whether arbitrary 2-DPC paths can be successfully constructed under the partitioned edge fault (PEF) model, especially in the case of a large number of link failures. Through this investigation, the paper aims to address the network fault tolerance issues related to path embedding problems. Theoretical proofs show that under the partitioned edge fault model, BCube exhibits exponential fault tolerance for constructing 2-DPC paths. This study, on one hand, expands the application of the partitioned edge fault model, and on the other hand, contributes to enhancing BCube's ability to achieve large-scale edge fault tolerance.

Paired 2-disjoint path covers of Bcube under the partitioned edge fault model

TL;DR

This work addresses the problem of embedding paired -disjoint path covers (-DPC) in BCube under partitioned edge faults. It develops the -PEF fault model and proves that for all , , the BC graph remains Hamiltonian-connected and supports paired -DPC for any two fault-free source-sink pairs when fault counts satisfy or . This establishes exponential fault tolerance for -DPC embedding in BCube and demonstrates the efficacy of the framework in enhancing large-scale edge fault resilience in DCN topologies.

Abstract

BCube network, as a typical distributed data center network topology, has significant advantages in fault tolerance, load balancing, and efficient routing due to its unique hierarchical structure. In terms of efficient routing, paired many-to-many m-disjoint path cover (m-DPC) plays an important role in message passing. To explore the capability of BCube in constructing paired many-to-many m-DPCs, this paper investigates whether arbitrary 2-DPC paths can be successfully constructed under the partitioned edge fault (PEF) model, especially in the case of a large number of link failures. Through this investigation, the paper aims to address the network fault tolerance issues related to path embedding problems. Theoretical proofs show that under the partitioned edge fault model, BCube exhibits exponential fault tolerance for constructing 2-DPC paths. This study, on one hand, expands the application of the partitioned edge fault model, and on the other hand, contributes to enhancing BCube's ability to achieve large-scale edge fault tolerance.
Paper Structure (6 sections, 5 theorems, 1 equation, 8 figures, 1 table)

This paper contains 6 sections, 5 theorems, 1 equation, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Terminology and Notation
  4. Bcube
  5. Paired 2-disjoint path covers in Bcube
  6. Summary

Key Result

lemma 1

For $n \geq 4, k \geq 0$, let $F$ be any faulty set of $BC_{n,k}$, where the nodes in $F$ satisfy the following conditions: (1) $f_0 = 0$ (2) For $1 \leq i \leq k$: In the absence of ambiguity, the above conditions will be referred to as l-PEF. Assume that in any subgraph $BC[m]$, there exist two pairwise disjoint paths, denoted by $P_1$ and $P_2$. Then, in $E(P_1) \cup E(P_2)$, one can find a pa

Figures (8)

  • Figure 1: $BC_{3,1}$
  • Figure 2: Suppose $s_1, s_2, t_1, t_2$ belong to the same subgraph $BC[l_{s_1}]$
  • Figure 3: Suppose $s_1, s_2, t_1,$ belong to the same subgraph $BC[l_{s_1}]$
  • Figure 4: Suppose $t_1, s_2, t_2,$ belong to the same subgraph $BC[l_{s_2}]$
  • Figure 5: $l_{t_1} = l_{t_2}$, and $4 \leq n \leq 9$
  • ...and 3 more figures

Theorems & Definitions (10)

  • definition 1
  • definition 2
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • theorem 1
  • theorem 2
  • proof