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Packing internally disjoint Steiner paths of data center networks

Wen-Han Zhu, Rong-Xia Hao, Jou-Ming Chang, Jaeun Lee

TL;DR

The maximum number t of edge-disjoint paths P1,P2,…,Pt spanning proved the maximum number t of edge-disjoint paths P1,P2,…,Pt.

Abstract

Let $S\subseteq V(G)$ and $π_{G}(S)$ denote the maximum number $t$ of edge-disjoint paths $P_{1},P_{2},\ldots,P_{t}$ in a graph $G$ such that $V(P_{i})\cap V(P_{j})=S$ for any $i,j\in\{1,2,\ldots,t\}$ and $i\neq j$. If $S=V(G)$, then $π_{G}(S)$ is the maximum number of edge-disjoint spanning paths in $G$. It is proved [Graphs Combin., 37 (2021) 2521-2533] that deciding whether $π_G(S)\geq r$ is NP-complete for a given $S\subseteq V(G)$. For an integer $r$ with $2\leq r\leq n$, the $r$-path connectivity of a graph $G$ is defined as $π_{r}(G)=$min$\{π_{G}(S)|S\subseteq V(G)$ and $|S|=r\}$, which is a generalization of tree connectivity. In this paper, we study the $3$-path connectivity of the $k$-dimensional data center network with $n$-port switches $D_{k,n}$ which has significate role in the cloud computing, and prove that $π_{3}(D_{k,n})=\lfloor\frac{2n+3k}{4}\rfloor$ with $k\geq 1$ and $n\geq 6$.

Packing internally disjoint Steiner paths of data center networks

TL;DR

The maximum number t of edge-disjoint paths P1,P2,…,Pt spanning proved the maximum number t of edge-disjoint paths P1,P2,…,Pt.

Abstract

Let and denote the maximum number of edge-disjoint paths in a graph such that for any and . If , then is the maximum number of edge-disjoint spanning paths in . It is proved [Graphs Combin., 37 (2021) 2521-2533] that deciding whether is NP-complete for a given . For an integer with , the -path connectivity of a graph is defined as min and , which is a generalization of tree connectivity. In this paper, we study the -path connectivity of the -dimensional data center network with -port switches which has significate role in the cloud computing, and prove that with and .
Paper Structure (4 sections, 9 theorems, 10 equations, 10 figures)

This paper contains 4 sections, 9 theorems, 10 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and preparation
  3. The $3$-path-connectivity of data center networks
  4. Conclusion

Key Result

Lemma 2.2

Let $D_{k,n}$ be the $k$-dimensional data center network with $n$-port switches for $k\geq 0$ and $n\geq 2$. Then the following four conditions hold. (1)$D_{k,n}$ is $(n+k-1)$-regular and $\kappa(D_{k,n})=\lambda(D_{k,n})=n+k-1$. (2) For $k\geq 1$, $D_{k,n}$ consists of $t_{k-1,n}+1$ copies of $D_{k

Figures (10)

  • Figure 1: Several DCells
  • Figure 2: The illustration of Case $2$ in Theorem \ref{['thm2']}
  • Figure 3: The illustration of Case $3$ in Theorem \ref{['thm2']}
  • Figure 4: The illustration of Claim $1$ in Theorem \ref{['thm3']}
  • Figure 5:
  • ...and 5 more figures

Theorems & Definitions (17)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 7 more