[Paper Review] Super edge-connectivity and matching preclusion of data center networks

Huazhong Lü, Tingzeng Wu|arXiv (Cornell University)|Jul 13, 2018

Interconnection Networks and Systems4 citations

TL;DR

This paper establishes the super edge-connectivity and matching preclusion properties of DCell data center networks, proving that $D_{k,n}$ is super-λ for $k \geq 2, n \geq 2$, super-λ₂ for $k \geq 3, n \geq 2$ or $k=2,n=2$, and super-λ₃ for $k \geq 4, n \geq 3$. It further determines the matching preclusion number $mp(D_{k,n}) = n+k-1$ and conditional matching preclusion number $mp_1(D_{k,n}) = 2n+2k-5$ for $k \geq 3, n \geq 3$, showing $D_{k,n}$ is super matched and conditionally super matched under specific conditions.

ABSTRACT

Edge-connectivity is a classic measure for reliability of a network in the presence of edge failures. $k$-restricted edge-connectivity is one of the refined indicators for fault tolerance of large networks. Matching preclusion and conditional matching preclusion are two important measures for the robustness of networks in edge fault scenario. In this paper, we show that the DCell network $D_{k,n}$ is super-$λ$ for $k\geq2$ and $n\geq2$, super-$λ_2$ for $k\geq3$ and $n\geq2$, or $k=2$ and $n=2$, and super-$λ_3$ for $k\geq4$ and $n\geq3$. Moreover, as an application of $k$-restricted edge-connectivity, we study the matching preclusion number and conditional matching preclusion number, and characterize the corresponding optimal solutions of $D_{k,n}$. In particular, we have shown that $D_{1,n}$ is isomorphic to the $(n,k)$-star graph $S_{n+1,2}$ for $n\geq2$.

Motivation & Objective

• To analyze the $k$-restricted edge-connectivity of DCell networks $D_{k,n}$ for reliability and fault tolerance.
• To determine the matching preclusion number $mp(D_{k,n})$ and conditional matching preclusion number $mp_1(D_{k,n})$ as measures of robustness under edge failures.
• To characterize optimal matching preclusion sets and establish whether $D_{k,n}$ is super matched or conditionally super matched.
• To clarify the structural relationship between $D_{1,n}$ and the $(n+1,2)$-star graph $S_{n+1,2}$, showing isomorphism for $n \geq 2$.

Proposed method

• Proved $D_{k,n}$ is super-λ for $k \geq 2, n \geq 2$ using edge-cut isolation properties and connectivity bounds.
• Established super-λ₂ and super-λ₃ properties via $k$-restricted edge-cut analysis and minimum edge-cut characterization.
• Used $k$-restricted edge-connectivity results to derive bounds on matching preclusion numbers, leveraging $\lambda_k(G) = \xi_k(G)$ for $\lambda_k$-optimal graphs.
• Applied Theorems 4.2–4.6 to link edge-connectivity properties to matching preclusion, using $\alpha(G)$ (independence number) bounds to exclude non-trivial obstruction sets.
• Defined and analyzed semi-trivial matching preclusion sets for $D_{1,n}$, showing their role in optimal solutions for even $n$.
• Used structural properties of $D_{k,n}$, including regularity ($r = n+k-1$) and component behavior after edge deletion, to prove super-matching properties.

Experimental results

Research questions

• RQ1 For which values of $k$ and $n$ is the DCell network $D_{k,n}$ super-λ, super-λ₂, or super-λ₃?
• RQ2 What is the exact value of the matching preclusion number $mp(D_{k,n})$ for $D_{k,n}$, and when is the network super matched?
• RQ3 What is the conditional matching preclusion number $mp_1(D_{k,n})$, and under what conditions is $D_{k,n}$ conditionally super matched?
• RQ4 How does the structure of $D_{1,n}$ relate to the $(n+1,2)$-star graph $S_{n+1,2}$, and what implications does this isomorphism have for matching preclusion?
• RQ5 Can non-trivial optimal matching preclusion sets exist for $D_{k,n}$, and under what conditions are all optimal sets trivial or semi-trivial?

Key findings

• The DCell network $D_{k,n}$ is super-λ for all $k \geq 2$ and $n \geq 2$, indicating high fault tolerance under edge failures.
• $D_{k,n}$ is super-λ₂ for $k \geq 3, n \geq 2$ or $k=2, n=2$, and super-λ₃ for $k \geq 4, n \geq 3$, confirming refined robustness against multiple edge failures.
• The matching preclusion number is $mp(D_{k,n}) = n + k - 1$ for $k \geq 2, n \geq 2$, and $D_{k,n}$ is super matched, meaning all optimal solutions are trivial.
• For $k \geq 3, n \geq 3$, the conditional matching preclusion number is $mp_1(D_{k,n}) = 2n + 2k - 5$, and $D_{k,n}$ is conditionally super matched.
• For $k \geq 2, n = 2$, the conditional matching preclusion number is $mp_1(D_{k,2}) = 2n + 2k - 4$, and $D_{k,2}$ is conditionally super matched for $k \geq 3$.
• $D_{1,n}$ is isomorphic to the $(n+1,2)$-star graph $S_{n+1,2}$ for $n \geq 2$, and $mp(D_{1,n}) = n$, with super-matching behavior depending on parity of $n$.