Constructing Completely Independent Spanning Trees in Data Center Network Based on Augmented Cube

• Published in: IEEE transactions on parallel and distributed systems Vol. 32; no. 3; pp. 665 - 673
• Main Authors: Chen, Guo, Cheng, Baolei, Wang, Dajin
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.03.2021; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Augmented cube; Broadcasting; completely independent spanning trees; Computer centers; data center network; Data centers; edge-disjoint hamiltonian paths; Fault tolerance; Fault tolerant systems; Graph theory; hamiltonian cycle; Hypercubes; Nodes; Scalability; Servers
• ISSN: 1045-9219; 1558-2183
• DOI: 10.1109/TPDS.2020.3029654

A set of spanning trees <inline-formula><tex-math notation="LaTeX">T_1,T_2,\ldots, T_k</tex-math> <mml:math><mml:mrow><mml:msub><mml:mi>T</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>...</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="cheng-ieq1-3029654.gif"/> </inline-formula> in a network <inline-formula><tex-math notation="LaTeX">G</tex-math> <mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="cheng-ieq2-3029654.gif"/> </inline-formula> are Completely Independent Spanning Trees (CISTs) if for any two nodes <inline-formula><tex-math notation="LaTeX">u</tex-math> <mml:math><mml:mi>u</mml:mi></mml:math><inline-graphic xlink:href="cheng-ieq3-3029654.gif"/> </inline-formula> and <inline-formula><tex-math notation="LaTeX">v</tex-math> <mml:math><mml:mi>v</mml:mi></mml:math><inline-graphic xlink:href="cheng-ieq4-3029654.gif"/> </inline-formula> in <inline-formula><tex-math notation="LaTeX">V(G)</tex-math> <mml:math><mml:mrow><mml:mi>V</mml:mi><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="cheng-ieq5-3029654.gif"/> </inline-formula>, the paths between <inline-formula><tex-math notation="LaTeX">u</tex-math> <mml:math><mml:mi>u</mml:mi></mml:math><inline-graphic xlink:href="cheng-ieq6-3029654.gif"/> </inline-formula> and <inline-formula><tex-math notation="LaTeX">v</tex-math> <mml:math><mml:mi>v</mml:mi></mml:math><inline-graphic xlink:href="cheng-ieq7-3029654.gif"/> </inline-formula> in any two trees have no common edges and no common internal nodes. CISTs have important applications in data center networks, such as fault-tolerant multi-node broadcasting, fault-tolerant one-to-all broadcasting, reliable broadcasting, secure message distribution, and so on. The augmented cube <inline-formula><tex-math notation="LaTeX">AQ_n</tex-math> <mml:math><mml:mrow><mml:mi>A</mml:mi><mml:msub><mml:mi>Q</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="cheng-ieq8-3029654.gif"/> </inline-formula> is a prominent variant of the well-known hypercube <inline-formula><tex-math notation="LaTeX">Q_n</tex-math> <mml:math><mml:msub><mml:mi>Q</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="cheng-ieq9-3029654.gif"/> </inline-formula>, and having the important property of scalability, and both <inline-formula><tex-math notation="LaTeX">Q_n</tex-math> <mml:math><mml:msub><mml:mi>Q</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math><inline-graphic xlink:href="cheng-ieq10-3029654.gif"/> </inline-formula> and <inline-formula><tex-math notation="LaTeX">AQ_n</tex-math> <mml:math><mml:mrow><mml:mi>A</mml:mi><mml:msub><mml:mi>Q</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="cheng-ieq11-3029654.gif"/> </inline-formula> have been proposed as the underlying structure for a data center network. The data center network based on <inline-formula><tex-math notation="LaTeX">AQ_n</tex-math> <mml:math><mml:mrow><mml:mi>A</mml:mi><mml:msub><mml:mi>Q</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="cheng-ieq12-3029654.gif"/> </inline-formula> is denoted by <inline-formula><tex-math notation="LaTeX">AQDN_n</tex-math> <mml:math><mml:mrow><mml:mi>A</mml:mi><mml:mi>Q</mml:mi><mml:mi>D</mml:mi><mml:msub><mml:mi>N</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="cheng-ieq13-3029654.gif"/> </inline-formula>, and the logic graph of <inline-formula><tex-math notation="LaTeX">AQDN_n</tex-math> <mml:math><mml:mrow><mml:mi>A</mml:mi><mml:mi>Q</mml:mi><mml:mi>D</mml:mi><mml:msub><mml:mi>N</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="cheng-ieq14-3029654.gif"/> </inline-formula> is denoted by <inline-formula><tex-math notation="LaTeX">L\text{-}AQDN_n</tex-math> <mml:math><mml:mrow><mml:mi>L</mml:mi><mml:mtext>-</mml:mtext><mml:mi>A</mml:mi><mml:mi>Q</mml:mi><mml:mi>D</mml:mi><mml:msub><mml:mi>N</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="cheng-ieq15-3029654.gif"/> </inline-formula>. In this article, we study how to construct <inline-formula><tex-math notation="LaTeX">n-1</tex-math> <mml:math><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="cheng-ieq16-3029654.gif"/> </inline-formula> CISTs in <inline-formula><tex-math notation="LaTeX">L\text{-}AQDN_n</tex-math> <mml:math><mml:mrow><mml:mi>L</mml:mi><mml:mtext>-</mml:mtext><mml:mi>A</mml:mi><mml:mi>Q</mml:mi><mml:mi>D</mml:mi><mml:msub><mml:mi>N</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="cheng-ieq17-3029654.gif"/> </inline-formula>. The constructed <inline-formula><tex-math notation="LaTeX">n-1</tex-math> <mml:math><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="cheng-ieq18-3029654.gif"/> </inline-formula> CISTs are optimal in the sense that <inline-formula><tex-math notation="LaTeX">n-1</tex-math> <mml:math><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math><inline-graphic xlink:href="cheng-ieq19-3029654.gif"/> </inline-formula> is the maximally allowed CISTs in <inline-formula><tex-math notation="LaTeX">L\text{-}AQDN_n</tex-math> <mml:math><mml:mrow><mml:mi>L</mml:mi><mml:mtext>-</mml:mtext><mml:mi>A</mml:mi><mml:mi>Q</mml:mi><mml:mi>D</mml:mi><mml:msub><mml:mi>N</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math><inline-graphic xlink:href="cheng-ieq20-3029654.gif"/> </inline-formula>. The correctness of our construction algorithm is proved. It is the first time a direct relationship is established between the dimension of a hypercube-family network and the number of CISTs it can host.