A Fast f(r,k+1)/kf(r,k+1)/k-Diagnosis for Interconnection Networks Under MM Model

• Published in: IEEE transactions on parallel and distributed systems Vol. 33; no. 7; pp. 1593 - 1604
• Main Authors: Huang, Yanze, Lin, Limei, Hsieh, Sun-Yuan
• Format: Journal Article
• Language: English
• Published: IEEE 01.07.2022
• Subjects: <inline-formula xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> <tex-math notation="LaTeX"> t/k</tex-math> <mml:math> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>/</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> <inline-graphic xlink:href="lin-ieq24-3122440.gif" xlink:type="simple"/> </inline-formula>-diagnosability; Computational modeling; Cyberspace; Data centers; Endogenous security; fault diagnosis; Hypercubes; interconnection networks; MM model; Multiprocessing systems; Program processors; reliability; Security
• ISSN: 1045-9219; 1558-2183
• DOI: 10.1109/TPDS.2021.3122440

Cyberspace is not a "vacuum space", and it is normal that there are inevitable viruses and worms in cyberspace. Cyberspace security threats stem from the problem of endogenous security, which is caused by the incompleteness of theoretical system and technology of the information field itself. Thus it is impossible and unnecessary for us to build an "aseptic" cyberspace. On the contrast, we must focus on improving the "self-immunity" of network. Literally, endogenous security is an endogenous effect from its own structural factors rather than external ones. The <inline-formula><tex-math notation="LaTeX">t/k</tex-math> <mml:math><mml:mrow><mml:mi>t</mml:mi><mml:mo>/</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="lin-ieq2-3122440.gif"/> </inline-formula>-diagnosis strategy plays a very important role in measuring endogenous network security without prior knowledge, which can significantly enhance the self-diagnosing capability of network. As far as we know, few research involves <inline-formula><tex-math notation="LaTeX">t/k</tex-math> <mml:math><mml:mrow><mml:mi>t</mml:mi><mml:mo>/</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="lin-ieq3-3122440.gif"/> </inline-formula>-diagnosis algorithm and <inline-formula><tex-math notation="LaTeX">t/k</tex-math> <mml:math><mml:mrow><mml:mi>t</mml:mi><mml:mo>/</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="lin-ieq4-3122440.gif"/> </inline-formula>-diagnosability of interconnection networks under MM* model. In this article, we propose a fast <inline-formula><tex-math notation="LaTeX">f(r,k+1)/k</tex-math> <mml:math><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="lin-ieq5-3122440.gif"/> </inline-formula>-diagnosis algorithm of complexity <inline-formula><tex-math notation="LaTeX">O(Nr^2)</tex-math> <mml:math><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="lin-ieq6-3122440.gif"/> </inline-formula>, say <inline-formula><tex-math notation="LaTeX">G</tex-math> <mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="lin-ieq7-3122440.gif"/> </inline-formula>MIS<inline-formula><tex-math notation="LaTeX">k</tex-math> <mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="lin-ieq8-3122440.gif"/> </inline-formula>DIAGMM*, for a general <inline-formula><tex-math notation="LaTeX">r</tex-math> <mml:math><mml:mi>r</mml:mi></mml:math><inline-graphic xlink:href="lin-ieq9-3122440.gif"/> </inline-formula>-regular network <inline-formula><tex-math notation="LaTeX">G</tex-math> <mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="lin-ieq10-3122440.gif"/> </inline-formula> under MM* model by designing a 0-comparison subgraph <inline-formula><tex-math notation="LaTeX">M_0(G)</tex-math> <mml:math><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="lin-ieq11-3122440.gif"/> </inline-formula>, where <inline-formula><tex-math notation="LaTeX">N</tex-math> <mml:math><mml:mi>N</mml:mi></mml:math><inline-graphic xlink:href="lin-ieq12-3122440.gif"/> </inline-formula> is the size of <inline-formula><tex-math notation="LaTeX">G</tex-math> <mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="lin-ieq13-3122440.gif"/> </inline-formula>. We determine that the <inline-formula><tex-math notation="LaTeX">t/k</tex-math> <mml:math><mml:mrow><mml:mi>t</mml:mi><mml:mo>/</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="lin-ieq14-3122440.gif"/> </inline-formula>-diagnosability <inline-formula><tex-math notation="LaTeX">(t(G)/k)^M</tex-math> <mml:math><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>M</mml:mi></mml:msup></mml:math><inline-graphic xlink:href="lin-ieq15-3122440.gif"/> </inline-formula> of <inline-formula><tex-math notation="LaTeX">G</tex-math> <mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="lin-ieq16-3122440.gif"/> </inline-formula> under MM* model is <inline-formula><tex-math notation="LaTeX">f(r,k+1)</tex-math> <mml:math><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="lin-ieq17-3122440.gif"/> </inline-formula> by <inline-formula><tex-math notation="LaTeX">G</tex-math> <mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="lin-ieq18-3122440.gif"/> </inline-formula>MIS<inline-formula><tex-math notation="LaTeX">k</tex-math> <mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="lin-ieq19-3122440.gif"/> </inline-formula>DIAGMM* algorithm. Moreover, we establish the <inline-formula><tex-math notation="LaTeX">(t(G)/k)^M</tex-math> <mml:math><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>M</mml:mi></mml:msup></mml:math><inline-graphic xlink:href="lin-ieq20-3122440.gif"/> </inline-formula> of some interconnection networks under MM* model, including BC networks, <inline-formula><tex-math notation="LaTeX">(n,l)</tex-math> <mml:math><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>l</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="lin-ieq21-3122440.gif"/> </inline-formula>-star graph networks, and data center network DCells. Finally, we compare <inline-formula><tex-math notation="LaTeX">(t(G)/k)^M</tex-math> <mml:math><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>M</mml:mi></mml:msup></mml:math><inline-graphic xlink:href="lin-ieq22-3122440.gif"/> </inline-formula> with diagnosability, conditional diagnosability, pessimistic diagnosability, extra diagnosability, and good-neighbor diagnosability under MM* model. It can be seen that <inline-formula><tex-math notation="LaTeX">(t(G)/k)^M</tex-math> <mml:math><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>M</mml:mi></mml:msup></mml:math><inline-graphic xlink:href="lin-ieq23-3122440.gif"/> </inline-formula> is greater than other fault diagnosabilities in most cases.