An algorithm for conditional-fault local diagnosis of multiprocessor systems under the MM⁎ model

Yali Lv, Cheng-Kuan Lin, D. Frank Hsu, Jianxi Fan

Abstract: Diagnosis is a crucial subject for maintaining the reliability of multiprocessor systems. Under the MM⁎ diagnosis model, Sengupta and Dahbura proposed a polynomial-time algorithm with time complexity O(N5)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">N</mi></mrow><mrow is="true"><mn is="true">5</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math> to diagnose a system with N processors. In this paper, we propose a (α,β)<math><mo stretchy="false" is="true">(</mo><mi is="true">α</mi><mo is="true">,</mo><mi is="true">β</mi><mo stretchy="false" is="true">)</mo></math>-trees combination S(u,X,α,β)<math><mi is="true">S</mi><mo stretchy="false" is="true">(</mo><mi is="true">u</mi><mo is="true">,</mo><mi is="true">X</mi><mo is="true">,</mo><mi is="true">α</mi><mo is="true">,</mo><mi is="true">β</mi><mo stretchy="false" is="true">)</mo></math> and give an algorithm to identify the fault or fault-free status of each processor for conditional local diagnosis under the MM⁎ model. According to our results, a connected network with a (α,β)<math><mo stretchy="false" is="true">(</mo><mi is="true">α</mi><mo is="true">,</mo><mi is="true">β</mi><mo stretchy="false" is="true">)</mo></math>-trees combination S(u,X,α,β)<math><mi is="true">S</mi><mo stretchy="false" is="true">(</mo><mi is="true">u</mi><mo is="true">,</mo><mi is="true">X</mi><mo is="true">,</mo><mi is="true">α</mi><mo is="true">,</mo><mi is="true">β</mi><mo stretchy="false" is="true">)</mo></math> for a node u is conditionally locally (α+2β−3)<math><mo stretchy="false" is="true">(</mo><mi is="true">α</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">+</mo><mn is="true">2</mn><mi is="true">β</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">3</mn><mo stretchy="false" is="true">)</mo></math>-diagnosable at node u and the time complexity of our algorithm to diagnose u is O(α2β+αβ2)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">α</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mi is="true">β</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">+</mo><mi is="true">α</mi><msup is="true"><mrow is="true"><mi is="true">β</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math>. As an application, we show that our algorithm can identify the status of each node of n-dimensional star graph Sn<math><msub is="true"><mrow is="true"><mi is="true">S</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub></math> if the faulty node number does not exceed 3n−8<math><mn is="true">3</mn><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">8</mn></math>. Compared with existing algorithms, our algorithm allows more faulty nodes in a multiprocessor system.