行政院國家科學委員會專題研究計畫 成果報告
配對合成網路的錯誤診斷研究
計畫類別： 個別型計畫 計畫編號： NSC93-2213-E-009-091- 執行期間： 93 年 08 月 01 日至 94 年 07 月 31 日 執行單位： 國立交通大學資訊科學學系(所) 計畫主持人： 譚建民 計畫參與人員： 楊明堅，徐國晃，滕元翔，施倫閔，吳宙耕 報告類型： 精簡報告 報告附件： 出席國際會議研究心得報告及發表論文 處理方式： 本計畫可公開查詢中 華 民 國 94 年 9 月 26 日
行政院國家科學委員會補助專題研究計畫成果報告
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※ 配對合成網路的錯誤診斷研究
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計畫類別：■個別型計畫 □整合型計畫
計畫編號：NSC93－2213－E－009－091－
執行期間：93 年 8 月 1 日至 94 年 7 月 31 日
計畫成果：本計畫其中之一結果由計畫主持人及博士班研究生共
同發表，已刊登在
2004 年 IEEE Transactions on
Computers 期刊:
P.-L. Lai, J.J.M. Tan, C.-H. Tsai, and L.-H. Hsu “The Diagnosability
of the Matching Composition Network under the Comparison
Diagnosis Model,” IEEE Trans. on Computers, vol. 53, no. 8,
pp.1064-1069 Aug. 2004.
計畫主持人：譚建民
成果報告類型(依經費核定清單規定繳交)：■精簡報告 □完整報告本成果報告包括以下應繳交之附件：
□赴國外出差或研習心得報告一份
□赴大陸地區出差或研習心得報告一份
□出席國際學術會議心得報告及發表之論文各一份
□國際合作研究計畫國外研究報告書一份
執行單位：國立交通大學資訊科學學系
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配對合成網路的錯誤診斷研究
The diagnosability of the matching composition network
計劃編號：NSC93-2213-E-009-091-
計劃期限：93/8/1～94/7/31
主持人：譚建民 國立交通大學資訊科學學系 教授
一、中英文摘要
在 這 個 計 劃 中 ， 我 們 研 究 multiprocessor system 的錯誤診斷問 題(fault diagnosis problem)，關於多 處理機系統的 錯誤自我診斷問題，在文 獻 中 已 有 幾 個 不 同 的 模 式 被 提 出 。 Preparata, Metze and Chien 三人最早 提 出 一 種 構 想 及 模 式 。 現 在 稱 為 PMC-Model。在此模式下，兩個相連接的 processor 可以互相偵測是否 faulty。 Maeng and Malek 在 之 後 提 出 一 種 comparison model 稱為 MM-model。他們 對 錯 誤 診 斷 的 基 本 構 想 是 由 一 個 processor 向相鄰的兩個 processors， 送出信號，然後由回收的訊號，比較並 判斷是否有 fault。為了要收集到最多的 資料以供錯誤診斷，在 MM*-model 下, 規定任一個 processor 都對其所有相鄰 的兩個 processors 作偵測及比較。 錯 誤 診 斷 在 IEEE Trans. on Computers 及 IEEE Trans. on Parallel and Distributed Computing 已有很多 文獻研究。我們近年也在這個領域作了 一 些 研 究 , 並 投 稿 至 IEEE Trans. on Computers ， 已 有 一 篇 在 做 最 後 revision。 在這個計畫中，我們研究一種配對 合 成 網 路 matching composition network (MCN) 的 診 斷 能 力 (diagnosability)，在 MM*-comparison model 下 ， 配 對 合 成 網 路 的 診 斷 能 力 (diagnosability) ，與它的組成成分之 間的 diagnosability and connectivity 有一定關係，我們研究其間的關係，可 將文獻中有關各種 cube 如 hypercube, crossed cube,twisted cube and Mobius cube 的 diagnosability 研究結果有一 個統一的解釋。我們已有一些研究成果, 撰文投稿 。 對 PMC- model 下的錯誤診斷也有所 研 究 。 我 們 定 義 出 一 種 ” strongly t-diagnosable systems ” 及 ” conditional diagnosability ” 的 觀 念。目前正在探究，期能繼續進行成果 收集。 關鍵詞：錯誤診斷(fault diagnosis)， Comparison-model, MM*-model, PMC-model 診 斷 能 力 (diagnosability).3
英文摘要
We propose to study the diagnosis problem of multiprocessor system. For the purpose of
self-diagnosis of a given system, several different models have been proposed in literature. Preparata, Metze, and Chien first introduced a model, so called PMC-model, for system level diagnosis in multiprocessor systems. In this model, it is assumed that a processor can test the faulty or fault-free status of another processor.
The comparison model, called MM model, proposed by Maeng and Malek, is considered to be another practical approach for fault diagnosis in multiprocessor systems. In this approach, the diagnosis is carried out by sending the same testing task to a pair {u, v} of processors and comparing their responses. The comparison is performed by a third processor w that has direct communication links to both processors u and v. The third processor w is called a comparator of u and v.
If the comparator is fault-free, a disagreement between the two responses is an indication of the existence of a faulty processor. To gain as much knowledge as possible about the faulty status of the system, it was assumed that a comparison is performed by each processor for each pair of distinct neighbors with which it can communicate directly. This special case of MM-model is referred to as the MM*-model. Sengupta and Dahbura studied the MM-model and the MM*-model, gave a characterization of diagnosable systems under the comparison approach, and proposed a polynomial time algorithm to determine faulty processors under MM*-model.
In this proposal, we consider the diagnosability of a family of networks, called the Matching Composition Network (MCN); two components are connected by a perfect matching. The diagnosability of MCN under the comparison model is shown to be one larger than that of the component, provided some connectivity constraints are satisfied. Applying our result, the diagnosability of the Hypercube Qn, the Crossed cube CQn, the Twisted cube TQn, and the Möbius cube MQn can all be proved to be n, for n ≥ 4.
We shall also study the diagnosis problem under PMC-model. We define some new concepts called “strongly t-diagnosable system” and “conditional diagnosability”, we are currently working on this subject. Keywords: fault diagnosis, comparison model, MM*-model, PMC-model, diagnosability.
二﹑計劃緣由及目的
With the rapid development of technology, the need for high-speed parallel processing systems has been continuously increasing. The reliability of the processors in parallel computing systems is therefore becoming an important issue. In order to maintain the reliability of a system, whenever a processor (node) is found faulty it should be replaced by a fault-free processor (node). The process of identifying all the faulty nodes is called the diagnosis of the system. The maximum number of faulty nodes that the system can guarantee to identify is called the diagnosability of the system.
We consider the diagnosability of the system under the comparison model, proposed by Malek and Maeng [16,17]. The
4 diagnosability of some well-known interconnection networks under the comparison model has been investigated. For example, Wang [21,22] showed that the diagnosability of an n-dimensional hypercube Qn is n for n ≥ 5, and the diagnosability of an n-dimensional enhanced hypercube is n+1 for n ≥6. Fan [12] proved that the diagnosability of an n-dimensional crossed cube is n for n ≥ 4. Araki proposed that the k-ary r-dimensional butterfly network BF(k, r) is 2k-diagnosable for k ≥2 and r ≥ 5. Besides, the diagnosability of the Hypercubes, the Crossed cubes, and the Möbius cubes under the PMC diagnostic model were also studied in [2,10,11,14].
We study the diagnosability of a family of interconnection networks, called the Matching Composition Networks (MCN), which can be recursively constructed. MCN includes many well-known interconnection networks as special cases, such as the Hypercube Qn, the Crossed cube CQn, the Twisted cube TQn, and the Möbius cube MQn. Basically, MCN and these mentioned cubes are all constructed from two graphs G1 and G2 with the same number of nodes, by adding a perfect matching between the nodes of G1 and G2. We shall call these two graphs G1 and G2 as the components of MCN.
Our main problem is the following. Suppose that the number of nodes in each component is at least t+2, the order of each node in Gi is t, and the connectivity of Gi is also t, i=1,2. We shall prove that the diagnosability of MCN constructed from G1 and G2 is t+1 under the comparison model, for t ≥2. In other words, the diagnosability of MCN is increased by one as compared with those of the components. Using our result, it is straightforward to see that the
diagnosability of the Hypercube Qn, the Crossed cube CQn, the Twisted cube TQn, and the Möbius cube MQn are n for n ≥4. Some of these particular applications are previously known results [12,22], using rather lengthy proofs. Our approach unifies these special cases and our proof is much simpler. The diagnosability of the Twisted cube TQn and the Möbius cube MQn, as far as we know, are not yet resolved.
三﹑研究方法與成果
近年我們的研究領域集中在連結網路上， 我們對於各種著名的網路架構有深入的研 究。可以將我們所知的知識應用在錯誤診 斷(fault diagnosis)研究上。連結網路中 的觀念如 connectivity, conditional connectivity, hypercube and cube family 都與錯誤診斷的研究息息相關。藉 由這些關念，我定義出 conditional diagnosability 及 strongly t-diagnosable system 等新名詞，已有新 的研究成果投稿，並還在撰寫論文繼續努 力。我們的實驗室每週定時研討如下： 一、 收集文獻 我們會藉由網路、圖書館、及國內外研討 會等，來收集相關的文獻。 二、 探討文獻及發現問題 將收集到的文獻作一個初步的探討後，由 計畫中的成員分工合作將文獻做進一步的 分析研究後，每週定期報告其文獻的內 容，以及分析其文獻，進而從文獻中發現 可以做進一步研究之問題，由主持人帶領 成員選定研究之主題。 三、 解決問題、程式撰寫、及定理證明由 老師帶領博士班學生，並由博士班學生帶 領碩士班學生共同研究，並解決其問題， 在研究的過程中，常常需要撰寫程式來輔5 助定理的證明。我們的實驗室已自行發展 出一些軟體程式供測試例子。最後將共同 研究之成果整理好，完成完整的一篇論文。 四、 成果發表 近年我們已有多篇論文被國際知名期刊刊 登。而本計畫結果已刊登在 2004 年 IEEE Transactions on Computers 期刊:
P.-L. Lai, J.J.M. Tan, C.-H. Tsai, and L.-H. Hsu “The Diagnosability of the Matching Composition Network under the Comparison Diagnosis Model,” IEEE Trans. on Computers, vol. 53, no. 8, pp.1064-1069 Aug. 2004.
四、結論與討論
本計畫在主持人的帶領之下，每週定 時討論與報告所收集的資料，並且分析和 比較各種方法的優點及缺點，在多方面的 不斷討論下，使我們對一些連結網路的診 斷能力特性的問題與解決方式有了更清楚 的了解。 由於我們對於連結網路的診斷能力相 關問題，先前已經有了相當良好的基礎及 經驗，所以為本計劃的執行奠定了良好的 根基，且能順利完成預定的研究進度。每 個參與的研究人員都能對診斷能力的特性 有充分的了解，在學術上可以對連結網路 發展出新的診斷能力的特性，並且可以比 較分析各種不同的連結網路，它們之間診 斷能力的特性。希望我們所研究的這些相 關問題能提升有關這方面的領域。而我們 研究所得的經驗、知識以及一些尚未解決 或仍可以發揮之處，希望可做為日後更深 入的研究。五、參考文獻
[1] T. Araik and Y. Shibata, “Diagnosability of Butterfly Networks under the Comparison Approach,” IEICE Trans. Fundamentals, vol. E85-A, no. 5, pp.1, 152-1,160, May 2002.
[2] J. R. Armstrong and F. G. Gray, “Fault Diagnosis in a Boolean n Cube Array of Multiprocessors,” IEEE Trans. on Computers, vol. 30, no. 8, pp. 587-590, Aug. 1981.
[3] C. P. Chang, J. N. Wang, and L. H. Hsu, “Topological Properties of Twisted Cubes,” Information Sciences, vol. 113, Issue. 1-2, pp. 147-167, Jan. 1999.
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[12] J. Fan, “Diagnosability of Crossed Cubes under the Comparison Diagnosis Model,” IEEE Trans. on Parallel and Distributed Systems, vol. 13, no. 7, pp. 687-692, Jul. 2002.
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[15] P. Kulasinghe, “Connectivity of the Crossed Cube,” Information Processing Letters, vol. 61, Issue. 4, pp. 221-226, Feb. 1997.
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[17] M. Malek, “A Comparison Connection Assignment for Diagnosis of Multiprocessor Systems,” Proc. 7th Int’l Symp. Computer Architecture, pp. 31-35,1980.
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7 [21] D. Wang, “Diagnosability of Enhanced Hypercubes,” IEEE Trans. on Computers, vol.43, no.9, pp. 1,054-1,061, Sep. 1994.
[22] D. Wang, “Diagnosability of Hypercubes and Enhanced Hypercubes under the Comparison Diagnosis Model,” IEEE Trans. on Computers, vol.48, no.12, pp.1,369-1,374, Dec. 1999.
