在本論文中，我們對於(n, k)星狀圖設計容錯繞徑演算法，我們應用原本用於n星狀圖上繞徑的想法，收集有限制的全域資訊(limited global information)。首先，利用修改過的cycle pattern來建立機率式安全向量；然後，我們的演算法根據機率式安全向量決定路徑。為了增進壞掉點增加時的效能，我們動態調整繞徑演算法判斷是否通行時所應用的門檻值。繞徑演算法的效能評判標準為所決定的路徑的長度愈短愈好。與距離優先搜尋方法、安全層級方法比較，在模擬實驗中，我們的方法達到最好的效能。

In this thesis, we focus on the design of the fault-tolerant routing algorithm for the (n, k)-star graph. We apply the idea of collecting the limited global information used for routing on the n-star graph to the (n, k)-star graph. First, we build the probabilistic safety vector (PSV) with modified cycle patterns. Then, our routing algorithm decides the fault-free routing path with the help of PSV. In order to improve the routing performance with more faulty nodes, we dynamically assign the threshold for our routing algorithm. The performance is judged by the average length of routing paths. Compared with distance first search and safety level, we get the best performance in the simulations.

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 The n-star and (n, k)-star Graphs . . . . . . . . . . . . . . . . . . . . 3
2.2 Permutation Cycles of (n, k)-star Graphs . . . . . . . . . . . . . . . . 5
2.3 The Distance Computation of (n, k)-star Graphs . . . . . . . . . . . . 8
Chapter 3. Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1 Routing on Sn,k with Distance First Search (DFS) . . . . . . . . . . . 12
3.2 Routing on Sn with Safety Levels (SL) . . . . . . . . . . . . . . . . . 13
3.3 Routing on Sn with Probabilistic Safety Vectors (PSV) . . . . . . . . 14
Chapter 4. Fault-Tolerant Routing on the (n, k)-star Graph . . . . . 19
4.1 The Modified Cycle Pattern of the (n, k)-star Graph . . . . . . . . . . 19
4.2 The Computation of PSV in Sn,k . . . . . . . . . . . . . . . . . . . . 22
4.3 Routing on Sn,k with Probabilistic Safety Vectors . . . . . . . . . . . 24
Chapter 5. Experimental Results . . . . . . . . . . . . . . . . . . . . . . 27
5.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2 Dynamic Threshold Assignment for the PSV . . . . . . . . . . . . . . 28
Page
5.3 The Length of the Probabilistic Safety Vector . . . . . . . . . . . . . 34
5.4 Ranking the Probabilistic Safety Vector . . . . . . . . . . . . . . . . . 36
Chapter 6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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