這篇論文中我們將提出兩個高效率的演算法，來分別計算有向與無向雙環網路中的最短路徑問題。
其中無向雙環網路部分的演算法是由 Janez Zerrovnik 與 Tomaz Pisanski 兩位先生所提出之 packed basis
概念設計而成。
在預先計算時間複雜度 O(log N) 之下，這兩個演算法的即時運算部分都只需要 O(1) 就能計算出我們所需要的結果。
在此之前最好的結果是一個時間複雜度為 O(l) 的演算法，其中 l 是最短路徑的長度。
這些演算法在計算雙環網路的路徑時相當有用。當雙環網路的結構決定好之後
，計算最短路徑的參數也就可以決定完畢，而之後再計算網路訊息的傳遞方向時只需要常數時間即可。

In this thesis, we present two e cent algorithms to compute shortest path
between pair of vertices in directed and undirected double-loop networks.
The algorithm for undirected double-loop networks is based on the concept
"packed basis" proposed by Janez Zerrovnik and Toma z Pisanski. With
O(logN) preprocessing time, both algorithms need only constant time to
compute the shortest path between any pair of vertices in the network. This
is an improvement of the best known algorithm, which needs O(l) time, where
l is the length of the path in the directed double-loop networks. These algo-
rithms are useful in message routing in the double-loop networks. Once the
network has been constructed, the parameters for computing the shortest
paths can be computed. At the time a message is to be delivered, the algo-
rithm needs only constant time to determine which edge the message should
be sent.

1 導論 1
2 名詞介紹 3
2.1 多環網路. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 雙環網路. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 最小路徑圖 MDD (Minimum Distance Diagram) . . . . . . . . 5
2.4 幾個主要研究方向. . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4.1 路由演算法 (Routing algorithm) . . . . . . . . . . . . . 6
2.4.2 網路直徑最小化 (Minimum diameter) . . . . . . . . . . . 7
2.4.3 最小路徑圖的形狀問題 (Shape of minimum distance di-
agram) . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 L 型的的定義以及相關證明. . . . . . . . . . . . . . . . . . . . . 8
2.6 包覆基底 (Packed basis) . . . . . . . . . . . . . . . . . . . . . 9
3 之前的相關演算法 10
3.1 寬先搜尋法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Guan's Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 無向雙環網路的最短路徑演算法. . . . . . . . . . . . . . . . . . . 12
3.3.1 設計概念 . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3.2 演算法. . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3.3 正確性證明. . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3.4 時間複雜度證明. . . . . . . . . . . . . . . . . . . . . . . 14
4 有向雙環網路的最短路徑演算法 15
4.1 設計概念 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 演算法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3 正確性證明. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.4 時間複雜度與額外空間證明 . . . . . . . . . . . . . . . . . . . . . 19
4.4.1 預先計算部分的時間複雜度 . . . . . . . . . . . . . . . . . 19
4.4.2 即時計算部分的時間複雜度 . . . . . . . . . . . . . . . . . 20
4.4.3 演算法所需的額外空間. . . . . . . . . . . . . . . . . . . 20
5 討論以及三維推廣 21
5.1 有向三環網路的逐步決定演算法. . . . . . . . . . . . . . . . . . . 21
5.2 所得結果意涵. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.3 無向三環網路的討論-幾種 basis 的定義. . . . . . . . . . . . . . . 23
6 結論以及待解問題 26

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