自我穩定系統是一個容錯分散式網路. 當系統遇到無法預期的暫態錯誤時, 像是節點加入或是離開網路, 或者是單純的硬體故障, 它就可以假想為被暫態錯誤帶到一個隨意的初始狀態. 依照自我穩定的定義, 系統可以自發性地在有限時間之內回到合法狀態, 進而達到容錯的功能. 基於此, 自我穩定系統亦有自主管理, 易擴展, 以及適應環境的功能.
在本篇論文裡, 我們探討自我穩定系統中的塗色問題, 我們提出三個塗色演算法. 第一個演算法討論的是節點的塗色, 它可以快速地用七種顏色塗平面網路的節點, 它的時間複雜度是 O(log n), n 代表的是節點總數. 第二個演算法討論的是平面圖的鄰邊塗色. 它用到的顏色數是 (Delta + 4), 其中 Delta 表示網路的最大分支度. 它的時間複雜度是 O(n^2). 第三個演算法討論的是雙分圖的鄰邊塗色. 它只用了 Delta 個顏色, 因此它能找到網路的最佳鄰邊塗色; 它的時間複雜度則是 O(kn^2+m), k 代表的是初始狀態裡沒有正確顏色的鄰邊個數, 而 m 代表的是網路中的鄰邊總數. 由於時間複雜度和初始狀態有關, 第三個演算法有和 time-adaptive 以及 superstabilization 類似的特性.
這篇論文另有提出一個將完全雙分網路平面化的演算法. 我們的方法是挑選出一組鄰邊, 該組鄰邊會將原本網路平面化 (即不會形成 K5 和 K3,3 的 subdivision, 這裡的 K5 和 K3,3 分別代表五個點和完全連通圖及三對三的完全雙分圖). 我們的方法可以在 O(n) 以內的時間就可以找到這樣的一組鄰邊. 該方法可以和其他的塗色演算法結合並找到一個網路的部分塗色.

A self-stabilizing system is a fault-tolerant distributed network. When it encounters transient faults, including leave/join of nodes or hardware breakdown, it is onceptually
brought into an arbitrary initial configuration. It then always manages to return to legitimate configurations without human intervention and resumes operations. For being
able to spontaneously fix itself, a self-stabilizing algorithm is also autonomous, scalable,
and adaptable.

In this dissertation, we study the well-known coloring problems in self-stabilizing systems. We propose three coloring algorithms, one for the vertex coloring problem
and the other two for the edge coloring problem. The first one quickly 7-colors planar networks; its time complexity is O(log n), where n is the number of nodes. The second
one (Delta + 4)-edge colors planar networks; its time complexity is O(n^2), where Delta is the maximum degree of the nodes. The third one Delta-edge colors bipartite networks; its time complexity is O(kn^2m+m), where k is the number of edges not properly colored in the
initial configuration and m is the number of edges. Thus, it finds optimal edge colorings.
In addition, it has a property similar to time-adaptive stabilization or superstabilization because its time complexity depends on the quality of the initial configuration.

We also study the planarization problem and propose an algorithm that planarizes complete bipartite networks. By not forming a subdivision of K5 or K3,3, the algorithm
finds spanning planar graphs in O(n) time, where K5 is a complete graph and K3,3 is a complete bipartite graph. It can further combine with coloring algorithms for finding
partial colorings for the networks.

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