分散式排程下的解決極小控制集問題的高效率自我穩定演算法

Abstract

本篇論文考慮的是設計解決極小控制集（MDS）問題的更具有效率的自我穩定演算法（self-stabilizing algorithms）。設n為分散式系統裡的節點數目。若一個自我穩定演算法在給定的分散式系統執行至多t次動作後，即可到達合理狀態（legitimate configuration），則稱此自我穩定演算法為t-動作演算法（t-move algorithm）。在2007年，Turau提出了一個分散式排程下的解決MDS問題的9n-動作演算法。隨後，在2008年，Goddard等人提出一個分散式排程下的解決MDS問題的5n-動作演算法。設計一個執行動作少於5n次的分散式排程下的解決MDS問題的演算法，確實是一個挑戰。本篇論文的目的就在於設計出這樣的演算法。具體來說，我們提出了一個分散式排程下的解決MDS問題的4n-動作演算法；此外，採用我們演算法，需要4n-1個動作才可以到達合理狀態的例子也被提出。

This thesis considers designing efficient self-stabilizing algorithms for solving the minimal dominating set (MDS) problem. Let n denote the number of nodes in a distributed system. A self-stabilizing algorithm is said to be a t-move algorithm if when it is used, a given distributed system takes at most t moves to reach a legitimate configuration. In 2007, Turau proposed a 9n-move algorithm for the MDS problem under a distributed scheduler. Later, in 2008, Goddard et al. proposed a 5n-move algorithm for the MDS problem under a distributed scheduler. It is indeed a challenge to develop an algorithm that takes less than 5n moves under a distributed scheduler. The purpose of this thesis is to propose such an algorithm. In particular, we propose a 4n-move algorithm under a distributed scheduler; an example such that our algorithm takes 4n − 1 moves to reach a legitimate configuration has also been proposed.