Self-stabilizing minimal dominating set algorithms of distributed systems and the signed star domination number of Cayley graphs

• Main Authors: Chiu, Yu-Chieh, 邱鈺傑
• Other Authors: Chen, Chiu-yuan
• Format: Others
• Language: en_US
• Published: 2014
• Online Access: http://ndltd.ncl.edu.tw/handle/28741059665723204862

博士 === 國立交通大學 === 應用數學系所 === 102 === The study of the domination problem in graph theory began in the nineteen-sixties. A distributed system such as an ad hoc network can be modeled by an undirected simple graph G = (V;E), where V represents the set of nodes (i.e., processes) and E represents the set of interconnections between processes of the distributed system. A subset D of the vertex set V of G is a dominating set if each vertex v in V is either a member of D or adjacent to a vertex in D. A dominating set of G is a minimal dominating set (MDS) if none of its proper subsets is a dominating set of G. An MDS has an application of clustering in wireless networks and is maintained for minimizing the number of required resource centers. Self-stabilization is a concept of designing a distributed system for transient fault toleration and was introduced by Dijkstra in 1974. A distributed system is self-stabilizing if, regardless of its initial configuration, the system is guaranteed to reach a legitimate (i.e., correct) configuration in a finite time. Here the system configuration consists of the state of every process. A self-stabilizing algorithm comprises a collection of rules and each rule has a trigger precondition and an action. The action changes the state of the node by updating its variables. An execution of a rule is called a move. The performance of the proposed algorithms of this thesis is measured by the total number of moves executed by an algorithm. Various execution models have been used in self-stabilizing algorithms and these are encapsulated with the notion of daemons. A daemon can be fair or unfair. It is well-known that an unfair distributed daemon is more practical than other types of daemons. Let n denote the number of nodes (processes) in a given distributed system. In 2007, Turau proposed the first linear-time self-stabilizing algorithm for the MDS problem under an unfair distributed daemon; this algorithm stabilizes in at most 9n moves. In 2008, Goddard et al. improved the result to a 5n-move algorithm. It is interesting to develop an algorithm that takes less moves than the best known result—5n moves using an unfair distributed daemon. In this thesis, we will present a 4n-move self-stabilizing MDS algorithm using an unfair distributed daemon. It is desired that an MDS algorithm is MDS-silent, which means that if the original configuration of the distributed system is already an MDS, then the algorithm should not make any move. Note that in the normal model, a node can only access the information of its 1-hop neighbors and we call such information distance-1 information. Unfortunately, in this thesis we will prove that distance-1 information is not sufficient for building up an MDS-silent algorithm for a distributed system. In this thesis, we will discuss this problem and propose a new performance measure, called stableness, for self-stabilizing MDS algorithms. We also generalize this result to categorize all self-stabilizing algorithms into four levels. In particular, we will show that a self-stabilizing MDS-silent algorithm can be built up under the distance-2 model and the stabilizing time is upper bounded by 2n. Let G be a simple connected graph with vertex set V (G) and edge set E(G). A function f : E(G)→{-1,1} is called a signed star dominating function (SSDF) on G if Σ_{e in E}(v) f(e) >=1 for every v in V(G), where E(v) is the set of all edges incident to v. The signed star domination number of G is defined as SS(G) = min weight sum of f which is an SSDF on G. Let Ω be a symmetric generating subset of nonidentity elements of Γ. The Cayley graph Cay(Γ; Ω) corresponding to Γ and Ω is the ordinary graph with vertex set Γ and edge set E. In this thesis, we obtain exact values for the signed star domination number of all Cayley digraphs CayD(Γ; S) and certain classes of Cayley graphs Cay(Γ; Ω), which is later generalized to f2; 1g-factorable graphs. Note that these solutions are from a joint work with Chelvam and Kalaimurugan.