Self-stabilizing minimal dominating set algorithms of distributed systems and the signed star domination number of Cayley graphs
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 se...
Main Authors: | , |
---|---|
Other Authors: | |
Format: | Others |
Language: | en_US |
Published: |
2014
|
Online Access: | http://ndltd.ncl.edu.tw/handle/28741059665723204862 |
id |
ndltd-TW-102NCTU5507073 |
---|---|
record_format |
oai_dc |
collection |
NDLTD |
language |
en_US |
format |
Others
|
sources |
NDLTD |
description |
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.
|
author2 |
Chen, Chiu-yuan |
author_facet |
Chen, Chiu-yuan Chiu, Yu-Chieh 邱鈺傑 |
author |
Chiu, Yu-Chieh 邱鈺傑 |
spellingShingle |
Chiu, Yu-Chieh 邱鈺傑 Self-stabilizing minimal dominating set algorithms of distributed systems and the signed star domination number of Cayley graphs |
author_sort |
Chiu, Yu-Chieh |
title |
Self-stabilizing minimal dominating set algorithms of distributed systems and the signed star domination number of Cayley graphs |
title_short |
Self-stabilizing minimal dominating set algorithms of distributed systems and the signed star domination number of Cayley graphs |
title_full |
Self-stabilizing minimal dominating set algorithms of distributed systems and the signed star domination number of Cayley graphs |
title_fullStr |
Self-stabilizing minimal dominating set algorithms of distributed systems and the signed star domination number of Cayley graphs |
title_full_unstemmed |
Self-stabilizing minimal dominating set algorithms of distributed systems and the signed star domination number of Cayley graphs |
title_sort |
self-stabilizing minimal dominating set algorithms of distributed systems and the signed star domination number of cayley graphs |
publishDate |
2014 |
url |
http://ndltd.ncl.edu.tw/handle/28741059665723204862 |
work_keys_str_mv |
AT chiuyuchieh selfstabilizingminimaldominatingsetalgorithmsofdistributedsystemsandthesignedstardominationnumberofcayleygraphs AT qiūyùjié selfstabilizingminimaldominatingsetalgorithmsofdistributedsystemsandthesignedstardominationnumberofcayleygraphs AT chiuyuchieh fēnsànshìxìtǒngzhīzìwǒwěndìngjíxiǎokòngzhìjíyǎnsuànfǎyǔkǎishìtúzhīzhèngfùhàoxīngkòngzhìshù AT qiūyùjié fēnsànshìxìtǒngzhīzìwǒwěndìngjíxiǎokòngzhìjíyǎnsuànfǎyǔkǎishìtúzhīzhèngfùhàoxīngkòngzhìshù |
_version_ |
1718088841806479360 |
spelling |
ndltd-TW-102NCTU55070732015-10-14T00:18:21Z http://ndltd.ncl.edu.tw/handle/28741059665723204862 Self-stabilizing minimal dominating set algorithms of distributed systems and the signed star domination number of Cayley graphs 分散式系統之自我穩定極小控制集演算法與凱氏圖之正負號星控制數 Chiu, Yu-Chieh 邱鈺傑 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. Chen, Chiu-yuan 陳秋媛 2014 學位論文 ; thesis 80 en_US |