Student: Shihyu Tsai Advisor: Chiuyuan Chen

1. An efficient self-stabilizing algorithm for the minimal dominating set problem under a distributed scheduler Student: Shihyu Tsai Advisor: Chiuyuan Chen Department of Applied Mathematics National Chiao Tung University

2. Outline • Introduction • History • Our Algorithm • An example • Correction and Convergence • Comparison • Conclusion

3. Introduction

4. Dominating Set; Minimal Dominating Set • a distributed system: • let G = (V , E)be a given graph • |V| = n, |E| = m (vertex => node) • A set D V is called a dominating set if every node v Visa member of D or is adjacent to a member of D. • A dominating set D is minimal if any proper subset of D is not a dominating set. (MDS) • optimize the number and location of resource centers in a network.

5. Self-stabilization • a distributed system is self-stabilizing if it has the following two properties: • (1) convergence property –incorrect configuration correct configuration finite time • (2) closure property – correct configuration correct configuration • tolerate arbitrary transient faults

6. Distributed system vsCentralized system a coordinator duty in a single node single point of control (failure) one leader / easy • every node is autonomous • concurrent nodes • multiple points of control (failure) • cooperate / difficult

7. Preliminary • the form of rules: ⟨precondition⟩ → ⟨statement⟩ • move: the execution of a statement is called a move • minimize the number of moves – save the electric power • round: self-stabilizing systems operate in rounds • a rule is called enabled if its precondition evaluates to be true • a node is called enabledif at least one of its rules enabled

8. Three common schedulers • central scheduler • only one enabled node can make a move at one time • synchronous scheduler • all enabled nodes will make a move at the time • distributed scheduler • a subset of the enabled nodes make a move at the same time

9. History

10. History • self-stabilizing algorithms for finding a minimal dominating set • Note: there is only [7] does not require the identification. We do it!

11. Our Algorithm

12. Our Algorithm • Each node has • a local distinct id, • a variable state • OUT / IN • not in the MDS / in the MDS • A variable dependent • 0 / an id / • no neighbor in IN/ a unique neighbor in IN/ more than one neighbor in IN • Need not any initialization. Just run it.

14. An example with 12 moves

15. An example with 12 moves 7 5 node 1, 4, and 6 form a MDS 6 1 3 4 2

16. Correction and Convergence

17. Correction • Theorem 3.1. When each node has no enabled rules, the set D = {v | v.state = IN} is a minimal dominating set of G.

18. Convergence • Lemma 3.2. No two neighboring nodes will execute rule 1 at the same time. • Lemma 3.3. After a node v executes rule 1, v will not execute any other rules. • Lemma 3.4. Suppose v.state is OUT and a neighbor w of v executes rule 1. Then the only rule that v can execute is rule 4′′ and after v executes rule 4′′, v will not execute any other rules.

19. Convergence • Theorem 3.5. Our algorithm is a self-stabilizing algorithm for the minimal dominating set problem under a distributed scheduler. Furthermore, it stabilizes after at most 4n moves. • Proof: • initially v.state is OUT • initially v.state is IN • case by case with the help of previous lemmas By Theorem 3.1, it suffices to prove that every node takes at most 4 moves undera distributed scheduler.

20. Comparison Turau [8] Our Algorithm Goddard et al. [4] state (OUT/ IN) x (0/1) state (OUT/ WAIT/ IN) dependent (0/ an id/ ) c (0/ 1/ 2) dependent (an id/ )

21. Conclusion • A self-stabilizing algorithm under a distributed scheduler. • performance: • An example with 4n − 1 moves. • Future Work • better algorithm? • better lower bound?