*Theoretical Computer Science*, 412(40):5527–5540, September 2011. Stabilization, Safety and Security.

# TSSSA

Time- and Space-efficient Self-Stabilizing Algorithms

Contact | Prof. Dr. rer. nat. Volker Turau |

## Project Description

The principle of self-stabilization is a very general technique to design a distributed system to tolerate arbitrary transient faults. An algorithm is self-stabilizing if it can start at any possible global configuration and gain consistency in a finite number of steps by itself without any external intervention and remains in a consistent state.

Designing self-stabilizing algorithms focusses on the objective to minimize the time from an arbitrary initial state until reaching a consistent state. In doing so it is accepted that during this stabilization period a considerable part of the network cannot operate properly. To optimize the usage of resources, aside from minimizing the expenditure of time until stabilization of an algorithm low memory requirements are desired.

Estimating the complexity of self-stabilizing algorithms is generally not a trivial task. Hence, for a lot of algorithms there only exist upper bounds that are far from the worst case examples known today. Thus, apart from developing new algorithms, the analysis of existing algorithms is another key issue of this project.

## Publications

**Abstract**: The maximum weight matching problem is a fundamental problem in graph theory with a variety of important applications. Recently Manne and Mjelde presented the first self-stabilizing algorithm computing a 2-approximation of the optimal solution. They established that their algorithm stabilizes after O(2^n) (resp. O(3^n)) moves under a central (resp. distributed) scheduler. This paper contributes a new analysis improving these bounds considerably. In particular it is shown that the algorithm stabilizes after O(nm) moves under the central scheduler and that a modified version of the algorithm also stabilizes after O(nm) moves under the distributed scheduler. The paper presents a new proof technique based on graph reduction to analyze the complexity of self-stabilizing algorithms.

*Theoretical Computer Science*, 412(33):4361–4371, 2011.

**Abstract**: The non-computability of many distributed tasks in anonymous networks is well known. This paper presents a deterministic self-stabilizing algorithm to compute a 3 - (2 / (Delta+1))-approximation of a minimum vertex cover in anonymous networks. The algorithm operates under the distributed unfair scheduler, stabilizes after O(n+m) moves respectively O(Delta) rounds, and requires O(log n) storage per node. Recovery from a single fault is reached within a constant time and the contamination number is O(Delta). For trees the algorithm computes a 2-approximation of a minimum vertex cover.

*Parallel Processing Letters*, 20(2):173–186, 2010.

**Abstract**: This paper presents a deterministic self-stabilizing algorithm that approximates a minimum vertex cover in anonymous networks with ratio 2 using the distributed scheduler and the link-register model with composite atomicity. No algorithm with a better approximation ratio can exist. The algorithm stabilizes in O(min{n, Δ², Δ log3 n}) rounds and requires O(Δ) memory per node.

*Proceedings of the 11th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS'09)*, Springer, November 2009, pp. 341–353. Lyon, France.

**Abstract**: This paper presents a deterministic self-stabilizing algorithm that computes a 3-approximation vertex cover in anonymous networks. It reaches a legal state after O(n+m) moves or 2n + 1 rounds respectively and recovers from a single fault within a constant containment time. The contamination number is 2 Delta + 1. An enhanced version of this algorithm achieves a 2-approximation on trees.

*Information Processing Letters*, 109(14):763–767, 2009.

**Abstract**: This paper presents a new distributed self-stabilizing algorithm for the weakly connected minimal dominating set problem. It assumes a self-stabilizing algorithm to compute a breadth-first tree. Using an unfair distributed scheduler the algorithm stabilizes in at most O(nmA) moves, where A is the number of moves to construct a breadth-first tree. All previously known algorithms required an exponential number of moves.

*Worst-Case Analysis of a Self-Stabilizing Algorithm Computing a Weakly Connected Minimal Dominating Set*. Technical Report urn:nbn:de:gbv:830-tubdok-5126, Hamburg University of Technology, Hamburg, Germany, October 2008.

**Abstract**: Recently, Srimani and Xu presented a self-stabilizing algorithm that computes a weakly connected minimal dominating set. They prove an upper bound of O(2^n) until stabilization but they do not provide a lower bound. This paper verifies by giving an example that their algorithm indeed requires O(2^n) moves on a certain graph.

*Information Processing Letters.*, 103(3):88–93, July 2007.

**Abstract**: This paper presents distributed self-stabilizing algorithms for the maximal independent and the minimal dominating set problems. Using an unfair distributed scheduler the algorithms stabilizes in at most max{3n-5,2n} resp. 9n moves. All previously known algorithms required O(n2) moves.